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arxiv_ground_truth/0705.4297.md

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+# Splitting families and the Noetherian type of \(\beta\omega\setminus\omega\)
+
+David Milovich
+
+University of Wisconsin-Madison Mathematics Dept.
+milovich@math.wisc.edu
+
+(Date: March 26, 2008)
+
+###### Abstract.
+
+Extending some results of Malykhin, we prove several independence results about base properties of \(\beta\omega\setminus\omega\) and its powers, especially the Noetherian type \(Nt(\beta\omega\setminus\omega)\), the least \(\kappa\) for which \(\beta\omega\setminus\omega\) has a base that is \(\kappa\)-like with respect to containment. For example, \(Nt(\beta\omega\setminus\omega)\) is at least \(\mathfrak{s}\), but can consistently be \(\omega_{1}\), \(\mathfrak{c}\), \(\mathfrak{c}^{+}\), or strictly between \(\omega_{1}\) and \(\mathfrak{c}\). \(Nt(\beta\omega\setminus\omega)\) is also consistently less than the additivity of the meager ideal. \(Nt(\beta\omega\setminus\omega)\) is closely related to the existence of special kinds of splitting families.
+
+Support provided by an NSF graduate fellowship.
+
+## 1. Introduction
+
+**Definition 1.1****.**: Given a cardinal \(\kappa\), define a poset to be \(\kappa\)-_like_ (\(\kappa^{\mathrm{op}}\)-_like_) if no element is above (below) \(\kappa\)-many elements. Define a poset to be _almost_\(\kappa^{\mathrm{op}}\)-_like_ if it has a \(\kappa^{\mathrm{op}}\)-like dense subset.
+
+In the context of families of subsets of a topological space, we will always implicitly order by inclusion. We are particularly interested in \(\kappa^{\mathrm{op}}\)-like bases, \(\pi\)-bases, local bases, and local \(\pi\)-bases of the space \(\omega^{*}\) of nonprincipal ultrafilters on \(\omega\). Recall that a local base (local \(\pi\)-base) at a point in a space is a family of open neighborhoods of that point (family of nonempty open subsets) such that every neighborhood of the point contains an element of the family; a base (\(\pi\)-base) of a space is family of open sets that contains local bases (local \(\pi\)-bases) at every point. See Engelking [9] for the more background on bases and their cousins. Also recall the following basic cardinal functions. For more about these functions, see Juhász [12].
+
+**Definition 1.2****.**: Given a space \(X\), let the _weight_ of \(X\), or \(w(X)\), be the least \(\kappa\geq\omega\) such that \(X\) has a base of size at most \(\kappa\). Given \(p\in X\), let the _character_ of \(p\), or \(\chi(p,X)\), be the least \(\kappa\geq\omega\) such that there is a local base at \(p\) of size at most \(\kappa\). Let the character of \(X\), or \(\chi(X)\), be the supremum of the characters of its points. Analogously define _\(\pi\)-weight_ and _local \(\pi\)-character_, respectively denoting them using \(\pi\) and \(\pi\chi\).
+
+Now consider the following order-theoretic parallels.
+
+**Definition 1.3****.**: Given a space \(X\), let the _Noetherian type_ of \(X\), or \(Nt(X)\), be the least \(\kappa\geq\omega\) such that \(X\) has a base that is \(\kappa^{\mathrm{op}}\)-like. Given \(p\in X\), let the _local Noetherian type_ of \(p\), or \(\chi Nt(p,X)\), be the least \(\kappa\geq\omega\) such that there is a \(\kappa^{\mathrm{op}}\)-like local base at \(p\). Let the local Noetherian type of \(X\), or \(\chi Nt(X)\), be the supremum of the local Noetherian types of its points. Analogously define _Noetherian \(\pi\)-type_ and _local Noetherian \(\pi\)-type_, respectively denoting them using \(\pi Nt\) and \(\pi\chi Nt\).
+
+Noetherian type and Noetherian \(\pi\)-type were introduced by Peregudov [16]. Let \(\omega^{*}\) denote the space of nonprincipal ultrafilters on \(\omega\). Malykhin [15] proved that MA implies \(\pi Nt(\omega^{*})=\mathfrak{c}\) and CH implies \(Nt(\omega^{*})=\mathfrak{c}\). We extend these results by investigating \(Nt(\omega^{*})\), \(\pi Nt(\omega^{*})\), \(\chi Nt(\omega^{*})\), and \(\pi\chi Nt(\omega^{*})\) as cardinal characteristics of the continuum. For background on such cardinals, see Blass [7]. We also examine the sequence \(\langle Nt((\omega^{*})^{1+\alpha})\rangle_{\alpha\in{\mathrm{O}n}}\).
+
+**Definition 1.4****.**: Let \(\mathfrak{b}\) denote the minimum of \(\lvert\mathcal{F}\rvert\) where \(\mathcal{F}\) ranges over the subsets of \(\omega^{\omega}\) that have no upper bound in \(\omega^{\omega}\) with respect to eventual domination.
+
+**Definition 1.5****.**: A _tree \(\pi\)-base_ of a space \(X\) is a \(\pi\)-base that is a tree when ordered by containment. Let \(\mathfrak{h}\) be the minimum of the set of heights of tree \(\pi\)-bases of \(\omega^{*}\).
+
+Balcar, Pelant, and Simon [1] proved that tree \(\pi\)-bases of \(\omega^{*}\) exist, and that \(\mathfrak{h}\leq\min\{\mathfrak{b},\operatorname{cf}\mathfrak{c}\}\). They also proved that the above definition of \(\mathfrak{h}\) is equivalent to the more common definition of \(\mathfrak{h}\) as the distributivity number of \([\omega]^{\omega}\) ordered by \(\subseteq^{*}\).
+
+**Definition 1.6****.**: Given \(x,y\in[\omega]^{\omega}\), we say that \(x\)_splits_\(y\) if \(\lvert y\cap x\rvert=\lvert y\setminus x\rvert=\omega\). Let \(\mathfrak{r}\) be the minimum value of \(\lvert A\rvert\) where \(A\) ranges over the subsets of \([\omega]^{\omega}\) such that no \(x\in[\omega]^{\omega}\) splits every \(y\in A\). Let \(\mathfrak{s}\) be the minimum value of \(\lvert A\rvert\) where \(A\) ranges over the subsets of \([\omega]^{\omega}\) such that every \(x\in[\omega]^{\omega}\) is split by some \(y\in A\).
+
+It is known that \(\mathfrak{b}\leq\mathfrak{r}\) and \(\mathfrak{h}\leq\mathfrak{s}\). (See Theorems 3.8 and 6.9 of [7].)
+
+Clearly, \(Nt(\omega^{*})\leq w(\omega^{*})^{+}=\mathfrak{c}^{+}\). We will show that also \(\pi\chi Nt(\omega^{*})=\omega\) and \(\pi Nt(\omega^{*})=\mathfrak{h}\) and \(\mathfrak{s}\leq Nt(\omega^{*})\). Furthermore, \(Nt(\omega^{*})\) can consistently be \(\mathfrak{c}\), \(\mathfrak{c}^{+}\), or any regular \(\kappa\) satisfying \(2^{<\kappa}=\mathfrak{c}\). Also, \(Nt(\omega^{*})=\omega_{1}\) is relatively consistent with any values of \(\mathfrak{b}\) and \(\mathfrak{c}\). The relations \(\omega_{1}<\mathfrak{b}=\mathfrak{s}=Nt(\omega^{*})<\mathfrak{c}\) and \(\omega_{1}=\mathfrak{b}=\mathfrak{s}<Nt(\omega^{*})<\mathfrak{c}\) are also each consistent. We also prove some relations between \(\mathfrak{r}\) and \(Nt(\omega^{*})\), as well as some consistency results about the local Noetherian type of points in \(\omega^{*}\).
+
+## 2. Basic results
+
+The following proposition is essentially due to Peregudov (see Lemma 1 of [16]).
+
+**Proposition 2.1****.**: _Suppose a point \(p\) in a space \(X\) satisfies \(\pi\chi(p,X)<\operatorname{cf}\kappa\leq\kappa\leq\chi(p,X)\). Then \(Nt(X)>\kappa\)._
+
+Proof.: Let \(\mathcal{A}\) be a base of \(X\). Let \(\mathcal{U}_{0}\) and \(\mathcal{V}_{0}\) be, respectively, a local \(\pi\)-base at \(p\) of size at most \(\pi\chi(p,X)\) and a local base at \(p\) of size \(\chi(p,X)\). For each element of \(\mathcal{U}_{0}\), choose a subset in \(\mathcal{A}\), thereby producing local \(\pi\)-base \(\mathcal{U}\) at \(p\) that is a subset of \(\mathcal{A}\) of size at most \(\pi\chi(p,X)\). Similarly, for each element of \(\mathcal{V}_{0}\), choose a smaller neighborhood of \(p\) in \(\mathcal{A}\), thereby producing a local base \(\mathcal{V}\) at \(p\) that is a subset of \(\mathcal{A}\) of size \(\chi(p,X)\). Every element of \(\mathcal{V}\) contains an element of \(\mathcal{U}\). Hence, some element of \(\mathcal{U}\) is contained in \(\kappa\)-many elements of \(\mathcal{V}\); hence, \(\mathcal{A}\) is not \(\kappa^{\mathrm{op}}\)-like. ∎
+
+**Definition 2.2****.**: For all \(x\in[\omega]^{\omega}\), set \(x^{*}=\{p\in\omega^{*}:p\in x\}\).
+
+**Theorem 2.3****.**: _It is relatively consistent with any value of \(\mathfrak{c}\) satisfying \(\operatorname{cf}\mathfrak{c}>\omega_{1}\) that \(Nt(\omega^{*})=\mathfrak{c}^{+}\)._
+
+Proof.: We may assume \(\operatorname{cf}\mathfrak{c}>\omega_{1}\). By Exercise A10 on p. 289 of Kunen [14], there is a ccc generic extension \(V[G]\) such that \(\check{\mathfrak{c}}=\mathfrak{c}^{V[G]}\) and, in \(V[G]\), there exists \(p\in\omega^{*}\) such that \(\chi(p,\omega^{*})=\omega_{1}\). Henceforth work in \(V[G]\). Let \(\varphi\) be a bijection from \(\omega^{2}\) to \(\omega\). Define \(\psi\colon\omega^{*}\to\omega^{*}\) by
+
+\[x\mapsto\{E\subseteq\omega:\{m<\omega:\{n<\omega:\varphi(m,n)\in E\}\in p\}\in x\}.\]
+
+Since \(\pi\chi(p,\omega^{*})\leq\chi(p,\omega^{*})=\omega_{1}\), there exists \(\langle E_{\alpha}\rangle_{\alpha<\omega_{1}}\in([\omega]^{\omega})^{\omega_{1}}\) such that every neighborhood of \(p\) contains \(E_{\alpha}^{*}\) for some \(\alpha<\omega_{1}\). Hence, for all \(x\in\omega^{*}\), every neighborhood of \(\psi(x)\) contains \((\varphi``(\{m\}\times E_{\alpha}))^{*}\) for some \(m<\omega\) and \(\alpha<\omega_{1}\); whence, \(\pi\chi(\psi(x),\omega^{*})=\omega_{1}\). Since \(\psi\) is easily verified to be a topological embedding, \(\chi(x,\omega^{*})\leq\chi(\psi(x),\omega^{*})\) for all \(x\in\omega^{*}\). By a result of Pospišil [17], there exists \(q\in\omega^{*}\) such that \(\chi(q,\omega^{*})=\mathfrak{c}\). Hence, \(\pi\chi(\psi(q),\omega^{*})=\omega_{1}\) and \(\chi(\psi(q),\omega^{*})=\mathfrak{c}\). By Proposition 2.1, \(Nt(\omega^{*})>\chi(\psi(q),\omega^{*})=\mathfrak{c}\). ∎
+
+**Definition 2.4****.**: Given \(n<\omega\), let \(\mathfrak{s}\mathfrak{s}_{n}\) (\(\mathfrak{s}\mathfrak{s}_{\omega}\)) denote the least cardinal \(\kappa\) for which there exists a sequence \(\langle f_{\alpha}\rangle_{\alpha<\mathfrak{c}}\) of functions on \(\omega\) each with range contained in \(n\) (each with finite range) such that for all \(I\in[\mathfrak{c}]^{\kappa}\) and \(x\in[\omega]^{\omega}\) there exists \(\alpha\in I\) such that \(f_{\alpha}\) is not eventually constant on \(x\). (The notation \(\mathfrak{s}\mathfrak{s}\) was chosen with the phrase “supersplitting number” in mind.) Note that if such an \(\langle f_{\alpha}\rangle_{\alpha<\mathfrak{c}}\) does not exist for any \(\kappa\leq\mathfrak{c}\), then \(\mathfrak{s}\mathfrak{s}_{n}\) (\(\mathfrak{s}\mathfrak{s}_{\omega}\)) is by definition equal to \(\mathfrak{c}^{+}\).
+
+Clearly \(\mathfrak{s}\mathfrak{s}_{n}\geq\mathfrak{s}\mathfrak{s}_{n+1}\geq\mathfrak{s}\mathfrak{s}_{\omega}\) for all \(n<\omega\). Moreover, since \(\operatorname{cf}\mathfrak{c}>\omega\), we have \(\mathfrak{s}\mathfrak{s}_{\omega}=\mathfrak{s}\mathfrak{s}_{n}\) for some \(n<\omega\). However, for any particular \(n\in\omega\setminus 2\), it is not clear whether ZFC proves \(\mathfrak{s}\mathfrak{s}_{\omega}=\mathfrak{s}\mathfrak{s}_{n}\).
+
+**Definition 2.5****.**: Given \(\lambda\geq\kappa\geq\omega\) and a space \(X\), a \(\langle\lambda,\kappa\rangle\)-_splitter_ of \(X\) is a sequence \(\langle\mathcal{F}_{\alpha}\rangle_{\alpha<\lambda}\) of finite open covers of \(X\) such that, for all \(I\in[\lambda]^{\kappa}\) and \(\langle U_{\alpha}\rangle_{\alpha\in I}\in\prod_{\alpha\in I}\mathcal{F}_{\alpha}\), the interior of \(\bigcap_{\alpha\in I}U_{\alpha}\) is empty.
+
+**Lemma 2.6****.**: _Suppose \(X\) is a compact space with a base \(\mathcal{A}\) of size at most \(w(X)\) such that \(U\cap V\in\mathcal{A}\cup\{\emptyset\}\) for all \(U,V\in\mathcal{A}\). If \(\kappa\leq w(X)\) and \(X\) has a \(\langle w(X),\kappa\rangle\)-splitter, then \(\mathcal{A}\) contains a \(\kappa^{\mathrm{op}}\)-like base of \(X\). Hence, \(Nt(\omega^{*})\leq\mathfrak{s}\mathfrak{s}_{\omega}\)._
+
+Proof.: Set \(\lambda=w(X)\) and let \(\langle\mathcal{F}_{\alpha}\rangle_{\alpha<\lambda}\) be a \(\langle\lambda,\kappa\rangle\)-splitter of \(X\). For each \(\alpha<\lambda\), the cover \(\mathcal{F}_{\alpha}\) is refined by a finite subcover of \(\mathcal{A}\); hence, we may assume \(\mathcal{F}_{\alpha}\subseteq\mathcal{A}\). Let \(\mathcal{A}=\{U_{\alpha}:\alpha<\lambda\}\). For each \(\alpha<\lambda\), set \(\mathcal{B}_{\alpha}=\{U_{\alpha}\cap V:V\in\mathcal{F}_{\alpha}\}\). Set \(\mathcal{B}=\bigcup_{\alpha<\lambda}\mathcal{B}_{\alpha}\setminus\{\emptyset\}\). Then \(\mathcal{B}\) is easily seen to be a base of \(X\) and a \(\kappa^{\mathrm{op}}\)-like subset of \(\mathcal{A}\). ∎
+
+**Lemma 2.7****.**: _Let \(X\) be a compact space without isolated points and let \(\omega\leq\kappa\leq\lambda\leq\min_{p\in X}\chi(p,X)\). If \(X\) has no \(\langle\lambda,\kappa\rangle\)-splitter, then \(Nt(X)>\kappa\)._
+
+Proof.: Let \(\mathcal{A}\) be a base of \(X\). Construct a sequence \(\langle\mathcal{F}_{\alpha}\rangle_{\alpha<\lambda}\) of finite subcovers of \(\mathcal{A}\) as follows. Suppose we have \(\alpha<\lambda\) and \(\langle\mathcal{F}_{\beta}\rangle_{\beta<\alpha}\). For each \(p\in X\), choose \(V_{p}\in\mathcal{A}\) such that \(p\in V_{p}\not\in\bigcup_{\beta<\alpha}\mathcal{F}_{\beta}\). Let \(\mathcal{F}_{\alpha}\) be a finite subcover of \(\{V_{p}:p\in X\}\). Then \(\mathcal{F}_{\alpha}\cap\mathcal{F}_{\beta}=\emptyset\) for all \(\alpha<\beta<\lambda\). Suppose \(X\) has no \(\langle\lambda,\kappa\rangle\)-splitter. Then choose \(I\in[\lambda]^{\kappa}\) and \(\langle U_{\alpha}\rangle_{\alpha\in I}\in\prod_{\alpha\in I}\mathcal{F}_{\alpha}\) such that \(\bigcap_{\alpha\in I}U_{\alpha}\) has nonempty interior. Then there exists \(W\in\mathcal{A}\) such that \(W\subseteq\bigcap_{\alpha\in I}U_{\alpha}\). Thus, \(\mathcal{A}\) is not \(\kappa^{\mathrm{op}}\)-like. ∎
+
+**Definition 2.8****.**: Let \(\mathfrak{u}\) denote the minimum of the set of characters of points in \(\omega^{*}\). Let \(\pi\mathfrak{u}\) denote the minimum of the set of \(\pi\)-characters of points in \(\omega^{*}\).
+
+By a theorem of Balcar and Simon [2], \(\pi\mathfrak{u}=\mathfrak{r}\).
+
+**Theorem 2.9****.**: _Suppose \(\mathfrak{u}=\mathfrak{c}\). Then \(Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{\omega}\)._
+
+Proof.: By Lemma 2.6, \(Nt(\omega^{*})\leq\mathfrak{s}\mathfrak{s}_{\omega}\). Suppose \(\kappa\leq\mathfrak{c}\). Since every finite open cover of \(\omega^{*}\) is refined by a finite, pairwise disjoint, clopen cover, \(\omega^{*}\) has a \(\langle\mathfrak{c},\kappa\rangle\)-splitter if and only if \(\mathfrak{s}\mathfrak{s}_{\omega}\leq\kappa\). Hence, \(Nt(\omega^{*})\geq\mathfrak{s}\mathfrak{s}_{\omega}\) by Lemma 2.7. ∎
+
+**Lemma 2.10****.**: _Suppose \(\mathfrak{r}=\mathfrak{c}\). Then \(\mathfrak{s}\mathfrak{s}_{2}\leq\mathfrak{c}\)._
+
+Proof.: Let \(\langle x_{\alpha}\rangle_{\alpha<\mathfrak{c}}\) enumerate \([\omega]^{\omega}\). Construct \(\langle y_{\alpha}\rangle_{\alpha<\mathfrak{c}}\in([\omega]^{\omega})^{\mathfrak{c}}\) as follows. Given \(\alpha<\mathfrak{c}\) and \(\langle y_{\beta}\rangle_{\beta<\alpha}\), choose \(y_{\alpha}\) such that \(y_{\alpha}\) splits every element of \(\{x_{\alpha}\}\cup\{y_{\beta}:\beta<\alpha\}\). Suppose \(I\in[\mathfrak{c}]^{\mathfrak{c}}\) and \(\alpha<\mathfrak{c}\). Then \(x_{\alpha}\) is split by \(y_{\beta}\) for all \(\beta\in I\setminus\alpha\). Thus, \(\langle\{y_{\alpha},\omega\setminus y_{\alpha}\}\rangle_{\alpha<\mathfrak{c}}\) witnesses \(\mathfrak{s}\mathfrak{s}_{2}\leq\mathfrak{c}\). ∎
+
+**Theorem 2.11****.**: _The cardinals \(\mathfrak{r}\) and \(Nt(\omega^{*})\) are related as follows._
+
+1. (1)_If_ \(\mathfrak{r}=\mathfrak{c}\)_, then_ \(Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{\omega}\leq\mathfrak{c}\)_._
+2. (2)_If_ \(\mathfrak{r}<\mathfrak{c}\)_, then_ \(Nt(\omega^{*})\geq\mathfrak{c}\)_._
+3. (3)_If_ \(\mathfrak{r}<\operatorname{cf}\mathfrak{c}\)_, then_ \(Nt(\omega^{*})=\mathfrak{c}^{+}\)_._
+
+Proof.: Statement (1) follows from Lemma 2.10, Theorem 2.9, and \(\pi\mathfrak{u}=\mathfrak{r}\). The proof of Theorem 2.3 shows how to construct \(p\in\omega^{*}\) such that \(\pi\chi(p,\omega^{*})=\pi\mathfrak{u}=\mathfrak{r}\) and \(\chi(p,\omega^{*})=\mathfrak{c}\). Hence, (2) and (3) follow from Proposition 2.1. ∎
+
+**Definition 2.12****.**: A subset \(A\) of \([\omega]^{\omega}\) has the _strong finite intersection property_ (SFIP) if the intersection of every finite subset of \(A\) is infinite. Given \(A\subseteq[\omega]^{\omega}\) with the SFIP, define the _Booth forcing for_\(A\) to be \([\omega]^{<\omega}\times[A]^{<\omega}\) ordered by \(\langle\sigma_{0},F_{0}\rangle\leq\langle\sigma_{1},F_{1}\rangle\) if and only if \(F_{0}\supseteq F_{1}\) and \(\sigma_{1}\subseteq\sigma_{0}\subseteq\sigma_{1}\cup\bigcap F_{1}\). Define a _generic pseudointersection_ of \(A\) to be \(\bigcup_{\langle\sigma,F\rangle\in G}\sigma\) where \(G\) is a generic filter of \([\omega]^{<\omega}\times[A]^{<\omega}\).
+
+**Theorem 2.13****.**: _For all cardinals \(\kappa\) satisfying \(\kappa>\operatorname{cf}\kappa>\omega\), it is consistent that \(\mathfrak{r}=\mathfrak{u}=\operatorname{cf}\kappa\) and \(Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{2}=\mathfrak{c}=\kappa\)._
+
+Proof.: Assuming GCH in the ground model, construct a finite support iteration \(\langle\mathbb{P}_{\alpha}\rangle_{\alpha\leq\kappa}\) as follows. First choose some \(U_{0}\in\omega^{*}\). Then suppose we have \(\alpha<\kappa\) and \(\mathbb{P}_{\alpha}\) and \(\Vdash_{\alpha}U_{\alpha}\in\omega^{*}\). Let \(\mathbb{P}_{\alpha+1}\cong\mathbb{P}_{\alpha}*\mathbb{Q}_{\alpha}\) where \(\mathbb{Q}_{\alpha}\) is a \(\mathbb{P}_{\alpha}\)-name for the Booth forcing for \(U_{\alpha}\). Let \(x_{\alpha}\) be a \(\mathbb{P}_{\alpha+1}\)-name for a generic pseudointersection of \(U_{\alpha}\) added by \(\mathbb{Q}_{\alpha}\); let \(U_{\alpha+1}\) be a \(\mathbb{P}_{\alpha+1}\)-name for an element of \(\omega^{*}\) containing \(U_{\alpha}\cup\{x_{\alpha}\}\). For limit \(\alpha<\kappa\), let \(U_{\alpha}=\bigcup_{\beta<\alpha}U_{\beta}\).
+
+Let \(\langle\eta_{\alpha}\rangle_{\alpha<\operatorname{cf}\kappa}\) be an increasing sequence of ordinals with supremum \(\kappa\). Then \(\{x_{\eta_{\alpha}}:\alpha<\operatorname{cf}\kappa\}\) is forced to generate an ultrafilter in \(V^{\mathbb{P}_{\kappa}}\). Hence, \(\Vdash_{\kappa}\mathfrak{r}\leq\mathfrak{u}\leq\operatorname{cf}\kappa<\kappa=\mathfrak{c}\). Therefore, by Lemma 2.6 and Theorem 2.11, it suffices to show that \(\Vdash_{\kappa}\mathfrak{s}\mathfrak{s}_{2}\leq\kappa\). Every nontrivial finite support iteration of infinite length adds a Cohen real. Hence, we may choose for each \(\alpha<\kappa\) a \(\mathbb{P}_{\omega(\alpha+1)}\)-name \(y_{\alpha}\) for an element of \([\omega]^{\omega}\) that is Cohen over \(V^{\mathbb{P}_{\omega\alpha}}\). Then every name \(S\) for the range of a cofinal subsequence of \(\langle y_{\alpha}\rangle_{\alpha<\kappa}\) is such that
+
+\[\Vdash_{\kappa}\forall z\in[\omega]^{\omega}\ \,\exists w\in S\ \ w\text{ splits }z.\]
+
+Hence, \(\langle y_{\alpha}\rangle_{\alpha<\kappa}\) witnesses that \(\Vdash_{\kappa}\mathfrak{s}\mathfrak{s}_{2}\leq\kappa\). ∎
+
+**Theorem 2.14****.**: \(Nt(\omega^{*})\geq\mathfrak{s}\)_._
+
+Proof.: Suppose \(Nt(\omega^{*})=\kappa<\mathfrak{s}\). Since \(Nt(\omega^{*})<\mathfrak{c}\), we have \(\mathfrak{r}=\mathfrak{c}\) by Theorem 2.11. Hence, \(\mathfrak{u}=\mathfrak{c}\). By Theorem 2.9, it suffices to show that \(\mathfrak{s}\mathfrak{s}_{\omega}>\kappa\). Suppose \(\langle f_{\alpha}\rangle_{\alpha<\mathfrak{c}}\) is a sequence of functions on \(\omega\) with finite range and \(I\in[\mathfrak{c}]^{\kappa}\). Since \(\kappa<\mathfrak{s}\), there exists \(x\in[\omega]^{\omega}\) such that \(f_{\alpha}\) is eventually constant on \(x\) for all \(\alpha\in I\). Thus, \(\mathfrak{s}\mathfrak{s}_{\omega}>\kappa\). ∎
+
+**Lemma 2.15****.**: _Let \(\kappa\) be a cardinal and let \(P\) and \(Q\) be mutually dense subsets of a common poset. Then \(P\) is almost \(\kappa^{\mathrm{op}}\)-like if and only if \(Q\) is._
+
+Proof.: Suppose \(D\) is a \(\kappa^{\mathrm{op}}\)-like dense subset of \(P\). Then it suffices to construct a \(\kappa^{\mathrm{op}}\)-like dense subset of \(Q\). Define a partial map \(f\) from \(\lvert D\rvert^{+}\) to \(Q\) as follows. Set \(f_{0}=\emptyset\). Suppose \(\alpha<\lvert D\rvert^{+}\) and we have constructed a partial map \(f_{\alpha}\) from \(\alpha\) to \(Q\). Set \(E=\{d\in D:d\not\geq q\text{ for all }q\in\operatorname{ran}f_{\alpha}\}\). If \(E=\emptyset\), then set \(f_{\alpha+1}=f_{\alpha}\). Otherwise, choose \(q\in Q\) such that \(q\leq e\) for some \(e\in E\) and let \(f_{\alpha+1}\) be the smallest function extending \(f_{\alpha}\) such that \(f_{\alpha+1}(\alpha)=q\). For limit ordinals \(\gamma\leq\lvert D\rvert^{+}\), set \(f_{\gamma}=\bigcup_{\alpha<\gamma}f_{\alpha}\). Set \(f=f_{\lvert D\rvert^{+}}\).
+
+Let us show that \(\operatorname{ran}f\) is a \(\kappa^{\mathrm{op}}\)-like. Suppose otherwise. Then there exists \(q\in\operatorname{ran}f\) and an increasing sequence \(\langle\xi_{\alpha}\rangle_{\alpha<\kappa}\) in \(\operatorname{dom}f\) such that \(q\leq f(\xi_{\alpha})\) for all \(\alpha<\kappa\). By the way we constructed \(f\), there exists \(\langle d_{\alpha}\rangle_{\alpha<\kappa}\in D^{\kappa}\) such that \(f(\xi_{\beta})\leq d_{\beta}\not=d_{\alpha}\) for all \(\alpha<\beta<\kappa\). Choose \(p\in P\) such that \(p\leq q\). Then choose \(d\in D\) such that \(d\leq p\). Then \(d\leq d_{\beta}\not=d_{\alpha}\) for all \(\alpha<\beta<\kappa\), which contradicts that \(D\) is \(\kappa^{\mathrm{op}}\)-like. Therefore, \(\operatorname{ran}f\) is \(\kappa^{\mathrm{op}}\)-like.
+
+Finally, let us show that \(\operatorname{ran}f\) is a dense subset of \(Q\). Suppose \(q\in Q\). Choose \(p\in P\) such that \(p\leq q\). Then choose \(d\in D\) such that \(d\leq p\). By the way we constructed \(f\), there exists \(r\in\operatorname{ran}f\) such that \(r\leq d\); hence, \(r\leq q\). ∎
+
+**Theorem 2.16****.**: \(\pi Nt(\omega^{*})=\mathfrak{h}\)_._
+
+Proof.: First, we show that \(\pi Nt(\omega^{*})\leq\mathfrak{h}\). Let \(\mathcal{A}\) be a tree \(\pi\)-base of \(\omega^{*}\) such that \(\mathcal{A}\) has height \(\mathfrak{h}\) with respect to containment. Then \(\mathcal{A}\) is clearly \(\mathfrak{h}^{\mathrm{op}}\)-like. To show that \(\mathfrak{h}\leq\pi Nt(\omega^{*})\), let \(\mathcal{A}\) be as above and let \(\mathcal{B}\) be a \(\pi Nt(\omega^{*})^{\mathrm{op}}\)-like \(\pi\)-base of \(\omega^{*}\). Then \(\mathcal{A}\) and \(\mathcal{B}\) are mutually dense; hence, by Lemma 2.15, \(\mathcal{A}\) contains a \(\pi Nt(\omega^{*})^{\mathrm{op}}\)-like \(\pi\)-base \(\mathcal{C}\) of \(\omega^{*}\). Since \(\mathcal{C}\) is also a tree \(\pi\)-base, it has height at most \(\pi Nt(\omega^{*})\). Hence, \(\mathfrak{h}\leq\pi Nt(\omega^{*})\). ∎
+
+**Corollary 2.17****.**: _If \(\mathfrak{h}=\mathfrak{c}\), then \(\pi Nt(\omega^{*})=Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{2}=\mathfrak{c}\)._
+
+Proof.: Suppose \(\mathfrak{h}=\mathfrak{c}\). Then \(\mathfrak{r}=\mathfrak{c}\) because \(\mathfrak{h}\leq\mathfrak{b}\leq\mathfrak{r}\leq\mathfrak{c}\). Hence, by Theorem 2.16, Theorem 2.11, and Lemma 2.10, \(\mathfrak{c}\leq\pi Nt(\omega^{*})\leq Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{\omega}\leq\mathfrak{s}\mathfrak{s}_{2}\leq\mathfrak{c}\). ∎
+
+## 3. Models of \(Nt(\omega^{*})=\omega_{1}\)
+
+Adding \(\mathfrak{c}\)-many Cohen reals collapses \(\mathfrak{s}\mathfrak{s}_{2}\) to \(\omega_{1}\). By Lemma 2.6, it therefore also collapses \(Nt(\omega^{*})\) to \(\omega_{1}\). The same result holds for random reals and Hechler reals.
+
+**Theorem 3.1****.**: _Suppose \(\kappa^{\omega}=\kappa\) and \(\mathbb{P}=\mathcal{B}(2^{\kappa})/\mathcal{I}\) where \(\mathcal{B}(2^{\kappa})\) is the Borel alegebra of the product space \(2^{\kappa}\) and \(\mathcal{I}\) is either the meager ideal or the null ideal (with respect to the product measure). (In other words, \(\mathbb{P}\) adds \(\kappa\)-many Cohen reals or \(\kappa\)-many random reals in the usual way.) Then \(\mathbbm{1}_{\mathbb{P}}\Vdash\omega_{1}=\mathfrak{s}\mathfrak{s}_{2}\)._
+
+Proof.: Working in the generic extension \(V[G]\), we have \(\kappa=\mathfrak{c}\) and a sequence \(\langle x_{\alpha}\rangle_{\alpha<\kappa}\) in \([\omega]^{\omega}\) such that \(V[G]=V[\langle x_{\alpha}\rangle_{\alpha<\kappa}]\) and, if \(E\in\mathcal{P}(\kappa)\cap V\) and \(\alpha\in\kappa\setminus E\), then \(x_{\alpha}\) is Cohen or random over \(V[\langle x_{\beta}\rangle_{\beta\in E}]\). (See [13] for a proof.) Suppose \(I\in[\kappa]^{\omega_{1}}\) and \(y\in[\omega]^{\omega}\). Then \(y\in V[\langle x_{\alpha}\rangle_{\alpha\in J}]\) for some \(J\in[\kappa]^{\omega}\cap V\); hence, \(x_{\alpha}\) splits \(y\) for all \(\alpha\in I\setminus J\). Thus, \(\langle\{x_{\alpha},\omega\setminus x_{\alpha}\}\rangle_{\alpha<\kappa}\) witnesses \(\mathfrak{s}\mathfrak{s}_{2}=\omega_{1}\). ∎
+
+**Definition 3.2****.**: Let \(\mathfrak{d}\) denote the minimum of the cardinalities of subsets of \(\omega^{\omega}\) that are cofinal with respect to eventual domination.
+
+**Corollary 3.3****.**: _Every transitive model of ZFC has a ccc forcing extension that preserves \(\mathfrak{b}\), \(\mathfrak{d}\), and \(\mathfrak{c}\), and collapses \(\mathfrak{s}\mathfrak{s}_{2}\) to \(\omega_{1}\)._
+
+Proof.: Add \(\mathfrak{c}\)-many random reals to the ground model. Then every element of \(\omega^{\omega}\) in the extension is eventually dominated by an element of \(\omega^{\omega}\) in the ground model; hence, \(\mathfrak{b}\), \(\mathfrak{d}\), and \(\mathfrak{c}\) are preserved by this forcing, while \(\mathfrak{s}\mathfrak{s}_{2}\) becomes \(\omega_{1}\). ∎
+
+**Definition 3.4****.**: We say that a transfinite sequence \(\langle x_{\alpha}\rangle_{\alpha<\eta}\) of subsets of \(\omega\) is _eventually splitting_ if for all \(y\in[\omega]^{\omega}\) there exists \(\alpha<\eta\) such that for all \(\beta\in\eta\setminus\alpha\) the set \(x_{\beta}\) splits \(y\).
+
+**Theorem 3.5****.**: _Let \(\kappa=\kappa^{\omega}\). Then \(\mathfrak{s}\mathfrak{s}_{2}=\omega_{1}\) is forced by the \(\kappa\)-long finite support iteration of Hechler forcing._
+
+Proof.: Let \(\mathbb{P}\) be the \(\kappa\)-long finite support iteration of Hechler forcing. Let \(G\) be a generic filter of \(\mathbb{P}\). For each \(\alpha<\kappa\), let \(g_{\alpha}\) be the generic dominating function added at stage \(\alpha\); set \(x_{\alpha}=\{n<\omega:g_{\alpha}(n)\text{ is even}\}\). Suppose \(p\in G\) and \(I\) and \(y\) are names such that \(p\) forces \(I\in[\kappa]^{\omega_{1}}\) and \(y\in[\omega]^{\omega}\). Choose \(q\in G\) and a name \(h\) such that \(q\leq p\) and \(q\) forces \(h\) to be an increasing map from \(\omega_{1}\) to \(I\). For each \(\alpha<\omega_{1}\), set \(E_{\alpha}=\{\beta<\kappa:q\not\Vdash h(\alpha)\not=\check{\beta}\}\); let \(k_{\alpha}\) be a surjection from \(\omega\) to \(E_{\alpha}\). Let \(q\geq r\in G\) and \(n<\omega\) and \(\gamma\leq\kappa\) and \(J\) be a name such that \(r\) forces \(J\in[\omega_{1}]^{\omega_{1}}\) and \(\sup\operatorname{ran}h=\check{\gamma}\) and \(h(\alpha)=k_{\alpha}(n)^{\check{}}\) for all \(\alpha\in J\). Set \(F=\{k_{\alpha}(n):\alpha<\omega_{1}\}\cap\gamma\); let \(j\) be the order isomorphism from some ordinal \(\eta\) to \(F\). Then \(\operatorname{cf}\eta=\operatorname{cf}\gamma=\omega_{1}\). For all \(\alpha<\kappa\), the set \(x_{\alpha}\) is Cohen over \(V[\langle g_{\beta}\rangle_{\beta<\alpha}]\); hence, \(\langle x_{j(\alpha)}\rangle_{\alpha<\eta}\) is eventually splitting in \(V[\langle g_{\alpha}\rangle_{\alpha<\gamma}]\). By a result of Baumgartner and Dordal [5], \(\langle x_{j(\alpha)}\rangle_{\alpha<\eta}\) is also eventually splitting in \(V[G]\). Choose \(\beta<\eta\) such that \(x_{j(\alpha)}\) splits \(y_{G}\) for all \(\alpha\in\eta\setminus\beta\). Then there exist \(s\in G\) and \(\alpha\in\gamma\setminus j(\beta)\) such that \(r\geq s\Vdash\check{\alpha}\in h``J\). Hence, \(\alpha\in I_{G}\) and \(x_{\alpha}\) splits \(y_{G}\). Thus, \(\langle\{x_{\alpha},\omega\setminus x_{\alpha}\}\rangle_{\alpha<\kappa}\) witnesses \(\mathfrak{s}\mathfrak{s}_{2}=\omega_{1}\) in \(V[G]\). ∎
+
+**Definition 3.6****.**: Let \(\mathrm{add}(\mathcal{B})\) denote the additivity of the ideal of meager sets of reals.
+
+It is known that \(\mathrm{add}(\mathcal{B})\leq\mathfrak{b}\) and that it is consistent that \(\mathrm{add}(\mathcal{B})<\mathfrak{b}\). (See 5.4 and 11.7 of [7] and 7.3.D of [4]).
+
+**Corollary 3.7****.**: _If \(\kappa=\operatorname{cf}\kappa>\omega\), then it is consistent that \(\mathfrak{s}\mathfrak{s}_{2}=\omega_{1}\) and \(\mathrm{add}(\mathcal{B})=\mathfrak{c}=\kappa\)._
+
+Proof.: Starting with GCH in the ground model, perform a \(\kappa\)-long finite support iteration of Hechler forcing. This forces \(\mathrm{add}(\mathcal{B})=\mathfrak{c}=\kappa\) (see 11.6 of [7]). By Theorem 3.5, this also forces \(\mathfrak{s}\mathfrak{s}_{2}=\omega_{1}\). ∎
+
+## 4. Models of \(\omega_{1}<Nt(\omega^{*})<\mathfrak{c}\)
+
+To prove the consistency of \(\omega_{1}<Nt(\omega^{*})<\mathfrak{c}\), we employ generalized iteration of forcing along posets as defined by Groszek and Jech [10]. We will only use finite support iterations along well-founded posets. For simplicity, we limit our definition of generalized iterations to this special case.
+
+**Definition 4.1****.**: Suppose \(X\) is a well-founded poset and \(\mathbb{P}\) a forcing order consisting of functions on \(X\). Given any \(x\in X\), partial map \(f\) on \(X\), and down-set \(Y\) of \(X\), set \({\mathbb{P}\upharpoonright Y}=\{p\upharpoonright Y:p\in\mathbb{P}\}\), \({X\upharpoonright x}=\{y\in X:y<x\}\), \(X\upharpoonright_{\leq}x=\{y\in X:y\leq x\}\), \({\mathbb{P}\upharpoonright x}={\mathbb{P}\upharpoonright({X\upharpoonright x})}\), \(\mathbb{P}\upharpoonright_{\leq}x={\mathbb{P}\upharpoonright(X\upharpoonright_{\leq}x)}\), \({f\upharpoonright x}=f\upharpoonright({X\upharpoonright x})\), and \(f\upharpoonright_{\leq}x=f\upharpoonright(X\upharpoonright_{\leq}x)\). Then \(\mathbb{P}\) is a _finite support iteration along_\(X\) if there exists a sequence \(\langle\mathbb{Q}_{x}\rangle_{x\in X}\) satisfying the following conditions for all \(x\in X\) and all \(p,q\in\mathbb{P}\).
+
+1. (1)\({\mathbb{P}\upharpoonright x}\) is a finite support iteration along \({X\upharpoonright x}\).
+2. (2)\(\mathbb{Q}_{x}\) is a \(({\mathbb{P}\upharpoonright x})\)-name for a forcing order.
+3. (3)\(\mathbb{P}\upharpoonright_{\leq}x=\{p\cup\{\langle x,q\rangle\}:\langle p,q\rangle\in({\mathbb{P}\upharpoonright x})*\mathbb{Q}_{x}\}\).
+4. (4)\({\mathbbm{1}_{\mathbb{P}}\upharpoonright x}\Vdash\mathbbm{1}_{\mathbb{P}}(x)=\mathbbm{1}_{\mathbb{Q}_{x}}\).
+5. (5)\(\mathbb{P}\) is the set of functions \(r\) on \(X\) for which \(r\upharpoonright_{\leq}y\in\mathbb{P}\upharpoonright_{\leq}y\) for all \(y\in X\) and \(\mathbbm{1}_{{\mathbb{P}\upharpoonright z}}\Vdash r(z)=\mathbbm{1}_{\mathbb{Q}_{z}}\) for all but finitely many \(z\in X\).
+6. (6)\(p\leq q\) if and only if \({p\upharpoonright y}\leq{q\upharpoonright y}\) and \({p\upharpoonright y}\Vdash p(y)\leq q(y)\) for all \(y\in X\).
+
+Given a finite support iteration \(\mathbb{P}\) along \(X\) and \(x\in X\) and a filter \(G\) of \(\mathbb{P}\), set \({G}_{x}=\{p(x):p\in G\}\), \({G\upharpoonright x}=\{{p\upharpoonright x}:p\in G\}\), and \(G\upharpoonright_{\leq}x=\{p\upharpoonright_{\leq}x:p\in G\}\). Given any down-set \(Y\) of \(X\), set \({G\upharpoonright Y}=\{p\upharpoonright Y:p\in G\}\).
+
+_Remark__._: If \(\mathbb{P}\) is a finite support iteration along a well-founded poset \(X\) with down-set \(Y\), then \({\mathbb{P}\upharpoonright Y}\) is an iteration along \(Y\), and \(\mathbbm{1}_{{\mathbb{P}\upharpoonright Y}}=\mathbbm{1}_{\mathbb{P}}\upharpoonright Y\).
+
+**Definition 4.2****.**: Suppose \(\mathbb{P}\) is a finite support iteration along a well-founded poset \(X\) with down-sets \(Y\) and \(Z\) such that \(Y\subseteq Z\). Then there is a complete embedding \(j_{Y}^{Z}\colon{\mathbb{P}\upharpoonright Y}\to{\mathbb{P}\upharpoonright Z}\) given by \(j_{Y}^{Z}(p)=p\cup(\mathbbm{1}_{\mathbb{P}}\upharpoonright Z\setminus Y)\) for all \(p\in{\mathbb{P}\upharpoonright Y}\). This embedding naturally induces an embedding of the class of \(({\mathbb{P}\upharpoonright Y})\)-names, which in turn naturally induces an embedding of the class of atomic forumlae in the \(({\mathbb{P}\upharpoonright Y})\)-forcing language. Let \(j_{Y}^{Z}\) also denote these embeddings.
+
+**Proposition 4.3****.**: _Suppose \(\mathbb{P}\), \(Y\), and \(Z\) are as in the above definition, and \(\varphi\) is an atomic formula in the \(({\mathbb{P}\upharpoonright Y})\)-forcing language. Then, for all \(p\in{\mathbb{P}\upharpoonright Z}\), we have \(p\Vdash j_{Y}^{Z}(\varphi)\) if and only if \(p\upharpoonright Y\Vdash\varphi\)._
+
+Proof.: If \(p\upharpoonright Y\Vdash\varphi\), then \(p\leq j_{Y}^{Z}(p\upharpoonright Y)\Vdash j_{Y}^{Z}(\varphi)\). Conversely, suppose \(p\upharpoonright Y\not\Vdash\varphi\). Then we may choose \(q\leq p\upharpoonright Y\) such that \(q\Vdash\neg\varphi\). Hence, \(j_{Y}^{Z}(q)\Vdash\neg j_{Y}^{Z}(\varphi)\). Set \(r=q\cup(p\upharpoonright Z\setminus Y)\). Then \(j_{Y}^{Z}(q)\geq r\leq p\); hence, \(p\not\Vdash j_{Y}^{Z}(\varphi)\). ∎
+
+**Lemma 4.4****.**: _Suppose \(\mathbb{P}\) is a finite support iteration along a well-founded poset \(X\) and \(x\) is a maximal element of \(X\). Set \(Y=X\setminus\{x\}\). Then there is a dense embedding \(\phi\colon\mathbb{P}\to({\mathbb{P}\upharpoonright Y})*j_{{X\upharpoonright x}}^{Y}(\mathbb{Q}_{x})\) given by \(\phi(p)=\langle p\upharpoonright Y,\,j_{{X\upharpoonright x}}^{Y}(p(x))\rangle\). Hence, if \(G\) is a \(\mathbb{P}\)-generic filter, then \({G}_{x}\) is \((\mathbb{Q}_{x})_{{G\upharpoonright x}}\)-generic over \(V[{G\upharpoonright Y}]\)._
+
+Proof.: First, let us show that \(\phi\) is an order embedding. Suppose \(r,s\in\mathbb{P}\). Then \(r\leq s\) if and only if \(r\upharpoonright Y\leq s\upharpoonright Y\) and \({r\upharpoonright x}\Vdash r(x)\leq s(x)\). Also, \(\phi(r)\leq\phi(s)\) if and only if \(r\upharpoonright Y\leq s\upharpoonright Y\) and \(r\upharpoonright Y\Vdash j_{{X\upharpoonright x}}^{Y}(r(x)\leq s(x))\). By Proposition 4.3, \(r\upharpoonright Y\Vdash j_{{X\upharpoonright x}}^{Y}(r(x)\leq s(x))\) if and only if \({r\upharpoonright x}\Vdash r(x)\leq s(x)\); hence, \(r\leq s\) if and only if \(\phi(r)\leq\phi(s)\).
+
+Finally, let us show that \(\operatorname{ran}\phi\) is dense. Suppose \(\langle p,q\rangle\in({\mathbb{P}\upharpoonright Y})*j_{{X\upharpoonright x}}^{Y}(\mathbb{Q}_{x})\). Then there exist \(r\leq p\) and \(s\in\operatorname{dom}\bigl{(}j_{{X\upharpoonright x}}^{Y}(\mathbb{Q}_{x})\bigr{)}\) such that \(r\Vdash s=q\in j_{{X\upharpoonright x}}^{Y}(\mathbb{Q}_{x})\). Hence, \(\langle r,s\rangle\leq\langle p,q\rangle\). Also, \(s\) is a \((j_{{X\upharpoonright x}}^{Y}``({\mathbb{P}\upharpoonright x}))\)-name; hence, there exists a \(({\mathbb{P}\upharpoonright x})\)-name \(t\) such that \(j_{{X\upharpoonright x}}^{Y}(t)=s\). Hence, \(r\Vdash j_{{X\upharpoonright x}}^{Y}(t\in\mathbb{Q}_{x})\); hence, \({r\upharpoonright x}\Vdash t\in\mathbb{Q}_{x}\). Hence, \(r\cup\{\langle x,t\rangle\}\in\mathbb{P}\) and \(\phi(r\cup\{\langle x,t\rangle\})=\langle r,s\rangle\). Thus, \(\operatorname{ran}\phi\) is dense. ∎
+
+_Remark__._: Proposition 4.3 and Lemma 4.4 and their proofs remain valid for arbitrary iterations along posets as defined in [10].
+
+**Lemma 4.5****.**: _Let \(\mathbb{P}\) be a forcing order, \(A\) a subset of \([\omega]^{\omega}\) with the SFIP, \(\mathbb{Q}\) the Booth forcing for \(A\), \(x\) a \(\mathbb{Q}\)-name for a generic pseudointersection of \(A\), and \(B\) a \(\mathbb{P}\)-name such that \(\mathbbm{1}_{\mathbb{P}}\) forces \(\check{A}\subseteq B\subseteq[\omega]^{\omega}\) and forces \(B\) to have the SFIP. Let \(i\) and \(j\) be the canonical embeddings, respectivly, of \(\mathbb{P}\)-names and \(\mathbb{Q}\)-names into \((\mathbb{P}*\check{\mathbb{Q}})\)-names. Then \(\mathbbm{1}_{\mathbb{P}*\check{\mathbb{Q}}}\) forces \(i(B)\cup\{j(x)\}\) to have the SFIP._
+
+Proof.: Seeking a contradiction, suppose \(r_{0}=\langle p_{0},\langle\sigma,F\rangle^{\check{}}\rangle\in\mathbb{P}*\check{\mathbb{Q}}\) and \(n<\omega\) and \(p_{0}\Vdash H\in[B]^{<\omega}\) and \(r_{0}\Vdash j(x)\cap\bigcap i(H)\subseteq\check{n}\). Then \(p_{0}\) forces \(\check{F}\cup H\subseteq B\), which is forced to have the SFIP; hence, there exist \(p_{1}\leq p_{0}\) and \(m\in\omega\setminus n\) such that \(p_{1}\Vdash\check{m}\in\bigcap(\check{F}\cup H)\). Set \(r_{1}=\langle p_{1},\langle\sigma\cup\{m\},F\rangle^{\check{}}\rangle\). Then \(r_{0}\geq r_{1}\Vdash\check{m}\in j(x)\cap\bigcap i(H)\), contradicting how we chose \(r_{0}\). ∎
+
+**Lemma 4.6****.**: _Suppose \(\mathbb{P}\) and \(\mathbb{Q}\) are forcing orders such that \(\mathbb{P}\) is ccc and \(\mathbb{Q}\) has property (K). Then \(\mathbbm{1}_{\mathbb{P}}\) forces \(\check{\mathbb{Q}}\) to have property (K)._
+
+Proof.: Suppose the lemma fails. Then there exist \(p\in\mathbb{P}\) and \(f\) such that \(p\Vdash f\in\check{\mathbb{Q}}^{\omega_{1}}\) and \(p\Vdash\forall J\in[\omega_{1}]^{\omega_{1}}\ \exists\alpha,\beta\in J\ f(\alpha)\perp f(\beta)\). For each \(\alpha<\omega_{1}\), choose \(p_{\alpha}\leq p\) and \(q_{\alpha}\in\mathbb{Q}\) such that \(p_{\alpha}\Vdash f(\alpha)=\check{q}_{\alpha}\). Then there exists \(I\in[\omega_{1}]^{\omega_{1}}\) such that \(q_{\alpha}\not\perp q_{\beta}\) for all \(\alpha,\beta\in I\). Let \(J\) be the \(\mathbb{P}\)-name \(\{\langle\check{\alpha},p_{\alpha}\rangle:\alpha\in I\}\). Then \(p\Vdash\forall\alpha,\beta\in J\ f(\alpha)=\check{q}_{\alpha}\not\perp\check{q}_{\beta}=f(\beta)\). Hence, \(p\Vdash\lvert J\rvert\leq\omega\). Since \(\mathbb{P}\) is ccc, there exists \(\alpha\in I\) such that \(p\Vdash J\subseteq\check{\alpha}\). But this contradicts \(p\geq p_{\alpha}\Vdash\check{\alpha}\in J\). ∎
+
+**Lemma 4.7****.**: _Suppose \(\mathbb{P}\) is a finite support iteration along a well-founded poset \(X\) and \({\mathbbm{1}_{\mathbb{P}}\upharpoonright x}\) forces \(\mathbb{Q}_{x}\) to have property (K) for all \(x\in X\). Then \(\mathbb{P}\) has property (K)._
+
+Proof.: We may assume the lemma holds whenever \(X\) is replaced by a poset of lesser height. Let \(I\in[\mathbb{P}]^{\omega_{1}}\). We may assume \(\{\mathrm{supp}(p):p\in I\}\) is a \(\Delta\)-system; let \(\sigma\) be its root. Set \(Y_{0}=\bigcup_{x\in\sigma}{X\upharpoonright x}\). Then \({\mathbb{P}\upharpoonright Y_{0}}\) has property (K). Let \(n=\lvert\sigma\setminus Y_{0}\rvert\) and \(\langle x_{i}\rangle_{i<n}\) biject from \(n\) to \(\sigma\setminus Y_{0}\). Set \(Y_{i+1}=Y_{i}\cup\{x_{i}\}\) for all \(i<n\). Suppose \(i<n\) and \({\mathbb{P}\upharpoonright Y_{i}}\) has property (K). By Lemma 4.6, \(\mathbbm{1}_{{\mathbb{P}\upharpoonright Y_{i}}}\) forces \(j_{{X\upharpoonright x_{i}}}^{Y_{i}}(\mathbb{Q}_{x_{i}})\) to have property (K). Hence, \({\mathbb{P}\upharpoonright Y_{i+1}}\) has property (K), for it densely embeds into \({\mathbb{P}\upharpoonright Y_{i}}*j_{{X\upharpoonright x_{i}}}^{Y_{i}}(\mathbb{Q}_{x_{i}})\) by Lemma 4.4. By induction, \({\mathbb{P}\upharpoonright Y_{n}}\) has property (K); hence, there exists \(J\in[I]^{\omega_{1}}\) such that \(p\upharpoonright Y_{n}\not\perp q\upharpoonright Y_{n}\) for all \(p,q\in J\). Fix \(p,q\in J\) and choose \(r\) such that \(r\leq p\upharpoonright Y_{n}\) and \(r\leq q\upharpoonright Y_{n}\). Set \(s=r\cup(p\upharpoonright\mathrm{supp}(p)\setminus Y_{n})\cup(q\upharpoonright\mathrm{supp}(q)\setminus Y_{n})\) and \(t=s\cup(\mathbbm{1}_{\mathbb{P}}\upharpoonright X\setminus\operatorname{dom}s)\). Then \(t\leq p,q\). ∎
+
+**Lemma 4.8****.**: _Suppose \(\operatorname{cf}\kappa=\kappa\leq\lambda=\lambda^{<\kappa}\). Then there exists a \(\kappa\)-like, \(\kappa\)-directed, well-founded poset \(\Xi\) with cofinality and cardinality \(\lambda\)._
+
+Proof.: Let \(\{x_{\alpha}:\alpha<\lambda\}\) biject from \(\lambda\) to \([\lambda]^{<\kappa}\). Construct \(\langle y_{\alpha}\rangle_{\alpha<\lambda}\in([\lambda]^{<\kappa})^{\lambda}\) as follows. Given \(\alpha<\lambda\) and \(\langle y_{\beta}\rangle_{\beta<\alpha}\), choose \(\xi_{\alpha}\in\lambda\setminus\bigcup_{\beta<\alpha}y_{\beta}\) and set \(y_{\alpha}=x_{\alpha}\cup\{\xi_{\alpha}\}\). Let \(\Xi\) be \(\{y_{\alpha}:\alpha<\lambda\}\) ordered by inclusion. Then \(\Xi\) is cofinal with \([\lambda]^{<\kappa}\); hence, \(\Xi\) is \(\kappa\)-directed and has cofinality \(\lambda\). Also, \(\Xi\) is well-founded because \(\langle y_{\alpha}\rangle_{\alpha<\lambda}\) is nondecreasing. Finally, \(\Xi\) is \(\kappa\)-like because for all \(I\in[\lambda]^{\kappa}\) we have \(\lvert\bigcup_{\alpha\in I}y_{\alpha}\rvert\geq\lvert\{\xi_{\alpha}:\alpha\in I\}\rvert=\kappa\); whence, \(\{y_{\alpha}:\alpha\in I\}\) has no upper bound in \([\lambda]^{<\kappa}\). ∎
+
+**Definition 4.9****.**: A point \(q\) in a space \(X\) is a _\(P_{\kappa}\)-point_ if every intersection of fewer than \(\kappa\)-many neighborhoods of \(q\) contains a neighborhood of \(q\).
+
+**Definition 4.10****.**: For all \(x,y\subseteq\omega\), define \(x\subseteq^{*}y\) as \(\lvert x\setminus y\rvert<\omega\). Let \(\mathfrak{p}\) denote the minimum value of \(\lvert A\rvert\) where \(A\) ranges over the subsets of \([\omega]^{\omega}\) that have SFIP yet have no pseudointersection.
+
+_Remark__._: It easily seen that \(\omega_{1}\leq\mathfrak{p}\leq\mathfrak{h}\).
+
+**Theorem 4.11****.**: _Suppose \(\omega_{1}\leq\operatorname{cf}\kappa=\kappa\leq\lambda=\lambda^{<\kappa}\). Then there is a property (K) forcing extension in which_
+
+\[\mathfrak{p}=\pi Nt(\omega^{*})=Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{2}=\mathfrak{b}=\kappa\leq\lambda=\mathfrak{c}.\]
+
+_Moreover, in this extension \(\omega^{*}\) has \(P_{\kappa}\)-points; whence, \(\max_{q\in\omega^{*}}\chi Nt(q,\omega^{*})=\kappa\)._
+
+Proof.: Let \(\Xi\) be as in Lemma 4.8. Let \(\langle\sigma_{\alpha}\rangle_{\alpha<\lambda}\) biject from \(\lambda\) to \(\Xi\). Let \(\langle\langle\zeta_{\alpha},\eta_{\alpha}\rangle\rangle_{\alpha<\lambda}\) biject from \(\lambda\) to \(\lambda^{2}\). Given \(\alpha<\lambda\) and \(\langle\tau_{\zeta_{\beta},\eta_{\beta}}\rangle_{\beta<\alpha}\in\Xi^{\alpha}\), choose \(\tau_{\zeta_{\alpha},\eta_{\alpha}}\in\Xi\) such that \(\sigma_{\zeta_{\alpha}}<\tau_{\zeta_{\alpha},\eta_{\alpha}}\not\leq\tau_{\zeta_{\beta},\eta_{\beta}}\) for all \(\beta<\alpha\). We may so choose \(\tau_{\zeta_{\alpha},\eta_{\alpha}}\) because \(\Xi\) is directed and has cofinality \(\lambda\).
+
+Let us construct a finite support iteration \(\mathbb{P}\) along \(\Xi\). Since \(\Xi\) is well-founded, we may define \(\mathbb{Q}_{\sigma}\) in terms of \({\mathbb{P}\upharpoonright\sigma}\) for each \(\sigma\in\Xi\). Suppose \(\sigma\in\Xi\) and, for all \(\tau<\sigma\), we have \(\lvert\mathbb{P}\upharpoonright_{\leq}\tau\rvert<\kappa\) and \(\mathbbm{1}_{{\mathbb{P}\upharpoonright\tau}}\) forces \(\mathbb{Q}_{\tau}\) to have property (K). Then \({\mathbb{P}\upharpoonright\sigma}\) has property (K) by Lemma 4.7, and hence is ccc. Moreover, \(\lvert{\mathbb{P}\upharpoonright\sigma}\rvert<\kappa\) because \({\mathbb{P}\upharpoonright\sigma}\) is a finite support iteration along \({\Xi\upharpoonright\sigma}\) and \(\lvert{\Xi\upharpoonright\sigma}\rvert<\kappa\). Hence, \(\mathbbm{1}_{{\mathbb{P}\upharpoonright\sigma}}\Vdash\lvert\mathfrak{c}^{<\kappa}\rvert\leq((\kappa^{\omega})^{<\kappa})^{\check{}}\leq\lambda\). Let \(\mathcal{E}_{\sigma}\) be a \(({\mathbb{P}\upharpoonright\sigma})\)-name for the set of all \(E\) in the \(({\mathbb{P}\upharpoonright\sigma})\)-generic extension for which \(E\in[[\omega]^{\omega}]^{<\kappa}\) and \(E\) has the SFIP. Then we may choose a \(({\mathbb{P}\upharpoonright\sigma})\)-name \(f_{\sigma}\) such that \(\mathbbm{1}_{{\mathbb{P}\upharpoonright\sigma}}\) forces \(f_{\sigma}\) to be a surjection from \(\lambda\) to \(\mathcal{E}_{\sigma}\). We may assume we are given corresponding \(f_{\tau}\) for all \(\tau<\sigma\). If there exist \(\alpha,\beta<\lambda\) such that \(\sigma=\tau_{\alpha,\beta}\), then let \(\mathbb{Q}_{\sigma}\) be a \(({\mathbb{P}\upharpoonright\sigma})\)-name for \(\mathbb{Q}_{\sigma}^{\prime}\times\mathrm{Fn}(\omega,\,2)\) where \(\mathbb{Q}_{\sigma}^{\prime}\) is a \(({\mathbb{P}\upharpoonright\sigma})\)-name for the Booth forcing for \(f_{\sigma_{\alpha}}(\beta)\). If there are no such \(\alpha\) and \(\beta\), then let \(\mathbb{Q}_{\sigma}\) be a \(({\mathbb{P}\upharpoonright\sigma})\)-name for a singleton poset. Then \(\mathbbm{1}_{{\mathbb{P}\upharpoonright\sigma}}\) forces \(\mathbb{Q}_{\sigma}\) to have property (K). Also, we may assume \(\lvert\mathbb{Q}_{\sigma}\rvert<\kappa\). Hence, \(\lvert\mathbb{P}\upharpoonright_{\leq}\sigma\rvert<\kappa\).
+
+By induction, \(\lvert\mathbb{P}\upharpoonright_{\leq}\sigma\rvert<\kappa\) and \(\mathbbm{1}_{{\mathbb{P}\upharpoonright\sigma}}\) forces \(\mathbb{Q}_{\sigma}\) to have property (K) for all \(\sigma\in\Xi\). Hence, \(\mathbb{P}\) has property (K) by Lemma 4.7, and hence is ccc. Also, since \(\lvert\Xi\rvert\leq\lambda\) and \(\mathbb{P}\) is a finite support iteration, \(\lvert\mathbb{P}\rvert\leq\lambda\). Let \(G\) be a \(\mathbb{P}\)-generic filter. Then \(\mathfrak{c}^{V[G]}\leq\lambda^{\omega}=\lambda\). Moreover, \(\mathfrak{c}^{V[G]}\geq\lambda\) because \(\mathbb{P}\) adds \(\lambda\)-many Cohen reals.
+
+By Theorem 2.16 and Lemma 2.6, it suffices to show that \(\mathfrak{b}^{V[G]}\leq\kappa\leq\mathfrak{p}^{V[G]}\), that \(\mathfrak{s}\mathfrak{s}_{2}^{V[G]}\leq\kappa\), and that some \(q\in(\omega^{*})^{V[G]}\) is a \(P_{\kappa}\)-point. First, we prove \(\kappa\leq\mathfrak{p}^{V[G]}\). Suppose \(E\in([[\omega]^{\omega}]^{<\kappa})^{V[G]}\) and \(E\) has the SFIP. Then there exists \(\alpha<\lambda\) such that \(E\in V[{G\upharpoonright\sigma_{\alpha}}]\) because \(\Xi\) is \(\kappa\)-directed. Hence, there exists \(\beta<\lambda\) such that \((f_{\sigma_{\alpha}})_{{G\upharpoonright\sigma_{\alpha}}}(\beta)=E\). Hence, \(E\) has a pseudointersection in \(V[G\upharpoonright_{\leq}\tau_{\alpha,\beta}]\). Thus, \(\kappa\leq\mathfrak{p}^{V[G]}\).
+
+Second, let us show that \(\mathfrak{b}^{V[G]}\leq\kappa\). For each \(\alpha<\kappa\), let \(u_{\alpha}\) be the increasing enumeration of the Cohen real added by the \(\mathrm{Fn}(\omega,\,2)\) factor of \(\mathbb{Q}_{\tau_{0,\alpha}}\). Then it suffices to show that \(\{u_{\alpha}:\alpha<\kappa\}\) is unbounded in \((\omega^{\omega})^{V[G]}\). Suppose \(v\in(\omega^{\omega})^{V[G]}\). Then there exists \(\sigma\in\Xi\) such that \(v\in V[{G\upharpoonright\sigma}]\). Since \(\Xi\) is \(\kappa\)-like, there exists \(\alpha<\kappa\) such that \(\tau_{0,\alpha}\not\leq\sigma\). By Lemma 4.4, \(u_{\alpha}\) enumerates a real Cohen generic over \(V[{G\upharpoonright\sigma}]\); hence, \(u_{\alpha}\) is not eventually dominated by \(v\).
+
+Third, let us prove \(\mathfrak{s}\mathfrak{s}_{2}^{V[G]}\leq\kappa\). For each \(\alpha<\lambda\), let \(x_{\alpha}\) be the Cohen real added by the \(\mathrm{Fn}(\omega,\,2)\) factor of \(\mathbb{Q}_{\tau_{0,\alpha}}\). Suppose \(I\in([\lambda]^{\kappa})^{V[G]}\) and \(y\in([\omega]^{\omega})^{V[G]}\). Then there exists \(\sigma\in\Xi\) such that \(y\in V[{G\upharpoonright\sigma}]\). Since \(\Xi\) is \(\kappa\)-like, there exists \(\alpha\in I\) such that \(\tau_{0,\alpha}\not\leq\sigma\). By Lemma 4.4, \(x_{\alpha}\) is Cohen generic over \(V[{G\upharpoonright\sigma}]\), and therefore splits \(y\). Thus, \(\langle\{x_{\alpha},\omega\setminus x_{\alpha}\}\rangle_{\alpha<\lambda}\) witnesses \(\mathfrak{s}\mathfrak{s}_{2}^{V[G]}\leq\kappa\).
+
+Finally, let us construct a \(P_{\kappa}\)-point \(q\in(\omega^{*})^{V[G]}\). Let \(\sqsubseteq\) be an extension of the ordering of \(\Xi\) to a well-ordering of \(\Xi\). For each \(\sigma\in\Xi\), set \(Y_{\sigma}=\{\tau\in\Xi:\tau\sqsubset\sigma\}\). Set \(\rho=\min_{\sqsubseteq}\Xi\) and choose \(U_{\rho}\in(\omega^{*})^{V}\). Suppose \(\tau\in\Xi\) and \(\sigma\) is a final predecessor of \(\tau\) with respect to \(\sqsubseteq\) and \(U_{\sigma}\in(\omega^{*})^{V[{G\upharpoonright Y_{\sigma}]}}\). If there are no \(\alpha,\beta<\lambda\) such that \(\sigma=\tau_{\alpha,\beta}\) and \((f_{\sigma_{\alpha}})_{{G\upharpoonright\sigma_{\alpha}}}(\beta)\subseteq U_{\sigma}\), then choose \(U_{\tau}\in(\omega^{*})^{V[{G\upharpoonright Y_{\tau}]}}\) such that \(U_{\tau}\supseteq U_{\sigma}\). Now suppose such \(\alpha\) and \(\beta\) exist. Let \(v_{\sigma}\) be the pseudointersection of \((f_{\sigma_{\alpha}})_{{G\upharpoonright\sigma_{\alpha}}}(\beta)\) added by \(\mathbb{Q}_{\sigma}^{\prime}\).
+
+By Lemmas 4.4 and 4.5, \(U_{\sigma}\cup\{v_{\sigma}\}\) has the SFIP; hence, we may choose \(U_{\tau}\in(\omega^{*})^{V[{G\upharpoonright Y_{\tau}]}}\) such that \(U_{\tau}\supseteq U_{\sigma}\cup\{v_{\sigma}\}\). For \(\tau\in\Xi\) that are limit points with respect to \(\sqsubseteq\), choose \(U_{\tau}\in(\omega^{*})^{V[{G\upharpoonright Y_{\tau}}]}\) such that \(U_{\tau}\supseteq\bigcup_{\sigma\sqsubset\tau}U_{\sigma}\); set \(q=\bigcup_{\tau\in\Xi}U_{\tau}\). Then, arguing as in the proof of \(\kappa\leq\mathfrak{p}^{V[G]}\), we have that \(q\) is a \(P_{\kappa}\)-point in \((\omega^{*})^{V[G]}\). ∎
+
+The forcing extension of Theorem 4.11 can be modified to satisfy \(\mathfrak{b}=\mathfrak{s}<Nt(\omega^{*})<\mathfrak{c}\).
+
+**Definition 4.12****.**: Given a class \(\mathcal{J}\) of posets and a cardinal \(\kappa\), let \({\mathrm{M}A}(\kappa;\,\mathcal{J})\) denote the statement that, given any \(\mathbb{P}\in\mathcal{J}\) and fewer than \(\kappa\)-many dense subsets of \(\mathbb{P}\), there is a filter of \(\mathbb{P}\) intersecting each of these dense sets. We may replace \(\mathcal{J}\) with a descriptive term for \(\mathcal{J}\) when there is no ambiguity. For example, \({\mathrm{M}A}(\mathfrak{c};\,\text{ccc})\) is Martin’s axiom.
+
+**Theorem 4.13****.**: _Suppose \(\omega_{1}<\operatorname{cf}\kappa=\kappa\leq\lambda=\lambda^{<\kappa}\). Then there is a property (K) forcing extension in which_
+
+\[\omega_{1}=\pi Nt(\omega^{*})=\mathfrak{b}=\mathfrak{s}<Nt(\omega^{*})=\mathfrak{s}\mathfrak{s}_{2}=\kappa\leq\lambda=\mathfrak{c}.\]
+
+Proof.: Let \(\mathbb{P}\) be as in the proof of Theorem 4.11. Set \(\mathbb{R}=\mathbb{P}\times\mathrm{Fn}(\omega_{1},\,2)\), which has property (K) because \(\mathbb{P}\) does. Let \(K\) be a generic filter of \(\mathbb{R}\). Let \(\pi_{0}\) and \(\pi_{1}\) be the natural coordinate projections on \(\mathbb{R}\); let \(\pi_{0}\) and \(\pi_{1}\) also denote their respective natural extensions to the class of \(\mathbb{R}\)-names. Set \(G=\pi_{0}``K\) and \(H=\pi_{1}``K\). Then \(\mathfrak{c}^{V[K]}=\lambda\) clearly holds. Adding \(\omega_{1}\)-many Cohen reals to any model of ZFC forces \(\mathfrak{b}=\mathfrak{s}=\omega_{1}\), and \(\pi Nt(\omega^{*})=\mathfrak{h}\leq\mathfrak{b}\), so \(\pi Nt(\omega^{*})^{V[K]}=\mathfrak{b}^{V[K]}=\mathfrak{s}^{V[K]}=\omega_{1}\).
+
+For each \(\alpha<\lambda\), let \(x_{\alpha}\) be the Cohen real added by the \(\mathrm{Fn}(\omega,\,2)\) factor of \(\mathbb{Q}_{\tau_{0,\alpha}}\). Suppose \(I\in([\lambda]^{\kappa})^{V[K]}\) and \(y\in([\omega]^{\omega})^{V[K]}\). Then there exists \(\sigma\in\Xi\) such that \(y\in V[({G\upharpoonright\sigma})\times H]\). Since \(\Xi\) is \(\kappa\)-like, there exists \(\alpha\in I\) such that \(\tau_{0,\alpha}\not\leq\sigma\). By Lemma 4.4, \(x_{\alpha}\) is Cohen generic over \(V[{G\upharpoonright\sigma}]\); hence, \(x_{\alpha}\) is Cohen generic over \(V[({G\upharpoonright\sigma})\times H]\) and therefore splits \(y\). Thus, \(\langle\{x_{\alpha},\omega\setminus x_{\alpha}\}\rangle_{\alpha<\lambda}\)witnesses \(\mathfrak{s}\mathfrak{s}_{2}^{V[K]}\leq\kappa\).
+
+Therefore, it suffices to show that \(Nt(\omega^{*})^{V[K]}\geq\kappa\). Suppose \(\mu<\kappa\) and \(\mathcal{A}\) is an \(\mathbb{R}\)-name for a base of \(\omega^{*}\). Choose an \(\mathbb{R}\)-name \(q\) for an element of \(\omega^{*}\) with character \(\lambda\). Let \(f\) be a name for an injection from \(\lambda\) into \(\mathcal{A}\) such that \(q\in\bigcap\operatorname{ran}f\). Let \(g\) be a name for an element of \(([\omega]^{\omega})^{\lambda}\) such that \(q\in g(\alpha)^{*}\subseteq f(\alpha)\) for all \(\alpha<\lambda\). For each \(\alpha<\lambda\), let \(u_{\alpha}\) be a name for \(g(\alpha)\) such that \(u_{\alpha}=\{\{\check{n}\}\times A_{\alpha,n}:n<\omega\}\) where each \(A_{\alpha,n}\) is a countable antichain of \(\mathbb{R}\). Since \(\max\{\omega_{1},\mu\}<\lambda\), there exist \(\xi<\omega_{1}\) and \(J\in[\lambda]^{\mu}\) such that \(\operatorname{ran}\pi_{1}(u_{\alpha})\subseteq\mathrm{Fn}(\xi,\,2)\) for all \(\alpha\in J\). It suffices to show that \(\{(u_{\alpha})_{K}:\alpha\in J\}\) has a pseudointersection in \(V[K]\).
+
+For each \(\alpha\in J\), set \(v_{\alpha}=\{\langle\check{n},r\rangle:\langle\check{n},\langle p,r\rangle\rangle\in u_{\alpha}\text{ and }p\in G\}\). Set \(H_{0}=H\cap\mathrm{Fn}(\xi,\,2)\). By Bell’s Theorem [6], \({\mathrm{M}A}(\mathfrak{p};\,\sigma\text{-centered})\) is a theorem of ZFC. Hence, \(V[G]\) satisfies \({\mathrm{M}A}(\kappa;\,\sigma\text{-centered})\). By an argument of Baumgartner and Tall communicated by Roitman [18], adding a single Cohen real preserves \({\mathrm{M}A}(\kappa;\,\sigma\text{-centered})\). Since Booth forcing for \(\{(v_{\alpha})_{H_{0}}:\alpha\in J\}\) is \(\sigma\)-centered, \(\{(v_{\alpha})_{H_{0}}:\alpha\in J\}\), which is equal to \(\{(u_{\alpha})_{K}:\alpha\in J\}\), has a pseudointersection in \(V[G\times H_{0}]\). ∎
+
+## 5. Local Noetherian type and \(\pi\)-type
+
+**Definition 5.1****.**: For every infinite cardinal \(\kappa\), let \(u(\kappa)\) denote the space of uniform ultrafilters on \(\kappa\).
+
+Dow and Zhou [8] proved that there is a point in \(\omega^{*}\) that (along with satisfying some additional properties) has an \(\omega^{\mathrm{op}}\)-like local base. We present a simpler construction of an \(\omega^{\mathrm{op}}\)-like local base which also naturally generalizes to every \(u(\kappa)\). This construction is essentially due to Isbell [11], who was interested in actual intersections as opposed to pseudointersections.
+
+**Definition 5.2****.**: Given cardinals \(\lambda\geq\kappa\geq\omega\) and a point \(p\) in a space \(X\), a _local_\(\langle\lambda,\kappa\rangle\)-_splitter_ is a set \(\mathcal{U}\) of \(\lambda\)-many open neighborhoods of \(p\) such that \(p\) is not in the interior of \(\bigcap\mathcal{V}\) for any \(\mathcal{V}\in[\mathcal{U}]^{\kappa}\).
+
+**Lemma 5.3****.**: _Every poset \(P\) is almost \(\lvert P\rvert^{\mathrm{op}}\)-like._
+
+Proof.: Let \(\kappa=\lvert P\rvert\) and let \(\langle p_{\alpha}\rangle_{\alpha<\kappa}\) biject from \(\kappa\) to \(P\). Define a partial map \(f\colon\kappa\to P\) as follows. Suppose \(\alpha<\kappa\) and we have a partial map \(f_{\alpha}\colon\alpha\to P\). If \(\operatorname{ran}f_{\alpha}\) is dense in \(P\), then set \(f_{\alpha+1}=f_{\alpha}\). Otherwise, set \(\beta=\min\{\delta<\kappa:p_{\delta}\not\geq q\text{ for all }q\in\operatorname{ran}f_{\alpha}\}\) and set \(f_{\alpha+1}=f_{\alpha}\cup\{\langle\alpha,p_{\beta}\rangle\}\). For limit ordinals \(\gamma\leq\kappa\), set \(f_{\gamma}=\bigcup_{\alpha<\gamma}f_{\alpha}\). Set \(f=f_{\kappa}\). Then \(f\) is nonincreasing; hence, \(\operatorname{ran}f\) is \(\kappa^{\mathrm{op}}\)-like. Moreover, \(\operatorname{ran}f\) is dense in \(P\). ∎
+
+**Lemma 5.4****.**: _Suppose \(X\) is a space with a point \(p\) at which there is no finite local base. Then \(\chi Nt(p,X)\) is the least \(\kappa\geq\omega\) for which there is a local \(\langle\chi(p,X),\kappa\rangle\)-splitter at \(p\). Moreover, if \(\lambda>\chi(p,X)\), then \(p\) does not have a local \(\langle\lambda,\kappa\rangle\)-splitter at \(p\) for any \(\kappa<\lambda\) or \(\kappa\leq\operatorname{cf}\lambda\)._
+
+Proof.: By Lemma 5.3, \(\chi(p,X)\geq\chi Nt(p,X)\); hence, a \(\chi Nt(p,X)^{\mathrm{op}}\)-like local base at \(p\) (which necessarily has size \(\chi(p,X)\)) is a local \(\langle\chi(p,X),\chi Nt(p,X)\rangle\)-splitter at \(p\). To show the converse, let \(\lambda=\chi(p,X)\) and let \(\langle U_{\alpha}\rangle_{\alpha<\lambda}\) be a sequence of open neighborhoods of \(p\). Let \(\{V_{\alpha}:\alpha<\lambda\}\) be a local base at \(p\). For each \(\alpha<\lambda\), choose \(W_{\alpha}\in\{V_{\beta}:\beta<\lambda\}\) such that \(W_{\alpha}\subseteq U_{\alpha}\cap V_{\alpha}\). Then \(\{W_{\alpha}:\alpha<\lambda\}\) is a local base at \(p\). Let \(\kappa<\chi Nt(p,X)\). Then there exist \(\alpha<\lambda\) and \(I\in[\lambda]^{\kappa}\) such that \(W_{\alpha}\subseteq\bigcap_{\beta\in I}W_{\beta}\). Hence, \(p\) is in the interior of \(\bigcap_{\beta\in I}U_{\beta}\). Hence, \(\{U_{\alpha}:\alpha<\lambda\}\) is not a local \(\langle\lambda,\kappa\rangle\)-splitter at \(p\).
+
+To prove the second half of the lemma, suppose \(\lambda>\chi(p,X)\) and \(\mathcal{A}\) is a set of \(\lambda\)-many open neighborhoods of \(p\). Let \(\mathcal{B}\) be a local base at \(p\) of size \(\chi(p,X)\). Then, for all \(\kappa<\lambda\) and \(\kappa\leq\operatorname{cf}\lambda\), there exist \(U\in\mathcal{B}\) and \(\mathcal{C}\in[\mathcal{A}]^{\kappa}\) such that \(U\subseteq\bigcap\mathcal{C}\). Hence, \(\mathcal{A}\) is not a local \(\langle\lambda,\kappa\rangle\)-splitter at \(p\). ∎
+
+**Theorem 5.5****.**: _For each \(\kappa\geq\omega\), there exists \(p\in u(\kappa)\) such that \(\chi Nt(p,u(\kappa))=\omega\) and \(\chi(p,u(\kappa))=2^{\kappa}\)._
+
+Proof.: Let \(A\) be an independent family of subsets of \(\kappa\) of size \(2^{\kappa}\). Set \(B=\bigcup_{F\in[A]^{\omega}}\{x\subseteq\kappa:\forall y\in F\ \ \lvert x\setminus y\rvert<\kappa\}\). Since \(A\) is independent, we may extend \(A\) to an ultrafilter \(p\) on \(\kappa\) such that \(p\cap B=\emptyset\). For each \(x\subseteq\kappa\), set \(x^{*}=\{q\in u(\kappa):x\in q\}\). Then \(\{x^{*}:x\in A\}\) is a local \(\langle 2^{\kappa},\omega\rangle\)-splitter at \(p\). Since \(\chi(p,u(\kappa))\leq 2^{\kappa}\), it follows from Lemma 5.4 that \(\chi Nt(p,u(\kappa))=\omega\) and \(\chi(p,u(\kappa))=2^{\kappa}\). ∎
+
+**Definition 5.6****.**: Let \(\mathfrak{a}\) denote the minimum of the cardinalities of infinite, maximal almost disjoint subfamilies of \([\omega]^{\omega}\). Let \(\mathfrak{i}\) denote the minimum of the cardinalities of infinite, maximal independent subfamilies of \([\omega]^{\omega}\).
+
+It is known that \(\mathfrak{b}\leq\mathfrak{a}\) and \(\mathfrak{r}\leq\mathfrak{i}\geq\mathfrak{d}\geq\mathfrak{s}\). (See 8.4, 8.12, 8.13 and 3.3 of [7].) Because of Kunen’s result that \(\mathfrak{a}=\aleph_{1}\) in the Cohen model (see VIII.2.3 of [14]), it is consistent that \(\mathfrak{a}<\mathfrak{r}\). Also, Shelah [20] has constructed a model of \(\mathfrak{r}\leq\mathfrak{u}<\mathfrak{a}\).
+
+In ZFC, the best upper bound of \(\chi Nt(\omega^{*})\) of which we know is \(\mathfrak{c}\) by Lemma 5.3. We will next prove Theorem 5.10, which implies that, except for \(\mathfrak{c}\) and possibly \(\operatorname{cf}\mathfrak{c}\), all of the cardinal characteristics of the continuum with definitions included in Blass [7] can consistently be simultaneously strictly less than \(\chi Nt(\omega^{*})\).
+
+**Lemma 5.7****.**: _Suppose \(\kappa\), \(\lambda\), and \(\mu\) are regular cardinals and \(\kappa\leq\lambda>\mu\). Then \((\kappa\times\lambda)^{\mathrm{op}}\) is not almost \(\mu^{\mathrm{op}}\)-like._
+
+Proof.: Let \(I\) be a cofinal subset of \(\kappa\times\lambda\). Then it suffices to show that \(I\) is not \(\mu\)-like. If \(\kappa=\lambda\), then \(I\) is not \(\mu\)-like because it is \(\lambda\)-directed. Suppose \(\kappa<\lambda\). Then there exists \(\alpha<\kappa\) such that \(\lvert I\cap(\{\alpha\}\times\lambda)\rvert=\lambda\); hence, \(I\) has an increasing \(\lambda\)-sequence; hence, \(I\) is not \(\mu\)-like. ∎
+
+**Lemma 5.8****.**: _Given any infinite independent subfamily \(I\) of \([\omega]^{\omega}\), there exists \(J\subseteq[\omega]^{\omega}\) such that if \(x\) is a generic pseudointersection of \(J\) then \(I\cup\{x\}\) is independent, but \(I\cup\{x,y\}\) is not independent for any \(y\in[\omega]^{\omega}\cap V\setminus I\)._
+
+Proof.: See Exercise A12 on page 289 of Kunen [14]. ∎
+
+**Definition 5.9****.**: We say a \(P_{\kappa}\)-point in a space is _simple_ if it has a local base of order type \(\kappa^{\mathrm{op}}\).
+
+**Theorem 5.10****.**: _Suppose \(\omega_{1}\leq\operatorname{cf}\kappa=\kappa\leq\operatorname{cf}\lambda=\lambda=\lambda^{<\kappa}\). Then there is a property (K) forcing extension satisfying \(\mathfrak{p}=\mathfrak{a}=\mathfrak{i}=\mathfrak{u}=\kappa\leq\lambda=\chi Nt(\omega^{*})=\mathfrak{c}\)._
+
+Proof.: We will construct a finite support iteration \(\langle\mathbb{P}_{\alpha}\rangle_{\alpha\leq\lambda\kappa}\) where \(\lambda\kappa\) denotes the ordinal product of \(\lambda\) and \(\kappa\). It suffices to ensure that the iteration is at every stage property (K) and of size at most \(\lambda\), and that \(V^{\mathbb{P}_{\lambda\kappa}}\) satisfies \(\max\{\mathfrak{a},\mathfrak{i},\mathfrak{u}\}\leq\kappa\leq\mathfrak{p}\) and \(\lambda\leq\chi Nt(\omega^{*})\). Our strategy is to interleave an iteration of length \(\lambda\kappa\) and three iterations of length \(\kappa\). At every stage below \(\lambda\kappa\), add another piece of what will be an ultrafilter base that, ordered by \(\supseteq^{*}\), will be isomorphic to a cofinal subset of \(\kappa\times\lambda\). Also, at every stage we will add a pseudointersection, such that the final model satisfies \(\mathfrak{p}\geq\kappa\). After each limit stage of cofinality \(\lambda\), add an element to each of three objects that, when completed, will be a maximal almost disjoint family of size \(\kappa\), a maximal independent family of size \(\kappa\), and a base of a simple \(P_{\kappa}\)-point in \(\omega^{*}\).
+
+Let \(\varphi\colon\lambda^{2}\rightarrow\lambda\) be a bijection such that \(\varphi(\alpha,\beta)\geq\alpha\) for all \(\alpha,\beta<\lambda\). For each \(\langle\alpha,\beta\rangle\in\kappa\times\lambda\), set \(E_{\alpha,\beta}=\{\langle\gamma,\delta\rangle\in\kappa\times\lambda:\lambda\gamma+\delta<\lambda\alpha+\beta\}\). Suppose \(\langle\alpha,\beta\rangle\in\kappa\times\lambda\) and we have constructed \(\langle\mathbb{P}_{\gamma}\rangle_{\gamma\leq\lambda\alpha+\beta}\) to have property (K) and size at most \(\lambda\) at all of its stages, and a sequence \(\langle x_{\gamma,\delta}\rangle_{\langle\gamma,\delta\rangle\in E_{\alpha,\beta}}\) of \(\mathbb{P}_{\lambda\alpha+\beta}\)-names each forced to be in \([\omega]^{\omega}\). Set \(B=\{x_{\gamma,\delta}:\langle\gamma,\delta\rangle\in E_{\alpha,\beta}\}\). Let \(\langle S_{\gamma}\rangle_{\gamma<\kappa}\) be a partition of \(\lambda\) into \(\kappa\)-many stationary sets such that \(S_{0}\) contains all successor ordinals. Suppose we have constructed a sequence \(\langle\rho_{\gamma,\delta}\rangle_{\langle\gamma,\delta\rangle\in E_{\alpha,\beta}}\in\lambda^{E_{\alpha,\beta}}\) such that we always have \(\rho_{\gamma,\delta}\in S_{\gamma}\) and \(\rho_{\gamma,\delta_{0}}<\rho_{\gamma,\delta_{1}}\) whenever \(\delta_{0}<\delta_{1}\). Set \(D_{\alpha,\beta}=\{\langle\gamma,\rho_{\gamma,\delta}\rangle:\langle\gamma,\delta\rangle\in E_{\alpha,\beta}\}\). Further suppose that \(\{\langle\langle\gamma,\rho_{\gamma,\delta}\rangle,x_{\gamma,\delta}\rangle:\langle\gamma,\delta\rangle\in E_{\alpha,\beta}\}\) is forced to be an order embedding of \(D_{\alpha,\beta}\) into \(\langle[\omega]^{\omega},\supseteq^{*}\rangle\) and that its range \(B\) is forced to have the SFIP. Also suppose that we have the following if \(\alpha>0\).
+
+(5.1) \[\Vdash_{\lambda\alpha+\beta}\forall\sigma\in[B]^{<\omega}\ \exists\delta<\lambda\ \,\bigcap\sigma\not\subseteq^{*}x_{0,\delta}\]
+
+For each \(\varepsilon<\lambda\), set \(A_{\varepsilon}=\{x_{\gamma,\delta}:\langle\gamma,\delta\rangle\in E_{\alpha,\beta}\text{ and }\langle\gamma,\rho_{\gamma,\delta}\rangle<\langle\alpha,\varepsilon\rangle\}\).
+
+Let \(y_{\beta}\) be a \(\mathbb{P}_{\lambda\alpha+\beta}\)-name for a surjection from \(\lambda\) to \([\omega]^{\omega}\). We may assume that corresponding \(y_{\gamma}\) have already been constructed for all \(\gamma<\beta\). Let \(\varphi(\zeta,\eta)=\beta\).
+
+**Claim****.**: _If \(\alpha>0\), then we may choose \(z\in\{y_{\zeta}(\eta),\,\omega\setminus y_{\zeta}(\eta)\}\) such that_
+
+\[\Vdash_{\lambda\alpha+\beta}\forall\sigma\in[B]^{<\omega}\ \exists\delta<\lambda\ \,z\cap\bigcap\sigma\not\subseteq^{*}x_{0,\delta}.\]
+
+Proof.: Suppose not. Let \(\{z_{0},z_{1}\}=\{y_{\zeta}(\eta),\,\omega\setminus y_{\zeta}(\eta)\}\). Then, working in a generic extension by \(\mathbb{P}_{\lambda\alpha+\beta}\), there exist \(\sigma_{0},\sigma_{1}\in[B]^{<\omega}\) and such that \(z_{i}\cap\bigcap\sigma_{i}\subseteq^{*}x_{0,\delta}\) for all \(i<2\) and \(\delta<\lambda\). Hence, \(\bigcap\bigcup_{i<2}\sigma_{i}\subseteq^{*}x_{0,\delta}\) for all \(\delta<\lambda\), in contradiction with (5.1). ∎
+
+If \(\alpha>0\), then choose \(z\) as in the above claim; otherwise, choose \(z\) arbitrarily. If \(\alpha=0\), then set \(\rho_{\alpha,\beta}=\beta+1\). Otherwise, we may choose \(\rho_{\alpha,\beta}\in S_{\alpha}\) such that \(\rho_{\alpha,\beta}>\rho_{\alpha,\gamma}\) for all \(\gamma<\beta\) and
+
+\[\Vdash_{\lambda\alpha+\beta}\forall\sigma\in[A_{\rho_{\alpha,\beta}}]^{<\omega}\ \exists\delta<\rho_{\alpha,\beta}\ \,z\cap\bigcap\sigma\not\subseteq^{*}x_{0,\delta}.\]
+
+Set \(D_{\alpha,\beta+1}=D_{\alpha,\beta}\cup\{\langle\alpha,\rho_{\alpha,\beta}\rangle\}\). Let \(A^{\prime}\) be a \(\mathbb{P}_{\lambda\alpha+\beta}\)-name forced to satisfy \(A^{\prime}=A_{\rho_{\alpha,\beta}}\cup\{z\}\) if \(z\) splits \(B\) and \(A^{\prime}=A_{\rho_{\alpha,\beta}}\) otherwise. Let \(\mathbb{Q}_{0}\) be a name for the Booth forcing for \(A^{\prime}\cup\{\omega\setminus n:n<\omega\}\); let \(x_{\alpha,\beta}\) be a name for a generic pseudointersection of \(A^{\prime}\cup\{\omega\setminus n:n<\omega\}\). (The purpose of \(\{\omega\setminus n:n<\omega\}\) is to ensure that \(x_{\alpha,\beta}\) does not almost contain any element of \([\omega]^{\omega}\cap V^{\mathbb{P}_{\lambda\alpha+\beta}}\).)
+
+Let \(F_{\lambda\alpha+\beta}\) to be a \(\mathbb{P}_{\lambda\alpha+\beta}\)-name for a surjection from \(\lambda\) to the elements of \([[\omega]^{\omega}]^{<\kappa}\) that have the SFIP. We may assume that corresponding \(F_{\gamma}\) have already been constructed for all \(\gamma<\lambda\alpha+\beta\). Let \(\mathbb{Q}_{1}\) be a name for the Booth forcing for \(F_{\lambda\alpha+\zeta}(\eta)\).
+
+Further suppose we have constructed sequences \(\langle w_{\gamma}\rangle_{\gamma<\alpha}\) and \(\langle U_{\gamma}\rangle_{\gamma<\alpha}\) of \(\mathbb{P}_{\lambda\alpha}\)-names such that \(\Vdash_{\lambda\gamma}U_{\delta}\cup\{w_{\delta}\}\subseteq U_{\gamma}\in\omega^{*}\) for all \(\delta<\gamma<\alpha\), and such that \(w_{\gamma}\) is forced to be a pseudointersection \(U_{\gamma}\) for all \(\gamma<\alpha\). If \(\beta\not=0\), then let \(\mathbb{Q}_{2}\) be a name for the trivial forcing. If \(\beta=0\), then choose \(U_{\alpha}\) such that \(\Vdash_{\lambda\alpha}\bigcup_{\gamma<\alpha}U_{\gamma}\cup\{w_{\gamma}\}\subseteq U_{\alpha}\in\omega^{*}\), let \(\mathbb{Q}_{2}\) be a name for the Booth forcing for \(U_{\alpha}\), and let \(w_{\alpha}\) be a name for a generic pseudointersection of \(U_{\alpha}\).
+
+Further suppose we have constructed a sequence \(\langle a_{\gamma}\rangle_{\gamma<\alpha}\) of \(\mathbb{P}_{\lambda\alpha}\)-names whose range is forced to be an almost disjoint subfamily of \([\omega]^{\omega}\). If \(\beta\not=0\), then let \(\mathbb{Q}_{3}\) be a name for the trivial forcing. If \(\beta=0\), then let \(\mathbb{Q}_{3}\) be a name for the Booth forcing for \(\{\omega\setminus a_{\gamma}:\gamma<\alpha\}\), and let \(a_{\alpha}\) be a name for a generic pseudointersection of \(\{\omega\setminus a_{\gamma}:\gamma<\alpha\}\).
+
+Further suppose we have constructed a sequence \(\langle i_{\gamma}\rangle_{\gamma<\alpha}\) of \(\mathbb{P}_{\lambda\alpha}\)-names whose range is forced to be an independent subfamily of \([\omega]^{\omega}\). If \(\beta\not=0\), then let \(\mathbb{Q}_{4}\) be a name for the trivial forcing. If \(\beta=0\), then set \(I=\{i_{\gamma}:\gamma<\alpha\}\) and let \(J\) and \(x\) be as in Lemma 5.8; let \(\mathbb{Q}_{4}\) be a name for the Booth forcing for \(J\); let \(i_{\alpha}\) be a name for \(x\).
+
+Set \(\mathbb{P}_{\lambda\alpha+\beta+1}=\mathbb{P}_{\lambda\alpha+\beta}*\prod_{n<5}\mathbb{Q}_{n}\). We may assume \(\lvert\prod_{n<5}\mathbb{Q}_{n}\rvert\leq\lambda\); hence, \(\mathbb{P}_{\lambda\alpha+\beta+1}\) has property (K) and size at most \(\lambda\). Also, \(B\cup\{x_{\alpha,\beta}\}\) is forced to have the SFIP by \(\mathbb{Q}_{0}\)-genericity because for every \(b\in B\) we have that \(\{b\}\cup A^{\prime}\) is forced to have the SFIP because \(\{b\}\cup A^{\prime}\subseteq B\cup\{z\}\) if \(z\) splits \(B\) and \(\{b\}\cup A^{\prime}\subseteq B\) otherwise. Let us also show that (5.1) holds if we replace \(\beta\) with \(\beta+1\). We may assume \(\alpha>0\). Let \(\sigma\in[B]^{<\omega}\). Then there exists \(\delta<\lambda\) such that \(\Vdash_{\lambda\alpha+\beta}z\cap\bigcap(\sigma\cup\tau)\not\subseteq^{*}x_{0,\delta}\) for all \(\tau\in[A_{\rho_{\alpha,\beta}}]^{<\omega}\); hence, \(\bigl{\{}\bigl{(}\bigcap\sigma\bigr{)}\setminus x_{0,\delta}\bigr{\}}\cup A^{\prime}\) is forced to have the SFIP; hence, \(\Vdash_{\lambda\alpha+\beta+1}x_{\alpha,\beta}\cap\bigcap\sigma\not\subseteq^{*}x_{0,\delta}\) by \(\mathbb{Q}_{0}\)-genericity. Thus, (5.1) holds as desired.
+
+To complete our inductive construction of \(\langle\mathbb{P}_{\gamma}\rangle_{\gamma\leq\lambda\kappa}\), it suffices to show that \(\{\langle\langle\gamma,\rho_{\gamma,\delta}\rangle,x_{\gamma,\delta}\rangle:\langle\gamma,\delta\rangle\in E_{\alpha,\beta+1}\}\) is forced to be an order embedding of \(D_{\alpha,\beta+1}\) into \(\langle[\omega]^{\omega},\supseteq^{*}\rangle\). Suppose \(\langle\gamma,\delta\rangle\in E_{\alpha,\beta}\). Then \(\langle\alpha,\rho_{\alpha,\beta}\rangle\not\leq\langle\gamma,\rho_{\gamma,\delta}\rangle\) and \(\Vdash_{\lambda\alpha+\beta+1}x_{\alpha,\beta}\not\supseteq^{*}x_{\gamma,\delta}\) by \(\mathbb{Q}_{0}\)-genericity. If \(\langle\gamma,\rho_{\gamma,\delta}\rangle<\langle\alpha,\rho_{\alpha,\beta}\rangle\), then \(x_{\gamma,\delta}\in A^{\prime}\); whence, \(\Vdash_{\lambda\alpha+\beta+1}x_{\gamma,\delta}\supsetneq^{*}x_{\alpha,\beta}\). Suppose \(\langle\gamma,\rho_{\gamma,\delta}\rangle\not<\langle\alpha,\rho_{\alpha,\beta}\rangle\). Then \(\rho_{\alpha,\beta}<\rho_{\gamma,\delta}\); hence, \(\rho_{\gamma,\delta}\geq\rho_{\alpha,\beta}+1=\rho_{0,\rho_{\alpha,\beta}}\); hence, \(x_{\gamma,\delta}\subseteq^{*}x_{0,\rho_{\alpha,\beta}}\). By construction, \(A^{\prime}\cup\{\omega\setminus x_{0,\rho_{\alpha,\beta}}\}\) is forced to have the SFIP; hence, \(\Vdash_{\lambda\alpha+\beta+1}x_{\gamma,\delta}\subseteq^{*}x_{0,\rho_{\alpha,\beta}}\not\supseteq^{*}x_{\alpha,\beta}\) by \(\mathbb{Q}_{0}\)-genericity. Thus, \(\{\langle\langle\gamma,\rho_{\gamma,\delta}\rangle,x_{\gamma,\delta}\rangle:\langle\gamma,\delta\rangle\in E_{\alpha,\beta+1}\}\) is forced to be an embedding as desired.
+
+Let us show that \(V^{\mathbb{P}_{\lambda\kappa}}\) satisfies \(\lambda\leq\chi Nt(\omega^{*})\). Let \(G\) be a generic filter of \(\mathbb{P}_{\lambda\kappa}\) and set \(\mathcal{B}=\{(x_{\alpha,\beta})_{G}^{*}:\langle\alpha,\beta\rangle\in\kappa\times\lambda\}\). Then \(\mathcal{B}\) is a local base at some \(p\in(\omega^{*})^{V[G]}\) because every element of \(([\omega]^{\omega})^{V[G]}\) is handled by an appropriate \(\mathbb{Q}_{0}\). By Lemma 2.15, \(\mathcal{B}\) contains a \(\chi Nt(p,\omega^{*})^{\mathrm{op}}\)-like local base \(\{(x_{\alpha,\beta})_{G}^{*}:\langle\alpha,\beta\rangle\in I\}\) at \(p\) for some \(I\subseteq\kappa\times\lambda\). Set \(J=\{\langle\alpha,\rho_{\alpha,\beta}\rangle:\langle\alpha,\beta\rangle\in I\}\). Then \(J\) is cofinal in \(\kappa\times\lambda\); hence, by Lemma 5.7, \(J\) is not \(\nu\)-like for any \(\nu<\lambda\). Hence, \(\chi Nt(\omega^{*})^{V[G]}\geq\lambda\).
+
+Finally, let us show that \(V^{\mathbb{P}_{\lambda\kappa}}\) satisfies \(\max\{\mathfrak{a},\mathfrak{i},\mathfrak{u}\}\leq\kappa\leq\mathfrak{p}\). Working in \(V[G]\), notice that \(\mathfrak{u}\leq\kappa\) because \(\bigcup_{\alpha<\kappa}(U_{\alpha})_{G}\in\omega^{*}\) and \(\{(w_{\alpha})_{G}^{*}:\alpha<\kappa\}\) is a local base at \(\bigcup_{\alpha<\kappa}(U_{\alpha})_{G}\). Moreover, \(\{(a_{\alpha})_{G}:\alpha<\kappa\}\) and \(\{(i_{\alpha})_{G}:\alpha<\kappa\}\) witness that \(\mathfrak{a}\leq\kappa\) and \(\mathfrak{i}\leq\kappa\). For \(\mathfrak{p}\geq\kappa\), note that very element of \([[\omega]^{\omega}]^{<\kappa}\) with the SFIP is \(\left(F_{\lambda\alpha+\zeta}(\eta)\right)_{G}\) for some \(\alpha<\kappa\) and \(\zeta,\eta<\lambda\). By \(\mathbb{Q}_{1}\)-genericity, a pseudointersection of \(\left(F_{\lambda\alpha+\zeta}(\eta)\right)_{G}\) is added at stage \(\lambda\alpha+\varphi(\zeta,\eta)\). ∎
+
+**Theorem 5.11****.**: \(\pi\chi Nt(\omega^{*})=\omega\)_._
+
+Proof.: Fix \(p\in\omega^{*}\). By a result of Balcar and Vojtáš [3], there exists \(\langle y_{x}\rangle_{x\in p}\) such that \(y_{x}\in[x]^{\omega}\) for all \(x\in p\) and \(\{y_{x}\}_{x\in p}\) is an almost disjoint family. Clearly, \(\{y_{x}^{*}\}_{x\in p}\) is a pairwise disjoint—and therefore \(\omega^{\mathrm{op}}\)-like—local \(\pi\)-base at \(p\). ∎
+
+## 6. Powers of \(\omega^{*}\)
+
+**Definition 6.1****.**: A _box_ is a subset \(E\) of a product space \(\prod_{i\in I}X_{i}\) such that there exist \(\sigma\in[I]^{<\omega}\) and \(\langle E_{i}\rangle_{i\in\sigma}\) such that \(E=\bigcap_{i\in\sigma}\pi_{i}^{-1}E_{i}\). Let \(Nt_{\mathrm{box}}(\prod_{i\in I}X_{i})\) denote the least infinite \(\kappa\) such that \(\prod_{i\in I}X_{i}\) has a \(\kappa^{\mathrm{op}}\)-like base of open boxes.
+
+**Lemma 6.2******(Peregudov [16])**.**: _In any product space \(X=\prod_{i\in I}X_{i}\), we have \(Nt(X)\leq Nt_{\mathrm{box}}(X)\leq\sup_{i\in I}Nt(X_{i})\)._
+
+**Lemma 6.3******(Malykhin [15])**.**: _Let \(X=\prod_{i\in I}X_{i}\) where each \(X_{i}\) is a nonsingleton \(T_{1}\) space. If \(w(X)\leq\lvert I\rvert\), then \(Nt(X)=Nt_{\mathrm{box}}(X)=\omega\)._
+
+_Remark__._: In Lemma 6.3, the hypothesis that the factor spaces be nonsingleton and \(T_{1}\) can be weakened to merely require that each factor space is the union of two nontrivial open sets. Also, the conclusion of Lemma 6.3 may be amended with the statement that \(X\) has a \(\langle\lvert I\rvert,\omega\rangle\)-splitter: use \(\langle\{\pi^{-1}_{i}U_{i},\pi^{-1}_{i}V_{i}\}\rangle_{i\in I}\) where each \(\{U_{i},V_{i}\}\) is a nontrivial open cover of \(X_{i}\).
+
+**Theorem 6.4****.**: _The sequence \(\langle Nt((\omega^{*})^{\omega+\alpha})\rangle_{\alpha\in{\mathrm{O}n}}\) is nonincreasing. Moreover, \(Nt((\omega^{*})^{\mathfrak{c}})=\omega\)._
+
+Proof.: Note that if \(\omega\leq\alpha\leq\beta\), then \((\omega^{*})^{\beta}\cong((\omega^{*})^{\alpha})^{\beta}\). Then apply Lemmas 6.2 and 6.3. ∎
+
+**Lemma 6.5****.**: _Let \(0<n<\omega\) and \(X\) be a space. Then \(Nt_{\mathrm{box}}(X^{n})=Nt(X)\)._
+
+Proof.: Set \(\kappa=Nt_{\mathrm{box}}(X^{n})\). By Lemma 6.2, \(\kappa\leq Nt(X)\). Let us show that \(Nt(X)\leq\kappa\). Let \(\mathcal{A}\) be a \(\kappa^{\mathrm{op}}\)-like base of \(X^{n}\) consisting only of boxes. Let \(\mathcal{B}\) denote the set of all nonempty open \(V\subseteq X\) for which there exists \(\prod_{i<n}U_{i}\in\mathcal{A}\) such that \(V=\bigcap_{i<n}U_{i}\). Then \(\mathcal{B}\) is a base of \(X\) because if \(p\in U\) and \(U\) is an open subset of \(X\), then there exists \(\prod_{i<n}U_{i}\in\mathcal{A}\) such that \(\langle p\rangle_{i<n}\in\prod_{i<n}U_{i}\subseteq U^{n}\); whence, \(p\in\bigcap_{i<n}U_{i}\subseteq U\) and \(\bigcap_{i<n}U_{i}\in\mathcal{B}\).
+
+It suffices to show that \(\mathcal{B}\) is \(\kappa^{\mathrm{op}}\)-like. Suppose not. Then there exist \(\prod_{i<n}U_{i}\in\mathcal{A}\) and \(\langle\prod_{i<n}V_{\alpha,i}\rangle_{\alpha<\kappa}\in\mathcal{A}^{\kappa}\) such that
+
+\[\emptyset\not=\bigcap_{i<n}U_{i}\subseteq\bigcap_{i<n}V_{\alpha,i}\not=\bigcap_{i<n}V_{\beta,i}\]
+
+for all \(\alpha<\beta<\kappa\). Clearly, \(\prod_{i<n}V_{\alpha,i}\not=\prod_{i<n}V_{\beta,i}\) for all \(\alpha<\beta<\kappa\). Choose \(U\in\mathcal{A}\) such that \(U\subseteq(\bigcap_{i<n}U_{i})^{n}\). Then \(U\subseteq\prod_{i<n}V_{\alpha,i}\) for all \(\alpha<\kappa\), in contradiction with how we chose \(\mathcal{A}\). ∎
+
+**Lemma 6.6****.**: _If \(0<n<\omega\) and \(X\) is a compact space such that \(\chi(p,X)=w(X)\) for all \(p\in X\), then \(Nt(X)=Nt(X^{n})\)._
+
+Proof.: By Lemma 6.5, it suffices to show that \(Nt_{\mathrm{box}}(X^{n})\leq Nt(X^{n})\). By Lemma 2.7, either \(X^{n}\) has a \(\langle w(X^{n}),Nt(X^{n})\rangle\)-splitter, or \(Nt(X^{n})=w(X^{n})^{+}\). Hence, by Lemma 2.6, \(Nt_{\mathrm{box}}(X^{n})\leq Nt(X^{n})\). ∎
+
+**Theorem 6.7****.**: _If \(0<n<\omega\), then \(Nt(\omega^{*})\geq Nt((\omega^{*})^{n})\geq\min\{Nt(\omega^{*}),\mathfrak{c}\}\). Moreover, \(\max\{\mathfrak{u},\operatorname{cf}\mathfrak{c}\}=\mathfrak{c}\) implies \(Nt(\omega^{*})=Nt((\omega^{*})^{n})\)._
+
+Proof.: Lemma 6.2 implies \(Nt(\omega^{*})\geq Nt((\omega^{*})^{n})\). To prove the rest of the theorem, first consider the case \(\mathfrak{r}<\mathfrak{c}\). As in the proof of Theorem 2.3, construct a point \(p\in\omega^{*}\) such that \(\pi\chi(p,\omega^{*})=\mathfrak{r}\) and \(\chi(p,\omega^{*})=\mathfrak{c}\). Then \(\pi\chi(\langle p\rangle_{i<n},(\omega^{*})^{n})=\mathfrak{r}\) and \(\chi(\langle p\rangle_{i<n},(\omega^{*})^{n})=\mathfrak{c}\); hence, \(Nt((\omega^{*})^{n})\geq\mathfrak{c}\) by Theorem 2.1. Moreover, if \(\operatorname{cf}\mathfrak{c}=\mathfrak{c}\), then \(Nt((\omega^{*})^{n})=Nt(\omega^{*})=\mathfrak{c}^{+}\). If \(\mathfrak{u}=\mathfrak{c}\), then \(Nt(\omega^{*})=Nt((\omega^{*})^{n})\) by Lemma 6.6. Finally, in the case \(\mathfrak{r}=\mathfrak{c}\), we have \(\mathfrak{u}=\mathfrak{c}\), which again implies \(Nt(\omega^{*})=Nt((\omega^{*})^{n})\). ∎
+
+**Corollary 6.8****.**: _Suppose \(\max\{\mathfrak{u},\operatorname{cf}\mathfrak{c}\}=\mathfrak{c}\). Then \(\langle Nt((\omega^{*})^{1+\alpha})\rangle_{\alpha\in{\mathrm{O}n}}\) is nonincreasing._
+
+Proof.: By Theorem 6.7 and Lemma 6.2, \(Nt((\omega^{*})^{n})=Nt(\omega^{*})\geq Nt((\omega^{*})^{\alpha})\) whenever \(0<n<\omega\leq\alpha\). The rest follows from Theorem 6.4. ∎
+
+**Theorem 6.9****.**: _Suppose \(\mathfrak{u}=\mathfrak{c}\). Then \(Nt((\omega^{*})^{1+\alpha})=Nt(\omega^{*})\) for all \(\alpha<\operatorname{cf}\mathfrak{c}\)._
+
+Proof.: Let \(\lambda\) be an arbitrary infinite cardinal less than \(Nt(\omega^{*})\). By Lemma 2.7, it suffices to show that \((\omega^{*})^{1+\alpha}\) does not have a \(\langle\mathfrak{c},\lambda\rangle\)-splitter. Seeking a contradiction, suppose \(\langle\mathcal{F}_{\beta}\rangle_{\beta<\mathfrak{c}}\) is such a \(\langle\mathfrak{c},\lambda\rangle\)-splitter. We may assume \(\bigcup_{\beta<\mathfrak{c}}\mathcal{F}_{\beta}\) consists only of open boxes because we can replace each \(\mathcal{F}_{\beta}\) with a suitable refinement. Since \(\alpha<\operatorname{cf}\mathfrak{c}\), there exist \(\sigma\in[1+\alpha]^{<\omega}\) and \(I\in[\mathfrak{c}]^{\mathfrak{c}}\) such that, for every \(U\in\bigcup_{\beta\in I}\mathcal{F}_{\beta}\), there exists \(\varphi(U)\subseteq(\omega^{*})^{\sigma}\) such that \(U=\pi^{-1}_{\sigma}\varphi(U)\). Let \(j\) be a bijection from \(\mathfrak{c}\) to \(I\). Then \(\langle\varphi``\mathcal{F}_{j(\beta)}\rangle_{\beta<\mathfrak{c}}\) is a \(\langle\mathfrak{c},\lambda\rangle\)-splitter of \((\omega^{*})^{\sigma}\). Hence, \(Nt((\omega^{*})^{\sigma})\leq\lambda<Nt(\omega^{*})\) by Lemma 2.6. But \(Nt((\omega^{*})^{\sigma})<Nt(\omega^{*})\) contradicts Theorem 6.7. ∎
+
+**Lemma 6.10****.**: _Suppose a space \(X\) has a \(\langle\operatorname{cf}w(X),\operatorname{cf}w(X)\rangle\)-splitter. Then \(Nt(X)\leq w(X)\)._
+
+Proof.: Set \(\kappa=\operatorname{cf}w(X)\) and \(\lambda=w(X)\). Let \(\langle\mathcal{F}_{\alpha}\rangle_{\alpha<\kappa}\) be a \(\langle\kappa,\kappa\rangle\)-splitter of \(X\). Let \(h:\lambda\rightarrow\kappa\) satisfy \(\lvert h^{-1}\{\alpha\}\rvert<\lambda\) for all \(\alpha<\kappa\). Then \(\langle\mathcal{F}_{h(\alpha)}\rangle_{\alpha<\lambda}\) is a \(\langle\lambda,\lambda\rangle\)-splitter because if \(I\in[\lambda]^{\lambda}\), then \(h``I\in[\kappa]^{\kappa}\). By Lemma 2.6, \(Nt(X)\leq\lambda\). ∎
+
+_Remark__._: The proof of the above lemma shows that for any infinite cardinal \(\kappa\), a space with a \(\langle\operatorname{cf}\kappa,\operatorname{cf}\kappa\rangle\)-splitter also has a \(\langle\kappa,\kappa\rangle\)-splitter.
+
+**Theorem 6.11****.**: \(Nt((\omega^{*})^{\operatorname{cf}\mathfrak{c}})\leq\mathfrak{c}\)_._
+
+Proof.: The sequence \(\langle\{\pi^{-1}_{\alpha}(\{2n:n<\omega\}^{*}),\pi^{-1}_{\alpha}(\{2n+1:n<\omega\}^{*})\}\rangle_{\alpha<\operatorname{cf}\mathfrak{c}}\) is a \(\langle\operatorname{cf}\mathfrak{c},\omega\rangle\)-splitter of \((\omega^{*})^{\operatorname{cf}\mathfrak{c}}\). Apply Lemma 6.10. ∎
+
+**Theorem 6.12****.**: _For all cardinals \(\kappa\) satisfying \(\kappa>\operatorname{cf}\kappa>\omega_{1}\), it is consistent that \(\mathfrak{c}=\kappa\) and \(\mathfrak{r}<\operatorname{cf}\mathfrak{c}\). The last inequality implies \(Nt((\omega^{*})^{1+\alpha})=\mathfrak{c}^{+}\) for all \(\alpha<\operatorname{cf}\mathfrak{c}\) and \(Nt((\omega^{*})^{\beta})=\mathfrak{c}=\kappa\) for all \(\beta\in\mathfrak{c}\setminus\operatorname{cf}\mathfrak{c}\)._
+
+Proof.: Starting with \(\mathfrak{c}=\kappa\) in the ground model, the proof of Theorem 2.3 shows how to force \(\mathfrak{r}=\mathfrak{u}=\omega_{1}\) while preserving \(\mathfrak{c}\). Now suppose \(\mathfrak{r}<\operatorname{cf}\mathfrak{c}\). Fix \(\alpha<\operatorname{cf}\mathfrak{c}\) and \(\beta\in\mathfrak{c}\setminus\operatorname{cf}\mathfrak{c}\). By Theorems 6.11 and 6.4, \(Nt((\omega^{*})^{\beta})\leq\mathfrak{c}\). To see that \(Nt((\omega^{*})^{\beta})\geq\mathfrak{c}\), proceed as in the proof of Theorem 6.7, constructing a point with character \(\mathfrak{c}\) and \(\pi\)-character \(\lvert\beta\rvert\). Similarly prove \(Nt((\omega^{*})^{1+\alpha})=\mathfrak{c}^{+}\) by constructing a point with character \(\mathfrak{c}\) and \(\pi\)-character \(\lvert\mathfrak{r}+\alpha\rvert\). ∎
+
+**Lemma 6.13****.**: _Suppose \(\kappa\), \(\lambda\), and \(\mu\) are cardinals and \(p\) is a point in a product space \(X=\prod_{\alpha<\kappa}X_{\alpha}\) satisfying the following for all \(\alpha<\kappa\)._
+
+1. (1)\(0<\kappa<w(X)\) _and_ \(\omega\leq\lambda\leq w(X)\)_._
+2. (2)\(\kappa<\operatorname{cf}w(X)\) _or_ \(\lambda<w(X)\)_._
+3. (3)\(\mu<\lambda\) _or_ \(\mu=\operatorname{cf}\lambda\)_._
+4. (4)\(\chi(p(\alpha),X_{\alpha})<\lambda\) _or the intersection of any_ \(\mu\)_-many neighborhoods of_ \(p(\alpha)\) _has nonempty interior._
+
+_Then \(\chi(p,X)<w(X)\) or \(Nt(X)>\mu\)._
+
+Proof.: Let \(\mathcal{A}\) be a base of \(X\). Set \(\mathcal{B}=\{U\in\mathcal{A}:p\in U\}\). For each \(\alpha<\kappa\), let \(\mathcal{C}_{\alpha}\) be a local base at \(p(\alpha)\) of size \(\chi(p(\alpha),X_{\alpha})\). Set \(F=\bigcup_{r\in[\kappa]^{<\omega}}\prod_{\alpha\in r}\mathcal{C}_{\alpha}\). For each \(\sigma\in F\), set \(U_{\sigma}=\bigcap_{\alpha\in\operatorname{dom}\sigma}\pi_{\alpha}^{-1}\sigma(\alpha)\). For each \(V\in\mathcal{B}\), choose \(\sigma(V)\in F\) such that \(p\in U_{\sigma(V)}\subseteq V\). We may assume \(\chi(p,X)=w(X)\); hence, by (1) and (2), there exist \(r\in[\kappa]^{<\omega}\) and \(\mathcal{D}\in[\mathcal{B}]^{\lambda}\) such that \(\operatorname{dom}\sigma(V)=r\) for all \(V\in\mathcal{D}\). Set \(s=\{\alpha\in r:\chi(p(\alpha),X_{\alpha})<\lambda\}\) and \(t=r\setminus s\). By (3), there exist \(\tau\in\prod_{\alpha\in s}\mathcal{C}_{\alpha}\) and \(\mathcal{E}\in[\mathcal{D}]^{\mu}\) such that \(\sigma(V)\upharpoonright s=\tau\) for all \(V\in\mathcal{E}\). By (4), \(\bigcap_{V\in\mathcal{E}}\sigma(V)(\alpha)\) has nonempty interior for all \(\alpha\in t\). Hence, \(\bigcap\mathcal{E}\) has nonempty interior because it contains \(U_{\tau}\cap\bigcap_{\alpha\in t}\pi_{\alpha}^{-1}\bigcap_{V\in\mathcal{E}}\sigma(V)(\alpha)\). Thus, \(Nt(X)>\mu\). ∎
+
+**Theorem 6.14****.**: _Suppose \(0<\alpha<\mathfrak{c}\) and \(\langle X_{\beta}\rangle_{\beta<\alpha}\) is a sequence of spaces each with weight at most \(\mathfrak{c}\). Then \(Nt(\prod_{\beta<\alpha}(X_{\beta}\oplus\omega^{*}))>\nu\) for all regular \(\nu<\mathfrak{p}\)._
+
+Proof.: Let \(\nu\) be an arbitrary infinite regular cardinal less than \(\mathfrak{p}\). Set \(\kappa=\lvert\alpha\rvert\) and \(\lambda=\mu=\nu\). Choose \(q\in\omega^{*}\) such that \(\chi(q,\omega^{*})=\mathfrak{c}\); set \(p=\langle q\rangle_{\beta<\alpha}\). Applying Lemma 6.13, we have \(Nt(\prod_{\beta<\alpha}(X_{\beta}\oplus\omega^{*}))>\nu\). ∎
+
+**Corollary 6.15****.**: _Suppose \(\mathfrak{p}=\mathfrak{c}\). Then \(Nt((\omega^{*})^{1+\alpha})=\mathfrak{c}\) for all \(\alpha<\mathfrak{c}\)._
+
+Proof.: By Theorem 2.11, \(Nt(\omega^{*})\leq\mathfrak{c}\). Hence, by Corollary 6.8, \(Nt((\omega^{*})^{1+\alpha})\leq\mathfrak{c}\) for all \(\alpha\in{\mathrm{O}n}\). By Theorem 6.14, \(Nt((\omega^{*})^{1+\alpha})\geq\mathfrak{c}\) for all \(\alpha<\mathfrak{c}\). ∎
+
+**Corollary 6.16****.**: _Suppose \(\alpha<\mathfrak{c}\) and \(\langle X_{\beta}\rangle_{\beta<\alpha}\) is a sequence of spaces each with weight at most \(\mathfrak{c}\). Then \(\prod_{\beta<\alpha}(X_{\beta}\oplus\omega^{*})\) is not homeomorphic to a product of \(\mathfrak{c}\)-many nonsingleton spaces._
+
+Proof.: Combine Theorem 6.14 and Lemma 6.3. ∎
+
+## 7. Questions
+
+_Question 1__._: Is it consistent that \(Nt(\omega^{*})=\mathfrak{c}^{+}\) and \(\mathfrak{r}\geq\operatorname{cf}\mathfrak{c}\)?
+
+_Question 2__._: Is \(Nt(\omega^{*})<\mathfrak{s}\mathfrak{s}_{\omega}\) consistent? This inequality implies \(\mathfrak{u}<\mathfrak{c}\). Hence, by Theorem 2.11, the inequality further implies
+
+\[\operatorname{cf}\mathfrak{c}\leq\mathfrak{r}\leq\mathfrak{u}<\mathfrak{c}=Nt(\omega^{*})<\mathfrak{s}\mathfrak{s}_{\omega}=\mathfrak{c}^{+}.\]
+
+More generally, does any space \(X\) have a base that does not contain an \(Nt(X)^{\mathrm{op}}\)-like base?
+
+_Question 3__._: Is \(\mathfrak{s}\mathfrak{s}_{\omega}<\mathfrak{s}\mathfrak{s}_{2}\) consistent?
+
+_Question 4__._: Letting \(\mathfrak{g}\) denote the groupwise density number (see 6.26 of [7]), is \(Nt(\omega^{*})<\mathfrak{g}\) consistent? \(\chi Nt(\omega^{*})<\mathfrak{g}\)? In particular, what are \(Nt(\omega^{*})\) and \(\chi Nt(\omega^{*})\) in the Laver model (see 11.7 of [7])?
+
+_Question 5__._: Is \(\operatorname{cf}Nt(\omega^{*})<Nt(\omega^{*})<\mathfrak{c}\) consistent? \(\operatorname{cf}Nt(\omega^{*})=\omega\)?
+
+_Question 6__._: Is \(\operatorname{cf}\mathfrak{c}<Nt(\omega^{*})<\mathfrak{c}\) consistent?
+
+_Question 7__._: What is \(\chi Nt(\omega^{*})\) in the forcing extension of the proof of Theorem 4.13? More generally, is it consistent that \(\chi Nt(\omega^{*})<Nt(\omega^{*})\leq\mathfrak{c}\)?
+
+_Question 8__._: Is \(\chi Nt(\omega^{*})=\omega\) consistent? An affirmative answer would be a strengthening of Shelah’s result [19] that \(\omega^{*}\) consistently has no P-points. If the answer is negative, then which, if any, of \(\mathfrak{p}\), \(\mathfrak{h}\), \(\mathfrak{s}\), and \(\mathfrak{g}\) are lower bounds of \(\chi Nt(\omega^{*})\) in ZFC?
+
+_Question 9__._: Is \(\operatorname{cf}\mathfrak{c}<\chi Nt(\omega^{*})\) consistent? \(\operatorname{cf}\mathfrak{c}<\chi Nt(\omega^{*})<\mathfrak{c}\)?
+
+_Question 10__._: Does any Hausdorff space have uncountable local Noetherian \(\pi\)-type? (It is easy to construct such \(T_{1}\) spaces: give \(\omega_{1}+1\) the topology \(\{(\omega_{1}+1)\setminus(\alpha\cup\sigma):\alpha<\omega_{1}\text{ and }\sigma\in[\omega_{1}+1]^{<\omega}\}\cup\{\emptyset\}\).)
+
+_Question 11__._: Is it consistent that \(Nt((\omega^{*})^{1+\alpha})<\min\{Nt(\omega^{*}),\mathfrak{c}\}\) for some \(\alpha<\mathfrak{c}\)? Is it consistent that \(Nt((\omega^{*})^{1+\alpha})<Nt(\omega^{*})\) for some \(\alpha<\operatorname{cf}\mathfrak{c}\)?
+
+## References
+
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+

arxiv_ground_truth/0706.0028.md

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+# Py-Calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus
+
+Maor Rosenberg
+
+###### Abstract
+
+We show that Py-Calabi quasi-morphism on the group of Hamiltonian diffeomorphisms of surfaces of higher genus gives rise to a quasi-state.
+
+## 1 Introduction
+
+In [11] M. Entov and L. Polterovich establish an unexpected link between a group-theoretic notion of quasi-morphism, which has been found useful in symplectic geometry, and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures. In this paper we show this connection for a recently discovered, due to P. Py [18], Calabi quasi-morphism on orientable surfaces of higher genus. The proof relies on hyperbolic geometry tools, surprisingly combined with combinatorial tools such as Hall’s marriage theorem.
+
+### The Group \(\mbox{Ham}(M,\omega)\)
+
+**Definition 1.1****.**: Let \(M\) be a symplectic manifold equipped with a symplectic form \(\omega\). Let \(F_{t}(x):=F(x,t)\), \(F:M\times{\mathbb{R}}\rightarrow{\mathbb{R}}\) be a smooth function, called **Hamiltonian** function. The pointwise linear equation \(i_{v}w=-dF_{t}\) defines a vector field \(v\) on \(M\) denoted by \(sgradF_{t}\). The flow generated by the Hamiltonian vector field \(sgradF_{t}\) is denoted by \(f_{t}\). By assuming that the union over \(t\) of the supports of \(F_{t}\) is contained in a compact subset of \(M\), we can guarantee that the above equation has a well defined solution for all \(t\), and so \(f_{t}\) is well defined. The time-one map \(f_{1}\), denoted by \(\phi_{F}\), will be called **the Hamiltonian diffeomorphism generated by \(F\)**. The collection of Hamiltonian diffeomorphisms has a group structure and this group is denoted by \(\mbox{Ham}(M,\omega)\). For further details see [14, 17].
+
+### Algebraic Results on \(\mbox{Ham}(M,\omega)\)
+
+The following algebraic results on \(\mbox{Ham}(M,\omega)\) are due to Banyaga [5].
+
+**Theorem 1.2****.**: _Let \(M\) be a closed symplectic manifold, then \(\mbox{Ham}(M,\omega)\) is simple, i.e., it has no non-trivial normal subgroup._
+
+**Theorem 1.3****.**: _Let \(M\) be an open manifold with an exact symplectic structure, \(\omega=d\lambda\). Then \(\mbox{Ham}(M,\omega)\) admits the Calabi homomorphism:_
+
+\[Cal_{M}:\mbox{Ham}(M,\omega)\rightarrow{\mathbb{R}}\]
+
+_defined as_
+
+\[Cal_{M}(\phi_{F})=\int_{0}^{1}\int_{M}F(x,t){\omega}^{m}dt,\]
+
+_whose kernel is equal to the commutator subgroup of \(\mbox{Ham}(M,\omega)\). Furthermore, this kernel is a simple group._
+
+Note that \(Cal_{M}\) is defined by \(F\), but it can be shown that \(Cal_{M}\) depends only on \(\phi_{F}\) and not on the specific \(F\). Returning to the case where \(M\) is closed, one cannot hope to construct a non-trivial homomorphism to \({\mathbb{R}}\) since \(\mbox{Ham}(M,\omega)\) is simple. However, for certain manifolds, one can find a map which is ”locally” equal to the Calabi homomorphism and globally is a homomorphism up to a bounded error. This map is called a Calabi quasi-morphism. A precise definition is given in the following subsection.
+
+### Calabi Quasi-morphism
+
+**Definition 1.4****.**: Let \(G\) be a group, a function \(\mu:G\rightarrow{\mathbb{R}}\) is a called a **quasi-morphism** if there exists a constant \(C\), called the **defect** of \(\mu\), such that for each \(x,y\) in \(G\)
+
+\[|\mu(xy)-\mu(x)-\mu(y)|<C.\]
+
+If in addition \(\mu(x^{n})=n\mu(x)\) for each \(x\in G\) and \(n\in\mathbb{Z}\) then the quasi-morphism is called **homogeneous**. Given a quasi-morphism we can define a homogeneous quasi-morphism called its **homogenization**\({\mu}_{h}\) by
+
+\[{\mu}_{h}(x)=\lim_{n\to\infty}\frac{\mu(x^{n})}{n}.\]
+
+For further reading see e.g. [13].
+
+Let \(M\) be a closed manifold with a symplectic form \(\omega\). Let \(U\subset M\) be open and connected. Denote by \(\Gamma_{U}\) the subgroup of \(\mbox{Ham}(M,\omega)\) generated by Hamiltonians supported in \(U\). If \(\omega\) is exact on \(U\) then, by Theorem 1.3, \(\Gamma_{U}\) admits the Calabi homomorphism \(Cal_{U}\). A set \(U\subset M\) is called displaceable if there exists \(f\in~{}\mbox{Ham}(M,\omega)\) such that \(f(U)\cap\overline{U}=\emptyset\). The following question was posed by M. Entov and L. Polterovich in [10]. Can one construct a homogeneous quasi-morphism on \(\mbox{Ham}(M,\omega)\) such that its restriction to \(\Gamma_{U}\), for any open, connected, exact and displaceable \(U\subset M\), is equal to the Calabi homomorphism \(Cal_{U}\)?
+
+**Definition 1.5****.**: A homogeneous quasi-morphism with the above property is called a **Calabi quasi-morphism**.
+
+M. Entov and L. Polterovich [10, 6] have constructed Calabi quasi-morphisms for the case of the following symplectic manifolds: \(\mathbb{C}P^{n}\), a complex Grassmannian, \(\mathbb{C}P^{n_{1}}\times...\times\mathbb{C}P^{n_{k}}\) with a monotone product symplectic structure, the monotone symplectic blow-up of \(\mathbb{C}P^{2}\) at one point. Y. Ostrover extended it to some non-monotone manifolds [16].
+
+The following result is due to P. Py [18].
+
+**Theorem 1.6****.**: _Let \(M\) be an oriented closed surface of genus \(g\geq 2\), equipped with a symplectic form \(\omega\). Then there exists a homogeneous quasi-morphism_
+
+\[\mu:\mbox{Ham}(M,\omega)\rightarrow{\mathbb{R}}\]
+
+_such that the restriction to the subgroups \(\Gamma_{U}\) is equal to the Calabi homomorphism, where \(U\) is diffeomorphic to a disc or an annulus._
+
+In addition, P. Py has also constructed a Calabi quasi-morphism for the torus [19].
+
+**Definition 1.7****.**: A smooth function \(F:M\rightarrow{\mathbb{R}}\) is called a **Morse function** if all its critical points are non-degenerate. If in addition the critical values of \(F\) are distinct then \(F\) is called a **generic Morse function**.
+
+**Definition 1.8****.**: Let \(F:M\rightarrow{\mathbb{R}}\) be a generic Morse function. Let \({\mathcal{F}}\) be the space of smooth functions on \(M\) which commute with \(F\) in the Poisson sense, i.e.
+
+\[{\mathcal{F}}:=\{H:M\rightarrow{\mathbb{R}}|\{F,H\}=0\}.\]
+
+Note that \(F\) and \(H\) commute in the Poisson sense if and only if
+
+\[\omega(sgradF,sgradH)=0.\]
+
+Let \(\Gamma\) be the set of time one maps corresponding to the flows generated by the Hamiltonian functions in \({\mathcal{F}}\), i.e.
+
+\[\Gamma:=\{\phi_{H}|H\in{\mathcal{F}}\}.\]
+
+Clearly, \(\Gamma\) is an abelian subgroup of \(\mbox{Ham}(M,\omega)\). It is easy to show that if a homogeneous quasi-morphism is defined on an abelian group, then it is in fact a homomorphism. Hence, the restriction of \(\mu\), defined in Theorem 1.6, on the subgroup \(\Gamma\) is a homomorphism. In [18] P. Py has proved the following formula for \(\mu\) on \(\Gamma\), assuming that the total area of \(M\) is equal to \(2g-2\),
+
+\[\mu(\phi_{H})=\int_{M}H\omega-\sum_{x\in{\mathcal{V}}_{F}}H(x),\]
+
+where \(H\in{\mathcal{F}}\) and \({\mathcal{V}}_{F}\) is a certain subset of the critical points of \(F\). A precise formulation of this theorem will be given in Section 4.
+
+### Quasi-state
+
+The notion of a quasi-state originates in quantom mechanics [1, 2], and has been a subject of intensive study in recent years following the paper [3] by J. F. Aarnes. Here is the definition.
+
+**Definition 1.9****.**: Denote by \(C(M)\) the commutative (with respect to multiplication) Banach algebra of all continuous functions on \(M\) endowed with the uniform norm. For a function \(F\in C(M)\) denote by \({\mathcal{A}}_{F}\) the uniform closure of the set of functions of the form \(p\circ F\), where \(p\) is a real polynomial. A (not necessarily linear) functional \(\xi:C(M)\rightarrow{\mathbb{R}}\) is called a **quasi-state**, if it satisfies the following axioms:
+**Quasi-linearity.**\(\xi\) is linear on \({\mathcal{A}}_{F}\) for every \(F\in C(M)\).
+**Monotonicity.**\(\xi(F)\leq\xi(G)\) for \(F\leq G\).
+**Normalization.**\(\xi(1)=1\).
+
+It is easy to show that a quasi-state is Lipschitz continuous with respect to the \(C^{0}\)-norm.
+
+**Main Result.** In the following, \(M\) will be an oriented closed surface of genus \(g\geq 2\), equipped with a symplectic form \(\omega\), and \(\mu\) is Py’s quasi-morphism given in Theorem 1.6. In [11] M. Entov and L. Polterovich construct a quasi-state from a Calabi quasi-morphism, Our goal is to show that this procedure is applicable to Py’s Calabi quasi-morphism. In the following, we assume that the total area of \(M\), denoted by \(Vol(M)\), is equal to \(2g-2\). The quasi-state is obtained from \(\mu\) in the following way:
+
+**Definition 1.10****.**: For a smooth function \(F\) define
+
+\[\xi(F):=\frac{\int_{M}F\omega}{Vol(M)}-\frac{\mu(\phi_{F})}{Vol(M)}.\]
+
+The main result of the thesis is that the functional \(\xi\) related to Py’s quasi-morphism is a quasi-state.
+
+**Theorem 1.11****.**: _The functional \(\xi\) can be extended to \(C(M)\), and the extension is a quasi-state._
+
+Note that this result implies that \(\xi\) is Lipschitz continuous with respect to the \(C^{0}\)-norm.
+
+**Organization of the work.** In the following section we prove the main theorem assuming the monotonicity and continuity theorems, which are proved later on. In sections 3, 4, 5, 6 we make the preparations for the monotonicity theorem, which is proved in Section 7. In Section 3 we define the Reeb graph which is the base for the following constructions. In Section 4 we introduce the notion of essential critical points. In Section 5 we construct a pair of pants decomposition. In Section 6 we prove an intersection theorem of figure eights related to the pair of pants decomposition. In Section 8 we prove the continuity theorem by analyzing Py’s construction of the quasi-morphism, this section can be read independently.
+
+## 2 Main Steps
+
+The main ingredients of the proof are the following theorems.
+
+**Theorem 2.1****.**: _Let \(F,G:M\rightarrow{\mathbb{R}}\) be generic Morse functions, such that \(F\leq G\). Then \(\xi(F)\leq\xi(G)\)._
+
+**Theorem 2.2****.**: _The functional \(\xi:C^{\infty}(M)\rightarrow{\mathbb{R}}\) is continuous with respect to the \(C^{2}\)-topology._
+
+We will prove Theorem 1.11 assuming Theorem 2.1 and 2.2.
+
+Proof.: Normalization is due to the fact that \(\mu\) is a homogeneous quasi-morphism. Indeed, \(\phi_{1}\) corresponds to the identity element in the group \(\mbox{Ham}(M,\omega)\), so \(\mu(\phi_{1})=\mu(Id)=0\) by the homogeneity of \(\mu\), and it follows that \(\xi(1)=1\). Since \(\phi_{F+k}=\phi_{F}\) for any smooth function \(F\) and a real constant \(k\), we get from the definition of \(\xi\) that
+
+\[\xi(F+k)=\xi(F)+k.\] (2.1)
+
+Let \(\epsilon>0\) and \(F\), \(G\) be generic Morse functions, then
+
+\[|F-G|\leq\|F-G\|_{C^{0}},\]
+
+thus
+
+\[G-\|F-G\|_{C^{0}}\leq F\leq G+\|F-G\|_{C^{0}}.\]
+
+From the monotonicity of generic Morse functions (Theorem 2.1) and Equation 2.1 we get
+
+\[\xi(G)-\|F-G\|_{C^{0}}\leq\xi(F)\leq\xi(G)+\|F-G\|_{C^{0}}\]
+
+so
+
+\[|\xi(F)-\xi(G)|\leq\|F-G\|_{C^{0}}\]
+
+Thereby, \(\xi\) is Lipschitz continuous on generic Morse functions with respect to the \(C^{0}\)-norm. Generic Morse functions are \(C^{0}\)-dense in \(C(M)\), therefore there is a unique extension of \(\xi\) to a continuous map \(\widehat{\xi}:C(M)\rightarrow{\mathbb{R}}\). We will show that \(\widehat{\xi}|_{C^{\infty}(M)}\equiv\xi\). Indeed, for \(H\in C^{\infty}(M)\) we can find a sequence of Morse functions \(\{H_{n}\}\) that \(C^{2}\)-converges to \(H\). By Theorem 2.2, \(\lim_{n\rightarrow\infty}\xi(H_{n})=\xi(H)\). In particular \(\{H_{n}\}\)\(C^{0}\)-converges to \(H\), so \(\lim_{n\rightarrow\infty}\xi(H_{n})=\widehat{\xi}(H)\) by definition. Hence \(\widehat{\xi}(H)=\xi(H)\), as required.
+
+Monotonicity is easily extended to \(\widehat{\xi}\) in the following way: For \(F\), \(G\in~{}C(M)\) such that \(F\leq G\), choose generic Morse function sequences \(\{F_{n}\}\), \(\{G_{n}\}\) such that \(\|F-F_{n}\|_{C^{0}}<\frac{1}{n}\) , \(\|G-G_{n}\|_{C^{0}}<\frac{1}{n}\). Define the sequences \(\{F_{n}^{\prime}\}\), \(\{G_{n}^{\prime}\}\) as follows: \(F_{n}^{\prime}:=F_{n}-\frac{1}{n}\), \(G_{n}^{\prime}:=G_{n}+\frac{1}{n}\). Then for \(n\in{\mathbb{N}}\),
+
+\[F_{n}^{\prime}<F\leq G<G_{n}^{\prime}.\]
+
+By the monotonicity on Morse functions we obtain \(\xi(F_{n}^{\prime})\leq\xi(G_{n}^{\prime})\) and by taking limits we get \(\widehat{\xi}(F)\leq\widehat{\xi}(G)\).
+
+In order to show quasi-linearity we will first show a property called **strong quasi-additivity** which is defined as follows:
+\(\xi(F+G)=\xi(F)+\xi(G)\) for all smooth functions \(F\), \(G\) which commutes in the Poisson bracket, i.e. \(\{F,G\}=0\). The functional \(\widehat{\xi}\) satisfies this property since it coincides with \(\xi\) on smooth functions and the quasi-morphism \(\mu\) is linear on commuting elements. From the homogeneity of \(\mu\) we get that \(\xi\) is homogeneous and it is easily extended to \(\widehat{\xi}\). It is easy to see that strong quasi-additivity together with homogeneity yields quasi-linearity. ∎
+
+## 3 The Reeb Graph
+
+In this section we will define the Reeb graph [20] and prove a statement on its Euler characteristic. The Reeb graph is a simple yet very useful tool in this work, and we will use it in the following sections to define the set of essential critical points, and to construct the pair of pants decomposition.
+
+**Definition 3.1****.**: Let \(M\) be a closed oriented surface of genus \(g\). Let \(F:M\rightarrow{\mathbb{R}}\) be a generic Morse function. Let \(\{x_{1},x_{2},...,x_{n}\}\subset M\) be the set of critical points of \(F\), with critical values \(c_{i}=F(x_{i})\), for \(1\leq i\leq n\), such that \(c_{1}<c_{2}<...<c_{n}\). We will define the **Reeb graph of F**, \(\Gamma(V,E)\), in the following way:
+For each critical value \(c_{i}\), the connected component of \(F^{-1}(c_{i})\) that contains \(x_{i}\) can be:
+1) The critical point \(x_{i}\) in the case that its index is \(0\) or \(2\).
+2) An immersed closed curve with a unique transversal double point \(x_{i}\). This is the case when \(x_{i}\) is of index \(1\).
+We will assign a vertex \(v_{i}\) to the connected component of \(F^{-1}(c_{i})\) described above. Let \(C\) be the union of the above connected components, then \(M\backslash C\) doesn’t contain any critical points with respect to \(F\). Hence, by Morse theory [15], it is a disjoint finite union of open cylinders. The boundaries of each cylinder are contained in two connected components of \(C\). Define an edge associated with this cylinder between the two vertices that correspond to these components. By Morse theory, each cylinder can be parameterized by \(S^{1}\times(c_{k},c_{l})\) where \(c_{k}\) and \(c_{l}\) are the critical values of the critical points that bound the cylinder, and each \(S^{1}\times\{t\}\) is a connected component of the level curve \(F^{-1}(t)\). Parameterize the corresponding edge by the segment \((c_{k},c_{l})\). Hence, we can define in a natural way a projection map \(\pi_{\Gamma}:~{}M\rightarrow~{}\Gamma\), by sending each connected component that contains a critical point \(x_{i}\) to the vertex \(v_{i}\), and each connected component of the form \(S^{1}\times\{t\}\subset S^{1}\times(c_{k},c_{l})\) to the point \(t\) in the corresponding edge, parameterized by \((c_{k},c_{l})\).
+
+This is illustrated in Figure 1.
+
+Figure 1: \(\Gamma\) is the Reeb graph of \(F:M\rightarrow{\mathbb{R}}\).
+
+Note that if \(H:M\rightarrow{\mathbb{R}}\) is constant on each connected level curve of \(F\) then we can define \(H_{\Gamma}:\Gamma\rightarrow{\mathbb{R}}\) such that \(H=H_{\Gamma}\circ\pi_{\Gamma}\) by taking
+
+\[H_{\Gamma}(y)=\left\{\begin{array}[]{rl}H(F^{-1}(c_{i}))&y=v_{i}\\ H(F^{-1}(y))&y\mbox{ in edge $(c_{k},c_{l})$.}\end{array}\right.\]
+
+**Proposition 3.2****.**: _Let \(M\) be a closed oriented surface of genus \(g\). Let \(F:M\rightarrow{\mathbb{R}}\) be a generic Morse function. Let \(\Gamma\) be the Reeb graph of \(F\). Then the Euler characteristic \(\chi(\Gamma)\) is equal to \(1-g\)._
+
+Proof.: Let \(\{x_{1},...x_{n}\}\) be the set of critical points of \(F\). Recall the following formula for the calculation of the Euler characteristic
+
+\[\chi(M)=\sum_{i=1}^{n}ind_{x_{i}}(gradF),\]
+
+where \(ind_{x_{i}}(gradF)\) is the standard index of a critical point of the vector field \(gradF\). Observe that when \(F\) is a Morse function, \(ind_{x_{k}}(gradF)\) is \(+1\) for local maximum and minimum points, and \(-1\) for a saddle point. Denote by \(p\) the number of local maximum and minimum points, and by \(q\) the number of saddle points. Then by the above observation:
+
+\[\chi(M)=p-q.\]
+
+From the definition of the Reeb graph, the number of vertices in \(\Gamma\) is equal to the number of critical points in \(M\), therefore:
+
+\[\#V=p+q.\]
+
+Furthermore, the degree of each vertex associated with a local maximum or minimum point is \(1\), and the degree of each vertex associated with a saddle point is \(3\). Hence, the number of edges in \(\Gamma\) is
+
+\[\#E=\frac{1}{2}\sum_{v\in V}deg(v)=\frac{1}{2}(p+3q).\]
+
+We can now calculate the Euler characteristic of \(\Gamma\):
+
+\[\chi(\Gamma)=\#V-\#E =\]
+\[=p+q-\frac{1}{2}(p+3q)=\frac{1}{2}(p-q)=\frac{1}{2}\chi(M) = \frac{1}{2}(2-2g)=1-g.\]
+
+∎
+
+## 4 Essential Critical Points
+
+The notion of essential critical points is needed for the precise formulation of Py’s second theorem mentioned in Section 1.3. After defining the term and stating Py’s theorem, we will prove a statement regarding the cardinality of the set of essential critical points. The proof will clarify the concept, and its methods will also be used in Section 5 for the construction of the pair of pants decomposition. Similar methods have been used in [9].
+
+**Definition 4.1****.**: Let \(\Gamma\) be a connected graph. Given a vertex \(v\), \(\Gamma\backslash\{v\}\) is a disjoint union of connected topological spaces \(Y_{1}\bigsqcup Y_{2}\bigsqcup\dots\bigsqcup Y_{d}\). Let \(\overline{Y_{i}}:=Y_{i}\bigcup\{v\}\) be the subgraph of \(\Gamma\), which is composed of \(Y_{i}\) attached to the vertex \(v\). We will refer to \(\{\overline{Y_{1}},\dots,\overline{Y_{d}}\}\) as the **subgraphs associated with \(v\)**.
+
+**Definition 4.2****.**: A vertex \(v\) is called **non essential** if either one of the subgraphs associated with it, \(\{\overline{Y_{1}},\dots,\overline{Y_{d}}\}\), is a tree, or \(v\) is an endpoint.
+
+**Definition 4.3****.**: Let \(F\) be a generic Morse function defined on a closed surface of genus \(g\geq 2\), and denote by \(\Gamma_{F}\) the Reeb graph of \(F\). Define the **set of essential critical points of F**, \({\mathcal{V}}_{F}\), to be the critical points of \(F\) that correspond to essential vertices in \(\Gamma_{F}\).
+
+We can now state Py’s second result [18]. Let \(F:M\rightarrow{\mathbb{R}}\) be a generic Morse function, where \(M\) is a closed oriented surface of genus \(g\geq 2\) and of total area \(2g-2\). Let \({\mathcal{F}}\) be the space of smooth functions on \(M\) which commute with \(F\) (Definition 1.8), and \(\mu\) is Py’s Calabi quasi-morphism given in Theorem 1.6.
+
+**Theorem 4.4****.**: _For \(H\) in \({\mathcal{F}}\)_
+
+\[\mu(\phi_{H})=\int_{M}H\omega-\sum_{x\in{\mathcal{V}}_{F}}H(x)\]
+
+_where \({\mathcal{V}}_{F}\) is the set of essential critical points of \(F\)._
+
+In the rest of the section we will show that the number of essential critical points is equal to \(2g-2\), where \(g\geq 2\) is the genus of \(M\).
+
+**Lemma 4.5****.**: _Let \(\Gamma\) be a connected graph with vertices of degree \(1\) or \(3\), such that \(\chi(\Gamma)\leq-1\). Assume that \(\Gamma\) has at least one vertex of degree 1. Then \(\Gamma\) has more than two vertices._
+
+Proof.: The assumption that \(\Gamma\) has at least one vertex of degree \(1\) implies that there are at least two vertices in \(\Gamma\). Assume that there are only two vertices in \(\Gamma\) and denote them by \(v_{1}\) and \(v_{2}\). Up to graph isomorphism, there are only two connected graphs with one vertex of degree \(1\) and the other of degree \(1\) or \(3\). One graph is simply \(v_{1},v_{2}\) and an edge connecting them, and the other has an extra edge connected on both ends to one of the vertices. The Euler characteristic of the first graph is \(1\), and of the second is \(0\). But we assume that \(\chi(\Gamma)\leq-1\), hence \(\Gamma\) has more than two vertices. ∎
+
+**Construction algorithm. Let \(\Gamma\) be a connected graph with vertices of degree \(1\) or \(3\), such that \(\chi(\Gamma)\leq-1\). Assume that there is at least one vertex of degree \(1\). We will define a new graph \(\Gamma^{\prime}\) obtained from \(\Gamma\) in the following procedure. Choose a vertex \(v_{1}\) of degree \(1\). We will denote by \(e\) the edge adjacent to \(v_{1}\) and by \(v_{2}\) the vertex on the other end of \(e\). The degree of \(v_{2}\) is either \(1\) or \(3\). But if the degree is \(1\), it implies that \(\Gamma\) has only two vertices contradicting Lemma 4.5. Therefore, the degree of \(v_{2}\) is \(3\). Remove the vertex \(v_{1}\) along with the edge \(e\). The degree of \(v_{2}\) is now \(2\). Note that if \(v_{2}\) is adjacent to both ends of the same edge, it implies that \(\Gamma\) has only two vertices contradicting Lemma 4.5. Therefore \(v_{2}\) is adjacent to two different edges. Remove the vertex \(v_{2}\) and replace the two edges adjacent to it with one edge. The new graph is not empty since \(\Gamma\) has more than two vertices and we removed only two vertices. Define \(\Gamma^{\prime}\) to be the new graph.**
+
+Note that \(\Gamma^{\prime}\) is not uniquely defined since the endpoint to be removed can be chosen arbitrarily.
+
+**Lemma 4.6****.**: \(\Gamma^{\prime}\) _is a deformation retract of \(\Gamma\)._
+
+Proof.: From the topological point of view, \(\Gamma^{\prime}\) is obtained from \(\Gamma\) by contracting a line segment to a point. Hence \(\Gamma^{\prime}\) is a deformation retract of \(\Gamma\). ∎
+
+**Corollary 4.7****.**: \(\chi(\Gamma)=\chi(\Gamma^{\prime})\) _since the Euler characteristic is a topological invariant._
+
+**Lemma 4.8****.**: _The graph \(\Gamma^{\prime}\) is connected with vertices of degree \(1\) or \(3\)._
+
+Proof.: In the construction of \(\Gamma^{\prime}\), apart from the removed vertices \(v_{1}\) and \(v_{2}\), the rest of the vertices have the same degree as in \(\Gamma\). Therefore the vertices in \(\Gamma^{\prime}\) are of degree \(1\) or \(3\).
+
+Let \(v^{\prime}\),\(v^{\prime\prime}\) be any two vertices in \(\Gamma^{\prime}\). Since \(\Gamma\) is connected, there exists a path in \(\Gamma\) between \(v^{\prime}\) and \(v^{\prime\prime}\). The vertex \(v_{1}\) has degree \(1\), so obviously the path can be chosen not to pass through \(v_{1}\). If the path passes through \(v_{2}\) in \(\Gamma\), then in \(\Gamma^{\prime}\) it will pass through the new edge that replaced \(v_{2}\) and its two adjacent edges. Hence \(\Gamma^{\prime}\) is also connected. ∎
+
+**Lemma 4.9****.**: _Let \(v\in\Gamma\) be an essential vertex, then \(v\in\Gamma^{\prime}\) and \(v\) is essential in \(\Gamma^{\prime}\). The opposite also holds, if \(v\in\Gamma^{\prime}\) is essential in \(\Gamma^{\prime}\) then \(v\) is essential in \(\Gamma\)._
+
+Proof.: Let \(v\in\Gamma\) be an essential vertex. The vertices that can be removed in the process of constructing \(\Gamma^{\prime}\) are either endpoints, or vertices that are connected via an edge to an endpoint. The vertex \(v\) is essential, hence it can not be an endpoint. Furthermore, if \(v\) is connected via an edge to an endpoint, then there exists a subgraph associated with \(v\) which is a tree, namely, it is the subgraph that contains the endpoint and \(v\). Hence, we get a contradiction to the fact that \(v\) is essential. We conclude that \(v\in\Gamma^{\prime}\).
+Assume that \(v\) is not essential in \(\Gamma^{\prime}\). In the construction of \(\Gamma^{\prime}\), no new endpoints are created relative to those in \(\Gamma\). Now, \(v\) is not an endpoint in \(\Gamma\), hence it is not an endpoint in \(\Gamma^{\prime}\). If one of the subgraphs associated with \(v\) is a tree in \(\Gamma^{\prime}\), then it is also a tree in \(\Gamma\), since the addition of a free edge does not create a cycle. But \(v\) is essential in \(\Gamma\), hence we have a contradiction, and \(v\) is indeed essential in \(\Gamma^{\prime}\).
+Conversely, let \(v\) be essential in \(\Gamma^{\prime}\). The vertices of \(\Gamma^{\prime}\) are contained in those of \(\Gamma\), so obviously \(v\in\Gamma\). The vertex \(v\) is not an endpoint in \(\Gamma^{\prime}\) so in particular it is not an endpoint in \(\Gamma\). The subgraphs associated with \(v\) in \(\Gamma^{\prime}\) are all not trees, and the addition of a free edge does not change this property in \(\Gamma\). Hence \(v\) is essential in \(\Gamma\). ∎
+
+Figure 2: The essential vertices of \(\Gamma\) are \(\{v_{1},v_{2}\}\). \(\Gamma^{\prime\prime\prime}\) has only vertices of degree 3, which are all essential.
+
+**Definition 4.10****.**: By Corollary 4.7 and Lemma 4.8 we can apply the above algorithm on \(\Gamma\) recursively until there are no more vertices of degree \(1\). Denote the new graph by \(\widetilde{\Gamma}\). By Lemma 4.8, \(\widetilde{\Gamma}\) only has vertices of degree \(3\) (see Figure 2).
+
+**Proposition 4.11****.**: _Let \(\Gamma\) be a connected graph such that all vertices are of degree \(3\). Then all vertices of \(\Gamma\) are essential._
+
+Proof.: Let \(v\in\Gamma\). Obviously \(v\) can’t be an endpoint since its degree is \(3\). Consider the subgraphs associated with \(v\), \(\{\overline{Y_{1}},\dots,\overline{Y_{d}}\}\). Each subgraph has at most one vertex of degree \(1\) (Namely, the vertex \(v\)). But a non-trivial tree must have at least two vertices of degree \(1\). Therefore \(v\) is essential. ∎
+
+**Proposition 4.12****.**: _Let \(\Gamma\) be a connected graph with vertices of degree \(1\) or \(3\), such that \(\chi(\Gamma)\leq~{}-1\). Then the number of essential vertices in \(\Gamma\) is equal to \(-2\chi(\Gamma)\)._
+
+Proof.: Using Corollary 4.7 we get by induction \(\chi(\Gamma)=\chi(\widetilde{\Gamma})\). By Lemma 4.9, we can see that \(\Gamma\) and \(\widetilde{\Gamma}\) have the same essential vertices. Consequently, it is enough to prove the claim for \(\widetilde{\Gamma}\). Let \(\#V\) and \(\#E\) be the number of vertices and edges in \(\widetilde{\Gamma}\), respectively. Recall that the Euler characteristic of a graph is equal to \(\#V-\#E\). By Definition 4.10\(\widetilde{\Gamma}\) has only vertices of degree \(3\). Since every vertex is adjacent to three edges, and each edge is adjacent to two vertices, we get
+
+\[\#V=\frac{3}{2}\#E.\]
+
+Thus
+
+\[\chi(\widetilde{\Gamma})=\#V-\#E=-\frac{1}{2}\#V\]
+
+and
+
+\[\#V=-2\chi(\widetilde{\Gamma})\]
+
+as required. ∎
+
+**Corollary 4.13****.**: _Let \(\Gamma_{F}\) be the the Reeb graph of a generic Morse function \(F\), defined on a closed surface \(M\) of genus \(g\geq 2\). Then the number of essential vertices in \(\Gamma_{F}\) is equal to \(2g-2\)._
+
+Proof.: By Proposition 3.2\(\chi(\Gamma_{F})=1-g\leq-1\). Using Proposition 4.12 the number of essential vertices in \(\Gamma_{F}\) is equal to \(-2\chi(\Gamma_{F})\).
+
+Therefore
+
+\[-2\chi(\Gamma_{F})=-2(1-g)=2g-2\]
+
+∎
+
+## 5 The Pair of Pants Decomposition
+
+The pair of pants decomposition is crucial for our proof of the monotonicity. We will show here how to construct a pair of pants decomposition given a generic Morse function \(F:M\rightarrow{\mathbb{R}}\).
+
+Let \({\mathcal{V}}=\{x_{1},x_{2},...,x_{2g-2}\}\) be the set of essential critical points of \(F\). We will denote by \(\{v_{1},v_{2},...,v_{2g-2}\}\) the set of essential vertices in \(\Gamma\), the Reeb graph of \(F\). Let \(F_{\Gamma}\) be the function on the graph \(\Gamma\) induced by \(F\). Let \(c_{i}=F(x_{i})=F_{\Gamma}(v_{i})\) for \(i=1,...,2g-2\) be the critical values corresponding to the essential critical points. Without loss of generality, we may assume that \(c_{1}<c_{2}<...<c_{2g-2}\). Choose small enough \(\epsilon>0\) so that \([c_{i}-\epsilon,c_{i}+\epsilon]\) will only contain the critical value \(c_{i}\).
+
+**Proposition 5.1****.**: _Denote by \((F_{\Gamma}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) the connected component of \(F_{\Gamma}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon)\) that contains the vertex \(v_{i}\). Then \(\Gamma\backslash\bigcup_{i}(F_{\Gamma}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) is a disjoint union of trees, such that each tree has precisely two endpoints removed._
+
+Proof.: Let \(\widetilde{\Gamma}\) be the graph obtained from \(\Gamma\) by applying the algorithm described above iteratively (Definition 4.10). Recall that all vertices of \(\widetilde{\Gamma}\) are essential and their number is \(2g-2\geq 2\) for \(g\geq 2\). Hence \(\widetilde{\Gamma}\) cannot contain only one vertex with no edges.
+Note that \(\widetilde{\Gamma}\backslash\bigcup_{i}(F_{\widetilde{\Gamma}}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) is a collection of edges without the endpoints, which is of course also a collection of trees with exactly two endpoints removed. We will use induction on the reverse steps of the algorithm to show that \(\Gamma\backslash\bigcup_{i}(F_{\Gamma}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) is a collection of the required trees. The base of the induction is the case of \(\widetilde{\Gamma}\) which was shown above. Denote by \(\Gamma^{(n)}\) the graph obtained after the \(n\)-th iteration of the algorithm. Assume that the claim holds for \(\Gamma^{(n+1)}\) and we will show that it holds for \(\Gamma^{(n)}\).
+\(\Gamma^{(n+1)}\backslash\bigcup_{i}(F_{\Gamma^{(n+1)}}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) is a disjoint union of trees, such that each tree has precisely two endpoints removed. In the reverse step of the algorithm we attach a free edge to one of the edges in the graph. But an addition of a free edge to a tree is also a tree and there are still only two endpoints removed. Hence \(\Gamma^{(n)}\backslash\bigcup_{i}(F_{\Gamma^{(n)}}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) satisfies the inductive hypothesis. Therefore the claim holds for \(\Gamma=\Gamma^{(0)}\) as required. ∎
+
+**Proposition 5.2****.**: _Denote by \((F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{x_{i}}\) the connected component of \(F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon)\) that contains \(x_{i}\). Then \(M\backslash\bigcup_{i}(F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{x_{i}}\) is a disjoint union of cylinders with boundaries that corresponds to level sets of the form \(F^{-1}(c_{i}\pm\epsilon)\) for \(i=1,...,2g-2\)._
+
+Proof.: The connected components of \(M\backslash\bigcup_{i}(F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{x_{i}}\) correspond to the connected components of \(\Gamma\backslash\bigcup_{i}(F_{\Gamma}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) by the Reeb graph definition. By the previous claim, each connected component of \(\Gamma\backslash\bigcup_{i}(F_{\Gamma}^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{v_{i}}\) is a tree with two end-points removed. The analogue in \(M\backslash\bigcup_{i}(F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{x_{i}}\) is a surface of genus zero with two boundary components corresponding to the level sets of the form \(F^{-1}(c_{i}\pm\epsilon)\). The values \(c_{i}\pm\epsilon\) are regular, hence the boundary components have the structure of embedded circles. By classification of surfaces these components are cylinders. ∎
+
+**Definition 5.3****.**: By the term **pair of pants** we mean an embedding in \(M\) of a connected orientable surface of genus zero with three boundary components.
+
+**Pair of Pants Decomposition. For \(i=1,...,2g-2\), the critical point \(x_{i}\) is essential, hence it is of index \(1\). Furthermore, it is the single critical point in \((F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{x_{i}}\). Hence, by classification of surfaces, \((F^{-1}(c_{i}-~{}\epsilon,c_{i}+\epsilon))_{x_{i}}\) has the structure of an embedded surface of genus zero with three boundary components, or in other words, a pair of pants. Denote by \(l^{i}_{1},l^{i}_{2}\) and \(l^{i}_{3}\) the three boundary components of the pair of pants \((F^{-1}(c_{i}-\epsilon,c_{i}+\epsilon))_{x_{i}}\). By Proposition 5.2, the complement of the union of all pairs of pants is a disjoint union of cylinders (Figure 3, top). The boundary components of each cylinder correspond to some two boundary components \(l^{i}_{t}\) and \(l^{j}_{s}\). Hence, \(l^{i}_{t}\) and \(l^{j}_{s}\) are isotopic. Note that \(i\) can be equal to \(j\) in the case that both boundaries of the cylinder belong to the same pair of pants. There are \(2g-2\) pairs of pants, each has \(3\) boundary components. In total we have \(6g-6\) boundary components. Since each cylinder has \(2\) boundary components, we get that there are \(3g-3\) disjoint cylinders. For each cylinder, choose one of its boundary components and attach the cylinder to a pair of pants along this boundary component, denote the other boundary component of the cylinder by \(\gamma_{i}\) for \(i=1,...,3g-3\). By attaching a cylinder to a pair of pants we again get a pair of pants. As a result, \(M\backslash\bigcup_{i}\gamma_{i}\) is a disjoint union of \(2g-2\) pairs of pants. We will denote by \(P_{i}\) the pair of pants that contains the essential critical point \(x_{i}\) (Figure 3, bottom left).**
+
+Figure 3: The pair of pants decomposition.
+
+**Proposition 5.4****.**: _The circles in the collection \(\{\gamma_{1},...,\gamma_{3g-3}\}\) are disjoint, non-contractible, and pairwise non-isotopic._
+
+Proof.: The circles are disjoint since they are boundaries of disjoint cylinders. Let \(\gamma\in\{\gamma_{1},...,\gamma_{3g-3}\}\). If \(\gamma\) is contractible then it bounds a disc in \(M\). Let \(e\) be the edge in the Reeb graph \(\Gamma\), that contains the image of the level curve \(\gamma\) by the natural projection \(\pi:M\rightarrow\Gamma\), i.e. \(\pi(\gamma)\in e\). Then \(e\) is adjacent to an essential vertex \(v\) on one end, and to a tree on the other end, corresponding to the disc. But by the definition of an essential vertex, the subgraphs associated with \(v\) are not trees, leading to a contradiction. Therefore \(\gamma\) is not contractible.
+Let \(\gamma_{i},\gamma_{j}\in\{\gamma_{1},...,\gamma_{3g-3}\}\) for \(i\neq j\). Assume that \(\gamma_{i}\) is isotopic to \(\gamma_{j}\). Then \(\gamma_{i}\) and \(\gamma_{j}\) are the boundaries of some cylinder \(C\) in \(M\). But \(\gamma_{i}\) is also a boundary component of some pairs of pants \(P^{+},P^{-}\in\{P_{1},...,P_{2g-2}\}\) (maybe a priory equal). Thus, at least one of \(P^{+},P^{-}\) is contained in \(C\). This implies that at least one of the three boundary components of this pair of pants, denoted by \(\gamma_{k}\in\{\gamma_{1},...,\gamma_{3g-3}\}\), is contractible, contradicting the first claim. Hence, \(\{\gamma_{1},...,\gamma_{3g-3}\}\) are pairwise non-isotopic. ∎
+
+Take an auxiliary metric of negative curvature on \(M\). By a theorem [8], the submanifold that consists of the circles \(\{\gamma_{1},...,\gamma_{3g-3}\}\) is isotopic to a unique disjoint union of simple closed geodesics \(\{\delta_{1},...,\delta_{3g-3}\}\). Note that \(M\backslash\bigcup_{i}\delta_{i}\) is a disjoint union of pairs of pants since it is isotopic to \(M\backslash\bigcup_{i}\gamma_{i}\). We will denote by \(\overline{P_{i}}\) the pair of pants with geodesic boundaries isotopic to \(P_{i}\) (Figure 3, bottom right).
+
+**Definition 5.5****.**: We will call the collection \(\{\overline{P}_{1},...,\overline{P}_{2g-2}\}\), the **pair of pants decomposition of \(M\) associated with \(F\).**
+
+**Proposition 5.6****.**: _Given an auxiliary metric of negative curvature on \(M\), the pair of pants decomposition of \(M\) associated with \(F\) is well-defined._
+
+Proof.: The only choices we had in the definition, were the choice of a small enough \(\epsilon>0\), and which one of the boundary components of each cylinder will be denoted by \(\gamma_{i}\). Note that these choices do not affect the circles \(\{\gamma_{1},...,\gamma_{3g-3}\}\) up to isotopy. The geodesics \(\{\delta_{1},...,\delta_{3g-3}\}\) are isotopic to \(\{\gamma_{1},...,\gamma_{3g-3}\}\), and are uniquely determined, given the auxiliary metric. Hence they do not depend on the above choices, and the term is well-defined. ∎
+
+## 6 Figure Eight Intersections
+
+We will define a figure eight collection of a generic Morse function \(F:M\rightarrow{\mathbb{R}}\), and prove an intersection theorem, using hyperbolic geometry tools and Hall’s marriage theorem. This is the last step towards the proof of monotonicity.
+
+**Definition 6.1****.**: By the term **figure eight** we refer to an immersion in \(M\) of a closed curve with a unique transversal double point.
+
+**Definition 6.2****.**: Let \(M\) be a closed surface of genus \(g\geq 2\) and let \(F:M\rightarrow{\mathbb{R}}\) be a generic Morse function. Let \({\mathcal{V}}\) be the set of essential critical points of \(F\). For each \(x_{i}\in{\mathcal{V}}\), denote by \(c_{i}=F(x_{i})\) its critical value for \(i=1,...,2g-2\). We can assume \(c_{1}<c_{2}<...<c_{2g-2}\). Denote by \(e_{i}\) the connected component of \(F^{-1}(c_{i})\) that contains \(x_{i}\). Note that \(x_{i}\) is the only critical point in this level set, and its index is \(1\) since it is essential. By classification theory, \(e_{i}\) is an immersed closed curve with a unique transversal double point at \(x_{i}\). Thereby, \(e_{i}\) is a figure eight. We will call the collection \(\{e_{1},...,e_{2g-2}\}\) the **figure eight collection of \(F\)**.
+
+We will use the following preliminary results from hyperbolic geometry. Proofs can be found in [7].
+
+**Theorem 6.3****.**: _Let \(S\) be a compact hyperbolic surface and let \(\gamma\) be a non-contractible closed curve on \(S\). We will denote by \({\mathbb{D}}\) the Poincaré disc model, and by \(\widehat{S}\subset{\mathbb{D}}\) the universal cover of \(S\). Note that \(\widehat{S}\) is isometric to \({\mathbb{D}}\) when \(S\) is without boundary. Then the following hold: (i) \(\gamma\) is freely homotopic to a unique closed geodesic \(\eta\). (ii) For any lift \(\widehat{\eta}\) of \(\eta\) in the universal covering \(\widehat{S}\subset{\mathbb{D}}\) there exists a lift \(\widehat{\gamma}\) of \(\gamma\) such that \(\widehat{\gamma}\) and \(\widehat{\eta}\) have the same endpoints at the circle at infinity. (iii) \(\eta\) is either contained in \(\partial S\) or \(\eta\cap\partial S=\emptyset\)._
+
+**Theorem 6.4****.**: _Let \(\gamma_{1},\gamma_{2}\) be two non-contractible closed curves, and let \(\eta_{1}\), \(\eta_{2}\) be the unique closed geodesic curves freely homotopic to \(\gamma_{1}\) and \(\gamma_{2}\) respectively. Then if \(\eta_{1}\) and \(\eta_{2}\) have transversal intersection, it implies \(\gamma_{1}\cap\gamma_{2}\neq\emptyset\)._
+
+**Lemma 6.5****.**: _Let \(x\in{\mathcal{V}}=\{x_{1},...,x_{2g-2}\}\) be an essential critical point with critical value \(c=F(x)\). Let \(\overline{P}\in\{\overline{P}_{1},...,\overline{P}_{2g-2}\}\) be the corresponding geodesic pair of pants and \(e\in\{e_{1},...,e_{2g-2}\}\) the corresponding figure eight. Then the unique closed geodesic homotopic to \(e\), denoted by \(\eta\), is contained in \(\overline{P}\), and \(\overline{P}\backslash\eta\) is a disjoint union of three cylinders._
+
+Proof.: The figure eight \(e\) is defined as the connected component of the level set \(F^{-1}(c)\) that contains \(x\). Note that \(e\) is contained in the pair of pants \((F^{-1}(c-\epsilon,c+\epsilon))_{x}\), and \((F^{-1}(c-\epsilon,c+\epsilon))_{x}\backslash e\) is a disjoint union of three cylinders. In the definition of the pair of pants decomposition, we construct an intermediate pair of pants \(P\) by attaching cylinders to some (or none) of the boundary components of \((F^{-1}(c-\epsilon,c+\epsilon))_{x}\). Following the notation used in the definition, we denote the new boundary components of the pair of pants by \(\gamma_{1}\),\(\gamma_{2}\) and \(\gamma_{3}\). Note that \(\gamma_{2}\) and \(\gamma_{3}\) coincide in the case in which two boundary components of the initial pair of pants are isotopic. We only attached cylinders to boundaries of \((F^{-1}(c-\epsilon,c+\epsilon))_{x}\), therefore \(P\backslash e\) is also a disjoint union of three cylinders.
+
+By a theorem [8] there exists a homeomorphism \(h:M\to M\) isotopic to the identity such that \(\gamma_{1}\),\(\gamma_{2}\) and \(\gamma_{3}\) are sent to the unique closed geodesics \(\delta_{1}\), \(\delta_{2}\) and \(\delta_{3}\) respectively. The image of \(P\) by \(h\) is therefore \(\overline{P}\), the pair of pants bounded by \(\delta_{1}\), \(\delta_{2}\) and \(\delta_{3}\). The image of \(e\) by \(h\), denoted by \(\widetilde{e}:=h(e)\), is a figure eight homotopic to \(e\) and contained in \(\overline{P}\). Furthermore, \(h(P\backslash e)=\overline{P}\backslash\widetilde{e}\) is again a disjoint union of three cylinders since \(h\) is a homeomorphism. We can consider \(\overline{P}\cup\partial\overline{P}\) as a compact hyperbolic surface with boundary, and \(\widetilde{e}\) is a non-contractible closed curve on \(\overline{P}\). Hence, by Theorem 6.3, \(\widetilde{e}\) is freely homotopic to a unique closed geodesic \(\eta\), and \(\eta\) is either contained in \(\partial\overline{P}\) or \(\eta\cap\overline{P}=\emptyset\). Since \(\widetilde{e}\) is not homotopic to any of the boundary components of \(\overline{P}\), \(\eta\) cannot be contained in \(\partial\overline{P}\). Therefore, \(\eta\cap\partial\overline{P}=\emptyset\) and \(\eta\) is contained in \(\overline{P}\). Note that \(e\) is freely homotopic to \(\widetilde{e}\), so \(\eta\) is also the geodesic closed curve homotopic to \(e\) by uniqueness. Since \(\eta\) is in the homotopy class of \(\widetilde{e}\), together with the fact that a geodesic curve cannot bound any disc, it follows that \(\overline{P}\backslash\eta\) is also a disjoint union of three cylinders. ∎
+
+**Lemma 6.6****.**: _Let \(e_{1}\), \(e_{2}\) be two figure eights (not necessarily from the same figure eight collection), and let \(\eta_{1}\), \(\eta_{2}\) be the unique geodesic figure eights freely homotopic to \(e_{1}\) and \(e_{2}\), respectively. Then \(\eta_{1}\cap\eta_{2}\neq\emptyset\) implies \(e_{1}\cap e_{2}\neq\emptyset\)._
+
+Figure 4: The Poincaré disc \({\mathbb{D}}\).
+
+Proof.: Two geodesic curves with non-empty intersection either have transversal intersection or coincide. If \(\eta_{1}\) and \(\eta_{2}\) have transversal intersection, then the result follows immediately from Theorem 6.4, since \(e_{1}\) and \(e_{2}\) are in particular non-contractible closed curves. Assume that \(\eta_{1}\) and \(\eta_{2}\) coincide. Denote by \(x_{0}\) the unique transversal double point of the figure eight \(\eta_{1}\). Denote by \(\widehat{x}_{0}\) a lift of \(x_{0}\) to the universal cover of the surface, modeled by the Poincaré disc \({\mathbb{D}}\). In a small neighborhood of \(x_{0}\) there are two geodesic segments with transversal intersection. Lift this neighborhood to a neighborhood of \(\widehat{x}_{0}\) in the universal cover, and extend these two geodesics uniquely in \({\mathbb{D}}\). Denote them by \(\widehat{\eta}_{1}\) and \(\widehat{\eta}_{2}\). By Theorem 6.3, there exist lifts of \(e_{1}\) and \(e_{2}\), denoted by \(\widehat{e}_{1}\) and \(\widehat{e}_{2}\) respectively, such that for \(i=1,2\), \(\widehat{\eta}_{i}\) and \(\widehat{e}_{i}\) have the same endpoints at the circle at infinity (Figure 4). Since \(\widehat{\eta}_{1}\) and \(\widehat{\eta}_{2}\) have transversal intersection, the endpoints of \(\widehat{\eta}_{1}\) separate the endpoints of \(\widehat{\eta}_{2}\). As a result, \(\widehat{e}_{1}\) and \(\widehat{e}_{2}\) must intersect in \({\mathbb{D}}\), so their projection on the surface must also intersect as required. ∎
+
+**Lemma 6.7****.**: _Let \(P\) be a pair of pants, and let \(e_{1},e_{2}\) be two figure eights contained in \(P\), such that for each \(i\), \(P\backslash e_{i}\) is a disjoint union of three cylinders. Then \(e_{1}\cap e_{2}\neq\emptyset\)._
+
+Proof.: Assume that \(e_{1}\cap e_{2}=\emptyset\). Since \(e_{2}\) is connected, it is contained in one of the connected components of \(P\backslash e_{1}\). By our assumption this component is a cylinder, denoted by \(C\), so \(e_{2}\subset C\). The figure eight \(e_{2}\) is composed of two disjoint simple loops with a unique intersection point. But in a cylinder, each two non-contractible simple loops are freely homotopic to each other. Hence, \(e_{2}\) bounds a disc, in contradiction to the assumption that \(P\backslash e_{2}\) is a disjoint union of cylinders. Therefore \(e_{1}\cap e_{2}\neq\emptyset\). ∎
+
+Before the next proposition we wish to emphasize that we do not regard the boundary of a pair of pants as part of it, i.e. \(\partial P\cap P=\emptyset\).
+
+**Proposition 6.8****.**: _Let \(P_{1}\),\(P_{2}\subset M\) be two pairs of pants with geodesic boundaries such that \(P_{1}\cap P_{2}\neq\emptyset\). Then either the boundary components of \(P_{1}\) and \(P_{2}\) coincide, or there exists a boundary component of \(P_{1}\) that has transversal intersection with a boundary component of \(P_{2}\)._
+
+Proof.: We will assume that not all the boundary components of \(P_{1}\) coincide with those of \(P_{2}\), i.e. \((\partial P_{1}\backslash\partial P_{2})\cup(\partial P_{2}\backslash\partial P_{1})\neq\emptyset\). We will first show that \((\partial P_{1}\cap P_{2})\cup(\partial P_{2}\cap P_{1})\neq\emptyset\). Choose \(x\in P_{1}\cap P_{2}\) and \(y\in(\partial P_{1}\backslash\partial P_{2})\cup(\partial P_{2}\backslash\partial P_{1})\). Assume without loss of generality that \(y\in\partial P_{1}\backslash\partial P_{2}\). If \(y\in P_{2}\), then \(y\in(\partial P_{1}\cap P_{2})\cup(\partial P_{2}\cap P_{1})\) as required. Otherwise, i.e. \(y\notin P_{2}\), choose a path \(\gamma:[0,1]\to P_{1}\cup\partial P_{1}\) such that \(\gamma(0)=x\), \(\gamma([0,1))\subset P_{1}\), and \(\gamma(1)=y\in\partial P_{1}\). By the above assumptions, \(\gamma(1)\notin P_{2}\cup\partial P_{2}\) and \(\gamma(0)=x\in P_{1}\cap P_{2}\subset P_{2}\). Hence, there must exist \(t_{0}\in(0,1)\) such that \(\gamma(t_{0})\in\partial P_{2}\). But \(\gamma([0,1))\subset P_{1}\), so \(\gamma(t_{0})\in\partial P_{2}\cap P_{1}\subset(\partial P_{1}\cap P_{2})\cup(\partial P_{2}\cap P_{1})\) as required.
+Now, choose \(z\in(\partial P_{1}\cap P_{2})\cup(\partial P_{2}\cap P_{1})\) and assume without loss of generality that \(z\in\partial P_{1}\cap P_{2}\). Denote by \(\delta\) the boundary component in \(\partial P_{1}\) that contains \(z\). Note that \(\delta\) does not coincide with any of the boundary components of \(P_{2}\) since \(\partial P_{2}\cap P_{2}=\emptyset\) and if \(\delta\subset\partial P_{2}\) then \(\delta\cap P_{2}=\emptyset\) but \(z\in\delta\cap P_{2}\). We will show that \(\delta\cap\partial P_{2}\neq\emptyset\). Suppose that \(\delta\cap\partial P_{2}=\emptyset\), then since \(z\in\delta\cap P_{2}\) we get that \(\delta\) must be contained in \(P_{2}\). The boundary component \(\delta\) is a simple closed curve, and topologically, a pair of pants can be viewed as a sphere with three points removed, so \(\delta\) must bound a disc or a punctured disk. Hence, \(\delta\) is either contractible, or freely homotopic to one of the boundary components of \(P_{2}\). But \(\delta\) is a geodesic, thereby it is not contractible, and if it is homotopic to one of the geodesic boundary components of \(P_{2}\) then they must coincide by Theorem 6.3, in contradiction to the choice of \(\delta\). Therefore, \(\delta\cap\partial P_{2}\neq\emptyset\). If two geodesic curves intersect then they either coincide or have transversal intersection. We have already shown that they do not coincide, so the result follows.
+
+∎
+
+Figure 5: \(\gamma_{1}\) and \(\gamma_{2}\) are freely homotopic to \(\delta_{1}\) and \(\delta_{2}\) respectively. \(\gamma_{3}:=\gamma_{1}*\gamma_{2}^{-1}\) is freely homotopic to \(\delta_{3}\).
+
+**Proposition 6.9****.**: _Let \(P_{1},P_{2}\subset M\) be two pairs of pants with geodesic boundaries, such that \(P_{1}\cap P_{2}\neq\emptyset\). Let \(e_{1},e_{2}\) be two figure eights contained in \(P_{1}\) and \(P_{2}\), respectively, such that for \(i=1,2\)\(P_{i}\backslash e_{i}\) is a disjoint union of three cylinders. Then \(e_{1}\cap e_{2}\neq\emptyset\)._
+
+Proof.: We will first make the following observation. Let \(e\) be a figure eight contained in a pair of pants \(P\) with geodesic boundaries such that \(P\backslash e\) is a disjoint union of three cylinders. Let \(x_{0}\) be the unique transversal intersection point of the figure eight \(e\). The figure eight can be divided into two simple closed curves \(\gamma_{1}\) and \(\gamma_{2}\) with endpoints at \(x_{0}\). Define \(\gamma_{3}\) to be the (non-smooth) curve \(\gamma_{1}\) concatenated with \(\gamma_{2}\) in reverse orientation. Since \(P\backslash e\) is a disjoint union of three cylinders, we get that the curves \(\gamma_{1}\),\(\gamma_{2}\) and \(\gamma_{3}\) are freely homotopic to the three boundary components of \(P\) (Figure 5).
+
+Now, let \(P_{1}\) and \(P_{2}\) be two pairs of pants with geodesic boundaries such that \(P_{1}\cap P_{2}\neq\emptyset\). If \(P_{1}\) and \(P_{2}\) coincide then the result follows from Lemma 6.7. In the case that \(P_{1}\) and \(P_{2}\) do not coincide, then by Proposition 6.8 there exists a boundary component \(\delta\) of \(P_{1}\) that has transversal intersection with a boundary component \(\delta^{\prime}\) of \(P_{2}\). Let \(\gamma_{i}\) and \(\gamma_{i}^{\prime}\), \(i=1,2,3\) be the closed curves corresponding to the figure eights \(e_{1}\) and \(e_{2}\) respectively, as defined above. Let \(k,l\in\{1,2,3\}\) be such that the curves \(\gamma_{k}\) and \(\gamma_{l}^{\prime}\) are freely homotopic to \(\delta\) and \(\delta^{\prime}\) respectively. By Theorem 6.4 we get that \(\gamma_{k}\) intersects \(\gamma_{l}^{\prime}\) and since \(\gamma_{k}\) and \(\gamma_{l}^{\prime}\) are contained in \(e_{1}\) and \(e_{2}\) respectively, we get that \(e_{1}\cap e_{2}\neq\emptyset\) as required. ∎
+
+We will need the following definitions in order to state Hall’s marriage theorem.
+
+**Definition 6.10****.**: Let \(\mathcal{S}=\{S_{1},...,S_{n}\}\) be a collection of finite subsets of some larger set \(X\). A **system of distinct representatives** is a set \(R=\{r_{1},...,r_{n}\}\) of pairwise distinct elements of \(X\) with the property that for each \(i=1,...,n\), \(r_{i}\in S_{i}\).
+\(\mathcal{S}\) satisfies the **marriage condition** if for any subset \(\mathcal{T}=\{T_{i}\}\) of \(\mathcal{S}\), \(|\bigcup T_{i}|\geq|\mathcal{T}|\), i.e. any \(k\) subsets taken together have at least \(k\) elements.
+
+**Theorem 6.11****.**: _Hall’s marriage theorem[12]. Let \(\mathcal{S}=\{S_{1},...,S_{n}\}\) be a collection of finite subsets of some larger set. Then there exists a system of distinct representatives of \(\mathcal{S}\) if and only if \(\mathcal{S}\) satisfies the marriage condition._
+
+**Theorem 6.12****.**: _Let \(F,G:M\rightarrow{\mathbb{R}}\) be generic Morse functions. Let \(\{e_{1},...,e_{2g-2}\}\) and \(\{f_{1},...,f_{2g-2}\}\) be the figure eight collections of \(F\) and \(G\) respectively. Then there exists a permutation \(\sigma\in S(2g-2)\) such that \(e_{i}\cap f_{\sigma(i)}\neq\emptyset\) for each \(i=1,...,2g-2\)._
+
+Proof.: Let \(\{\overline{P}_{1},...,\overline{P}_{2g-2}\}\) and \(\{\overline{Q}_{1},...,\overline{Q}_{2g-2}\}\) be the pair of pants decompositions of \(M\) associated with \(F\) and \(G\) respectively. We will first show that there exists a permutation \(\sigma\in S(2g-2)\) such that \(\overline{P}_{i}\cap\overline{Q}_{\sigma(i)}\neq\emptyset\). We will use Hall’s marriage theorem in order to prove this. Define for \(i=1,...,2g-2\)
+
+\[S_{i}:=\{j|\overline{Q}_{j}\cap\overline{P}_{i}\neq\emptyset\}\]
+
+so \(S_{i}\) contains the indices of pairs of pants in \(\{\overline{Q}_{1},...,\overline{Q}_{2g-2}\}\) that intersect \(\overline{P}_{i}\). Define \(\mathcal{S}:=\{S_{1},...,S_{2g-2}\}\). We will prove the marriage condition for \(\mathcal{S}\). Let \(\mathcal{T}=\{S_{i}|i\in J\}\) be a subset of \(\mathcal{S}\), where \(J\subset\{1,...,2g-2\}\). Define \(\mathcal{P}:=\{\overline{P}_{i}|i\in J\}\). Note that the hyperbolic area of any pair of pants with geodesic boundaries is equal by Gauss-Bonnet to \(\frac{1}{2g-2}Vol(M)\). It follows that the union of pairs of pants in \(\mathcal{P}\) covers a total area of \(\frac{|J|}{2g-2}Vol(M)\). Thereby, this union must intersect at least \(|J|\) pairs of pants in \(\{\overline{Q}_{1},...,\overline{Q}_{2g-2}\}\). Equivalently, \(|\bigcup_{i\in J}S_{i}|\geq|J|=|\mathcal{T}|\) and the marriage condition is proved. Thus, there exists a system of distinct representatives \(R=\{r_{1},...,r_{2g-2}\}\) such that for \(i=1,...,2g-2\), \(r_{i}\in S_{i}\). We can now define a permutation \(\sigma\in S(2g-2)\) by \(\sigma(i):=r_{i}\) and for each \(i=1,...,2g-2\), \(\overline{P}_{i}\cap\overline{Q}_{\sigma(i)}\neq\emptyset\) as required.
+
+Now it is left to prove that \(e_{i}\cap f_{\sigma(i)}\neq\emptyset\). By Lemma 6.5 the unique closed geodesics homotopic to \(e_{i}\) and \(f_{\sigma(i)}\), denoted by \(\overline{e}_{i}\) and \(\overline{f}_{\sigma(i)}\), are contained in \(\overline{P}_{i}\) and \(\overline{Q}_{\sigma(i)}\) respectively. Furthermore, \(\overline{P}_{i}\backslash\overline{e}_{i}\) and \(\overline{Q}_{\sigma(i)}\backslash\overline{f}_{\sigma(i)}\), are both a disjoint union of three cylinders. By Proposition 6.9 we get that \(\overline{e}_{i}\cap\overline{f}_{\sigma(i)}\neq\emptyset\) and from Lemma 6.6 we conclude that \(e_{i}\cap f_{\sigma(i)}\neq\emptyset\). This completes the proof. ∎
+
+## 7 Monotonicity
+
+**Theorem 7.1****.**: _Let \(F,G:M\rightarrow{\mathbb{R}}\) be generic Morse functions, such that \(F\leq G\). Then \(\xi(F)\leq\xi(G)\)._
+
+Proof.: Let \({\mathcal{V}}_{F}=\{x_{1},...,x_{2g-2}\}\) and \({\mathcal{V}}_{G}=\{y_{1},...,y_{2g-2}\}\) be the sets of essential critical points of \(F\) and \(G\) respectively. Using Definition 1.10 of \(\xi\) and Theorem 4.4 we get that
+
+\[\xi(F)=\frac{1}{Vol(M)}\sum_{i=1}^{2g-2}F(x_{i})\]
+
+and
+
+\[\xi(G)=\frac{1}{Vol(M)}\sum_{i=1}^{2g-2}G(y_{i}).\]
+
+Let \(\{e_{1},...,e_{2g-2}\}\) and \(\{f_{1},...,f_{2g-2}\}\) be the figure eight collection of \(F\) and \(G\) respectively. By Theorem 6.12 there exists a permutation \(\sigma\in S(2g-2)\) such that for \(i=1,...,2g-2\) we have \(e_{i}\cap f_{\sigma(i)}\neq\emptyset\). For each \(i\), choose a point \(z_{i}\in e_{i}\cap f_{\sigma(i)}\). Then
+
+\[F(x_{i})=F(e_{i})=F(z_{i})\leq G(z_{i})=G(f_{\sigma(i)})=G(y_{\sigma(i)}),\]
+
+which implies
+
+\[\xi(F)=\frac{1}{Vol(M)}\sum_{i=1}^{2g-2}F(x_{i})\leq\frac{1}{Vol(M)}\sum_{i=1}^{2g-2}G(y_{\sigma(i)})=\xi(G)\]
+
+as required.
+
+∎
+
+## 8 Continuity
+
+In this section we will examine the construction of Py’s quasi-morphism as defined in [18] and show that it is continuous on time independent Hamiltonians, with respect to the \(C^{2}\)-topology. Lets recall the following definitions.
+
+**Definition 8.1****.**: A **contact form**\(\alpha\) on a \(2n+1\) dimensional manifold \(P\) is a 1-form with the property that \(\alpha\wedge(d\alpha)^{n}\neq 0\).
+
+**Definition 8.2****.**: Given a contact form \(\alpha\) on a manifold \(P\), the **Reeb vector field**\(X\) is defined to be the unique vector field that satisfies \(d\alpha(X,Z)=0\) for every \(Z\in TP\) and \(\alpha(X)=1\).
+
+**Definition 8.3****.**: A **principal G-bundle** is a fiber bundle \(\pi:P\to M\) together with a smooth right action \(P\times G\to P\) by a Lie group G such that G preserves the fibers of P and acts freely and transitively on them. The abstract fiber of the bundle is taken to be G itself.
+
+The following result is due to Banyaga [4].
+
+**Theorem 8.4****.**: _Let \(P\) be a closed connected manifold equipped with a contact form \(\alpha\). Let \(\pi:P\to M\) be a principal \(S^{1}\)-bundle, such that the Reeb vector field on \(P\) associated with \(\alpha\) coincides with the vector field generated by the action of \(S^{1}\), parameterized by \(\mathbb{R}/\mathbb{Z}\), on \(P\). Furthermore, we will assume that \(M\) has a symplectic form \(\omega\) that satisfies \(\pi^{*}\omega=d\alpha\). Then there exists a central extension by \(S^{1}\) of the group \(\mbox{Ham}(M,\omega)\),_
+
+\[0\to S^{1}\rightarrow\mbox{Diff}_{0}(P,\alpha)\rightarrow\mbox{Ham}(M,\omega)\to 0\]
+
+_where \(\mbox{Diff}_{0}(P,\alpha)\) stands for the group of diffeomorphisms on \(P\) which preserve \(\alpha\) and are isotopic to the identity via an isotopy that preserves \(\alpha\). Moreover, when \(\mbox{Ham}(M,\omega)\) is simply connected then the extension splits._
+
+In our case, \(M\) is a closed surface of genus \(g\geq 2\) hence \(\mbox{Ham}(M,\omega)\) is simply connected [11,19] and the extension splits.
+
+Let \(\phi_{H_{t}}\) be a Hamiltonian diffeomorphism generated by the Hamiltonian \(H_{t}\), where \(\int_{M}H_{t}\omega=0\) for every \(t\in[0,1]\). Define a vector field on \(P\),
+
+\[V_{t}:=\widehat{sgradH_{t}}+(H_{t}\circ\pi)X,\]
+
+where \(X\) is the Reeb vector field on \(P\), and \(\widehat{sgradH_{t}}\) is the horizontal lift of \(sgradH_{t}\), i.e. \(\alpha(\widehat{sgradH_{t}})=0\) and \(\pi_{*}(\widehat{sgradH_{t}})=sgradH_{t}\). Define \(\Theta(H_{t})\) to be the flow generated by \(V_{t}\). It can be shown that \(V_{t}\) preserves \(\alpha\) and that the homotopy class with fixed endpoints of \(\Theta(H_{t})\) depends only on the homotopy class with fixed endpoints of the flow generated by \(H_{t}\). Hence, \(\Theta:\widetilde{\mbox{Ham}}(M,\omega)\rightarrow\widetilde{\mbox{Diff}}_{0}(P,\alpha)\) (where \(\widetilde{G}\) stands for the universal cover of \(G\)) is well defined. Since \(\mbox{Ham}(M,\omega)\) is simply connected then \(\Theta\) can be defined on \(\mbox{Ham}(M,\omega)\), and by taking the time one map of \(\Theta(\phi_{H_{t}})\) we obtain the splitting map from \(\mbox{Ham}(M,\omega)\) to \(\mbox{Diff}_{0}(P,\alpha)\) of the extension in Theorem 8.4.
+
+Let \(H\in C^{\infty}(M)\), i.e. \(H\) is a time independent Hamiltonian. In order to apply \(\Theta\) on the flow generated by \(H\), we must first normalize \(H\), i.e. \(H\mapsto H-\frac{1}{Vol(M)}\int_{M}H\omega\). The normalization mapping is obviously smooth. The definition of the vector field \(V_{t}\) involves \(sgradH\), hence \(V_{t}\) is continuous as a function of \(H\) with respect to the \(C^{2}\)-topology on \(C^{\infty}(M)\), and so is the flow \((\Theta(H))_{t}\).
+
+**Construction of the quasi-morphism.** Let \(M\) be a closed surface of genus \(g\geq 2\), equipped with a symplectic form \(\omega\). We will assume that the total area of \(M\) is equal to \(2g-2\). Choose a metric with constant negative curvature on \(M\) such that its associated area form is equal to \(\omega\). Denote by \(P\) the unit tangent bundle of \(M\). We will use the Poincaré disc \({\mathbb{D}}\) as a model for the universal cover of \(M\), and denote by \(S^{1}{\mathbb{D}}\) the unit tangent bundle of \({\mathbb{D}}\). We can define a \(S^{1}\)-principal fiber bundle on \(P\) and \(S^{1}{\mathbb{D}}\) by rotating each vector in the unit tangent bundle by the same angle as defined by the metric. We will write \(S^{1}_{\infty}\) for the circle at infinity of \({\mathbb{D}}\) and \(p_{\infty}:S^{1}{\mathbb{D}}\to S^{1}_{\infty}\) for the natural projection, sending each unit vector in the tangent bundle of \({\mathbb{D}}\) to the limit at \(S^{1}_{\infty}\) of the unique geodesic tangent to it. Note that \(p_{\infty}\) is a smooth mapping. We will denote by \(\pi:P\to M\) the natural projection. Denote by \(X\) the vector field on \(P\) generated by the action of \(S^{1}\), parameterized by \(\mathbb{R}/\mathbb{Z}\), on \(P\). One can show that there exists a contact form \(\alpha\) on \(P\) such that \(\pi^{*}\omega=d\alpha\) and its Reeb vector field coincides with \(X\). Hence, according to Theorem 8.4 we can construct the homomorphism \(\Theta\) as defined above. Given an Hamiltonian \(H\) on \(M\), we can define an isotopy \((\Theta(H))_{t}\) on \(P\) as constructed above. Note that since \(P\) is closed, \((\Theta(H))_{t}\) is uniformly continuous on \(P\). Let \(\widehat{(\Theta(H))_{t}}:S^{1}{\mathbb{D}}\rightarrow\ S^{1}{\mathbb{D}}\) be a lift of \((\Theta(H))_{t}\) from \(P\) to \(S^{1}{\mathbb{D}}\). Thus, for every \(v\in S^{1}{\mathbb{D}}\) we can define a curve in \(S^{1}\), \(\gamma^{(H,v)}:[0,1]\to S^{1}\), by
+
+\[\gamma^{(H,v)}(t):=p_{\infty}(\widehat{(\Theta(H))_{t}}(v)).\]
+
+Parameterize \(S^{1}\) by \({\mathbb{R}}/{\mathbb{Z}}\) and let \(\widetilde{\gamma^{(H,v)}}\) be a lift to \({\mathbb{R}}\) of \(\gamma^{(H,v)}\). Define
+
+\[Rot(H,v):=\widetilde{\gamma^{(H,v)}}(1)-\widetilde{\gamma^{(H,v)}}(0).\]
+
+Note that \((\Theta(-))_{t}\) is continuous with respect to the \(C^{2}\)-topology, hence so is \(Rot(-,v)\) as a composition of continuous maps. Thereby, we can find \(\delta>0\), such that if \(H^{\prime}\in C^{\infty}\) and \(\|H-H^{\prime}\|_{C^{2}}<\delta\), then for every \(v\in S^{1}{\mathbb{D}}\),
+
+\[|Rot(H,v)-Rot(H^{\prime},v)|<1.\]
+
+Denote by \(\widetilde{\pi}\) the projection from \(S^{1}{\mathbb{D}}\) to \({\mathbb{D}}\). Now, define for every \(\widetilde{x}\in{\mathbb{D}}\)
+
+\[\widetilde{angle}(H,\widetilde{x})=-\inf_{\widetilde{\pi}(v)=\widetilde{x}}\lfloor Rot(H,v)\rfloor,\]
+
+where \(\lfloor x\rfloor\) is the integer part of \(x\). It is shown in [18] that if \(\widetilde{\pi}(v)=\widetilde{\pi}(w)\) then
+
+\[|\lfloor Rot(H,v)\rfloor-\lfloor Rot(H,w)\rfloor|\leq 2.\]
+
+Hence, for \(v\in S^{1}{\mathbb{D}}\) such that \(\widetilde{\pi}(v)=\widetilde{x}\)
+
+\[|\widetilde{angle}(H,\widetilde{x})-(-\lfloor Rot(H,v)\rfloor)|\leq 2.\]
+
+Obviously \(|\lfloor x\rfloor-x|\leq 1\), so altogether we obtain that
+
+\[|\widetilde{angle}(H,\widetilde{x})-\widetilde{angle}(H^{\prime},\widetilde{x})|\leq\]
+
+\[|\widetilde{angle}(H,\widetilde{x})-(-Rot(H,v))|+|Rot(H,v)-Rot(H^{\prime},v)|+\]
+
+\[+|(-Rot(H^{\prime},v))-\widetilde{angle}(H^{\prime},\widetilde{x})|\]
+
+\[\leq 3+1+3=7.\]
+
+The function \(\widetilde{angle}(H,-)\) is invariant by the action of the fundamental group of \(M\), so we can define a measurable bounded function \(angle(H,-)\) on \(M\). Define
+
+\[\mu_{1}(\phi_{H}):=\int_{M}angle(H,-)\omega.\]
+
+Note that
+
+\[|\mu_{1}(H)-\mu_{1}(H^{\prime})|\leq\int_{M}|angle(H,-)-angle(H^{\prime},-)|\omega\leq 7\cdot Vol(M).\]
+
+We will sum up the result.
+
+**Proposition 8.5****.**: _There exists a constant \(K\) (\(=7\cdot Vol(M)\)) such that for \(H\in C^{\infty}(M)\) there exists \(\delta>0\) such that for any \(H^{\prime}\in C^{\infty}(M)\) that satisfies \(\|H-H^{\prime}\|_{C^{2}}<\delta\) we have_
+
+\[|\mu_{1}(\phi_{H})-\mu_{1}(\phi_{H^{\prime}})|\leq K.\]
+
+**Definition 8.6****.**: For \(m\in{\mathbb{N}}\) and \(\phi_{H}\in\mbox{Ham}(M,\omega)\) define
+
+\[\mu_{m}(\phi_{H}):=\frac{1}{m}\mu_{1}(\phi_{H}^{m}).\]
+
+**Proposition 8.7****.**: _For \(H\in C^{\infty}(M)\) and \(m\in{\mathbb{N}}\), there exists \(\delta>0\) such that for \(H^{\prime}\in C^{\infty}(M)\) that satisfies \(\|H-H^{\prime}\|_{C^{2}}<\delta\) we have_
+
+\[|\mu_{m}(\phi_{H})-\mu_{m}(\phi_{H^{\prime}})|\leq\frac{K}{m}.\]
+
+Proof.: Let \(H\in C^{\infty}(M)\) and \(m\in{\mathbb{N}}\). Using Proposition 8.5 for the function \(mH\), there exists \(\delta^{\prime}>0\) such that for every \(G\in C^{\infty}(M)\) that satisfies \(\|G-mH\|_{C^{2}}<\delta^{\prime}\) we have \(|\mu_{1}(\phi_{G})-\mu_{1}(\phi_{mH})|\leq K\). Choose \(\delta:=\frac{\delta^{\prime}}{m}\) so for \(H^{\prime}\in C^{\infty}(M)\) such that \(\|H^{\prime}-H\|_{C^{2}}<\delta\) we have \(\|mH^{\prime}-mH\|_{C^{2}}<\delta^{\prime}\) which implies \(|\mu_{1}(\phi_{mH^{\prime}})-\mu_{1}(\phi_{mH})|\leq K\). With the observation that \(\phi_{mH}=\phi_{H}^{m}\) we have \(|\mu_{1}(\phi_{H^{\prime}}^{m})-\mu_{1}(\phi_{H}^{m})|\leq K\). Dividing by \(m\) we obtain
+
+\[|\mu_{m}(\phi_{H^{\prime}})-\mu_{m}(\phi_{H})|<\frac{K}{m},\]
+
+as required. ∎
+
+**Proposition 8.8****.**: _Let \(\mu_{1}:G\rightarrow{\mathbb{R}}\) be a quasi-morphism on a group \(G\) with defect \(C>0\). Define for \(m\in{\mathbb{N}}\) and \(x\in G\), \(\mu_{m}(x):=\frac{1}{m}\mu_{1}(x^{m})\). Let \(\mu_{\infty}(x):=\lim_{n\rightarrow\infty}\mu_{m}(x)\) be the homogenization of \(\mu_{1}\). Then for every \(x\in G\) and \(m\in{\mathbb{N}}\)_
+
+\[|\mu_{\infty}(x)-\mu_{m}(x)|\leq\frac{C}{m}.\]
+
+Proof.: Let \(x\in G\), \(m,p\in{\mathbb{N}}\). By the quasi-morphism property we get
+
+\[|\mu_{1}(x^{mp})-\mu_{1}(x^{m})-\mu_{1}(x^{m(p-1)})|<C.\]
+
+Using induction on \(p\) we obtain
+
+\[|\mu_{1}(x^{mp})-p\cdot\mu_{1}(x^{m})|<pC.\]
+
+Divide by \(mp\)
+
+\[|\frac{\mu_{1}(x^{mp})}{mp}-\frac{\mu_{1}(x^{m})}{m}|<\frac{C}{m}.\]
+
+Equivalently,
+
+\[|\mu_{mp}(x)-\mu_{m}(x)|<\frac{C}{m}.\]
+
+As \(p\) tends to infinity we get
+
+\[|\mu_{\infty}(x)-\mu_{m}(x)|\leq\frac{C}{m},\]
+
+as required. ∎
+
+**Theorem 8.9****.**: _The functional \(\xi:C^{\infty}(M)\rightarrow{\mathbb{R}}\) (see Definition 1.10) is continuous with respect to the \(C^{2}\)-topology._
+
+Proof.: Let \(H\in C^{\infty}(M)\) and \(\epsilon>0\). Let \(C>0\) be the defect of \(\mu_{1}\) and \(K>0\) the constant defined in Proposition 8.5. Choose \(N\in{\mathbb{N}}\) such that \(N>max(\frac{4C}{\epsilon},\frac{2K}{\epsilon})\). By Proposition 8.7 there exists \(\delta>0\) such that for \(H^{\prime}\) that satisfies \(\|H-H^{\prime}\|_{C^{\infty}}<\delta\) we get
+
+\[|\mu_{N}(\phi_{H})-\mu_{N}(\phi_{H^{\prime}})|<\frac{K}{N}.\]
+
+According to Proposition 8.8 we have
+
+\[|\mu_{\infty}(\phi_{H})-\mu_{N}(\phi_{H})|<\frac{C}{N}\]
+
+and
+
+\[|\mu_{\infty}(\phi_{H^{\prime}})-\mu_{N}(\phi_{H^{\prime}})|<\frac{C}{N}.\]
+
+Thus
+
+\[|\mu_{\infty}(\phi_{H})-\mu_{\infty}(\phi_{H^{\prime}})|\leq\]
+
+\[|\mu_{\infty}(\phi_{H})-\mu_{N}(\phi_{H})|+|\mu_{N}(\phi_{H})-\mu_{N}(\phi_{H^{\prime}})|+|\mu_{\infty}(\phi_{H^{\prime}})-\mu_{N}(\phi_{H^{\prime}})|<\]
+
+\[<\frac{2C}{N}+\frac{K}{N}<\epsilon.\]
+
+Py’s quasi-morphism \(\mu\) is defined to be \(\mu_{\infty}\), so the result follows from Definition 1.10 of \(\xi.\) ∎
+
+**Acknowledgements. I would like to express my sincere thanks to my thesis advisor, Professor Leonid Polterovich, for his dedicated and patient guiding, and for the time he spent sharing with me his knowledge and expertise. I would like to thank the Israel Science Foundation grant # 11/03 which partially supported this work.**
+
+## References
+
+* [1] J.F. Aarnes, _Physical states on a \(C^{*}\)-algebra._ Acta Math. 122, 161-172 (1969).
+* [2] J.F. Aarnes, _Quasi-states on \(C^{*}\)-algebras._ Trans. Amer. Math. Soc. 149, 601-625 (1970).
+* [3] J.F. Aarnes, _Quasi-states and quasi-measures._ Adv. Math 86, 41-67 (1991).
+* [4] A. Banyaga, _The group of diffeomorphisms preserving a regular contact form._ Topology and algebra (Proc. Colloq., Eidgenoss. Tech. Hochsch., Zurich, 1977), volume 26 of Monographs. Enseign. Math., pages 47-53. Univ. Genève, 1978.
+* [5] A. Banyaga, _Sur la structure du groupe des diffémorphismes qui préservent une forme symplectique._ Comment. Math. Helv. 53(no.2) :174-227, 1978.
+* [6] P. Biran, M. Entov, L. Polterovich, _Calabi quasimorphisms for the symplectic ball._ Commun. Contemp. Math., 6 :793-802, 2004.
+* [7] J. Buser, _Geometry and Spectra of Compact Riemann Surfaces._ Progress in Mathematics, Springer; 1 edition, 1992.
+* [8] A. J. Casson, S. A. Bleiler, _Automorphisms of Surfaces after Nielsen and Thurston._ London Mathematical Society Student Texts, 1988.
+* [9] K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci _Loops in Reeb Graphs of 2-Manifolds._ Discrete Comput. Geom. 32:231-244 (2004).
+* [10] M. Entov, L. Polterovich, _Calabi quasimorphisms and quantom homology._ Int. Math. Res. Not., (no.30) :1635-1676, 2003.
+* [11] M. Entov, L. Polterovich, _Quasi-states and symplectic intersections._ Comment. Math. Helv. 81 :75-99, 2006.
+* [12] Hall. P, _On Representatives of Subsets._ J. London Math. Soc. 10, 26-30, 1935.
+* [13] D. Kotschick, What is … a quasi-morphism? _Notices Amer. Math. Soc._ 51, 208-209, 2004.
+* [14] D. McDuff, D. Salamon, _Introduction to symplectic topology._ Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, second edition, 1998.
+* [15] Milnor, J. W. _Morse Theory._ Princeton, NJ: Princeton University Press, 1963.
+* [16] Y. Ostrover, _Calabi quasi-morphisms for some non-monotone symplectic manifolds._ Algebraic & Geometric Topology 6 (2006) 405-434.
+* [17] L. Polterovich, _Geometry of the Group of Symplectic Diffeomorphisms._ Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 2001.
+* [18] P. Py, _Quasi-morphismes et invariant de Calabi._ Ann. Sci. École Norm. Sup. 39, no.1 ,177-195 (2006).
+* [19] P. Py, _Quasi-morphismes de Calabi et graphe de Reeb sur le tore._ C. R. Acad. Sci. Paris, Ser. I vol.343 323-328 (2006).
+* [20] G. Reeb, _Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique._ C.R. Acad. Sci. Paris, 222 :847-849, 1946.
+

arxiv_ground_truth/0706.0954.md

@@ -0,0 +1,814 @@
+# Growth and mixing
+
+Krzysztof Frączek
+
+Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland, Institute of Mathematics Polish Academy of Science, Śniadeckich 8, 00-956 Warszawa, Poland
+fraczek@mat.uni.torun.pl
+
+Leonid Polterovich
+
+School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
+polterov@post.tau.ac.il
+
+(Date: July 2, 2024)
+
+###### Abstract.
+
+Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin-Shapiro sequence.
+
+Key words and phrases: Growth rate of homeomorphism, the rate of mixing 1991 Mathematics Subject Classification: 37A05, 37A25, 37C05 Research partially supported by Marie Curie "Transfer of Knowledge" program, project MTKD-CT-2005-030042 (TODEQ)
+
+## 1. Introduction and main results
+
+Let \((M,\rho,\mu)\) be a compact metric space endowed with a probability Borel measure \(\mu\) with \(\text{supp}(\mu)=M\). Denote by \(G\) the group of all bi-Lipschitz homeomorphisms of \((M,\rho)\) which preserve the measure \(\mu\). For \(\phi\in G\) write \(\Gamma(\phi)=\Gamma_{\rho}(\phi)\) for the maximum of the Lipschitz constants of \(\phi\) and \(\phi^{-1}\). Note that \(\Gamma(\phi)\) is a sub-multiplicative: \(\Gamma(\phi\psi)\leq\Gamma(\phi)\cdot\Gamma(\psi)\). Thus \(\log\Gamma\) is a pseudo-norm on \(G\), which enables us to consider the group \(G\) as a geometric object. In the present note we discuss a link between dynamics of \(\phi\in G\) (the rate of mixing) and geometry of the cyclic subgroup of \(G\) generated by \(\phi\) (the growth rate of \(\Gamma(\phi^{n})\) as \(n\to\infty\).) On the geometric side, we focus on the quantity
+
+\[{\widehat{\Gamma}}_{n}(\phi):=\max_{i=1,\ldots,n}\Gamma(\phi^{i})\;.\]
+
+**Notations****.**: We write \((f,g)_{L_{2}}\) for the \(L_{2}\)-scalar product on \(L_{2}(M,\mu)\). We denote by \(E\) the space of all Lipschitz functions on \(M\) with zero mean with respect to \(\mu\). We write \(||f||_{L_{2}}\) for the \(L_{2}\)-norm of a function \(f\), \({\operatorname{Lip}}(f)\) for the Lipschitz constant of \(f\) and \(||f||_{\infty}\) for the uniform norm of \(f\).
+
+**Definition 1.1****.**: We say that a diffeomorphism \(\phi\in G\)_mixes_ a function \(f\in E\) if \((f\circ\phi^{n},f)_{L_{2}}\to 0\) as \(n\to\infty\).
+
+It is known that there exist volume-preserving diffeomorphisms \(\phi\) of certain smooth closed manifolds \(M\) with arbitrarily slow growth of \({\widehat{\Gamma}}_{n}(\phi),n\to\infty\) (see e.g. Borichev [3] for \(M={\mathbb{T}}^{2}\) and Fuchs [9] for extension of Borichev’s results to manifolds admitting an effective \({\mathbb{T}}^{2}\)-action). As we shall see below, the situation changes if we assume that \(\phi\) mixes a Lipschitz function: in this case the growth rate of \({\widehat{\Gamma}}_{n}(\phi)\) admits a universal lower bound. Furthermore the bound becomes better provided the rate of mixing is decaying sufficiently fast.
+
+To state our first result we need the following invariant of the metric space \((M,\rho)\). Denote by \(E_{R,C}\), where \(R,C\geq 0\), the subset of functions \(f\in E\) with \({\operatorname{Lip}}(f)\leq R\) and \(\|f\|_{\infty}\leq C\). By the Arzela-Ascoli theorem \(E_{R,C}\) is compact with respect to the uniform norm. Denote by \(D(R,\epsilon,C)\) the minimal number of \(\epsilon/2\)-balls (in the uniform norm) needed to cover \(E_{R,C}\). Note that for fixed \(\epsilon\) and \(C\) the function \(D(R,\epsilon,C)\) is non-decreasing with \(R\). For \(t\geq D(0,1.4,C)=[C/0.7]+1\) put
+
+\[\tau(t,C):=\sup\{R\geq 0\;:\;D(R,1.4,C)\leq t\}\;.\]
+
+**Theorem 1.2****.**: _Assume that a bi-Lipschitz homeomorphism \(\phi\in G\) mixes a function \(f\in E\) with \(||f||_{L_{2}}=1\). Then there exists \(\alpha>0\) so that_
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\frac{\tau(\alpha n,\|f\|_{\infty})}{{\operatorname{Lip}}(f)}\]
+
+_for all sufficiently large \(n\)._
+
+The proof is given in Section 2.
+
+For a compact subset \(A\) of a metric space \((X,\rho_{1})\) and \(\epsilon>0\) denote by \(\mathcal{N}_{\epsilon}(A)\) the minimal number of open balls with radius \(\epsilon/2\) such that their union covers \(A\). Then the upper box dimension of \((A,\rho_{1})\) is defined as
+
+(1) \[\overline{\dim}_{B}(A)=\overline{\lim_{\epsilon\to 0}}\frac{\log\mathcal{N}_{\epsilon}(A)}{\log 1/\epsilon}.\]
+
+Let \((Y,\rho_{2})\) be a compact metric space and let \(\mathcal{D}^{A}_{R}(Y)\subset Y^{A}\) stand for the set of Lipschitz functions \(f:A\to Y\) with \({\operatorname{Lip}}(f)\leq R\), where \(Y^{A}\) is equipped with the uniform distance
+
+\[\text{dist}(f,g)=\sup_{x\in A}\rho_{2}(f(x),g(x))\;.\]
+
+It is easy to show (the proof is analogous to that of Theorem XXV in [11])
+
+(2) \[\mathcal{N}_{\epsilon}(\mathcal{D}^{A}_{R}(Y))\leq\mathcal{N}_{\epsilon/4}(Y)^{\mathcal{N}_{\epsilon/(4R)}(A)}.\]
+
+For the reader’s convenience, we present the proof in the Appendix.
+
+Assume now that the metric space \((M,\rho)\) satisfies the following condition:
+
+**Condition 1.3****.**: There exist positive numbers \(d\) and \(\kappa\) so that for every \(\delta>0\) one can find a \(\delta\)-net in \((M,\rho)\) consisting of at most \(\kappa\cdot\delta^{-d}\) points.
+
+This condition is immediately verified if \((M,\rho)\) is a smooth manifold of dimension \(d\) or if \(d>\overline{\dim}_{B}(M)\). Moreover, it is satisfied for some fractal sets \(M\subset{\mathbb{R}}^{n}\) where \(d\) is the fractal dimension \(M\), e.g. if \(M\) is a self–similar set (see Theorem 9.3 [5]).
+
+In what follows \([\alpha]\) denotes the integer part of \(\alpha\in{\mathbb{R}}\). Assume that Condition 1.3 holds. Since \(E_{R,C}=\mathcal{D}^{M}_{R}([-C,C])\), by (2), we have
+
+\[D(R,\epsilon,C)\leq\left(\left[\frac{4C}{\epsilon}\right]+1\right)^{\mathcal{N}_{\epsilon/(4R)}(M)}\\ \leq\left(\left[\frac{4C}{\epsilon}\right]+1\right)^{\kappa(\epsilon/(4R))^{-d}}.\]
+
+Therefore \(\tau(t,C)\geq\text{const}\cdot\log^{1/d}t.\) Thus Theorem 1.2 above yields the following:
+
+**Corollary 1.4****.**: _If \(\phi\in G\) mixes a Lipschitz function then there exists \(\lambda>0\) so that_
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\lambda\cdot\log^{\frac{1}{d}}n\]
+
+_for all sufficiently large \(n\)._
+
+This contrasts sharply with the situation when the growth of the sequence \(\Gamma(\phi^{n})\) is taken under consideration. In fact, for every slowly increasing function \(u:[0;+\infty)\to[0;+\infty)\) there exists a volume-preserving real-analytic diffeomorphism of the \(3\)–torus which mixes a real-analytic function and such that \(\Gamma(\phi^{n})\leq{\rm const}\cdot u(n)\) for infinitely many \(n\). Such diffeomorphisms are presented in Section 6.
+
+As a by-product of our proof of Theorem 1.2 we get the following result. Let \(\phi\) be a bi-Lipschitz homeomorphism of a compact metric space \(M\) satisfying Condition 1.3.
+
+**Theorem 1.5****.**: _If_
+
+(3) \[\liminf_{n\to\infty}\frac{{\widehat{\Gamma}}_{n}(\phi)}{\log^{1/d}n}=0\]
+
+_then the cyclic subgroup \(\{\phi^{n}\}\) has the identity map as its limit point with respect to \(C^{0}\)-topology._
+
+This theorem has the following application to bi-Lipschitz ergodic theory (the next discussion is stimulated by correspondence with A. Katok). Let \(T\) be an automorphism of a probability space \((X,\sigma)\). A _bi-Lipschitz realization_ of \((X,T,\sigma)\) is a metric isomorphism between \((X,T,\sigma)\) and \((M,\phi,\mu)\), where \(\phi\) is a bi-Lipschitz homeomorphism of a compact metric space \(M\) equipped with a Borel probability measure \(\mu\). An objective of bi-Lipschitz ergodic theory is to find restrictions on bi-Lipschitz realizations of various classes of dynamical systems \((X,T,\sigma)\). The class of interest for us is given by _non-rigid automorphisms_ which is defined as follows: Denote by \(U_{T}\) the induced Koopman operator \(f\mapsto f\circ T\) of \(L_{2}(X,\sigma)\). We say that \(T\) is _non-rigid_[10] if the closure of the cyclic subgroup generated by \(U_{T}\) with respect to strong operator topology does **not** contain the identity operator. Theorem 1.5 shows that _any bi-Lipschitz homeomorphism \(\phi\) satisfying condition (3) cannot serve as a bi-Lipschitz realization of a non-rigid dynamical system._
+
+Let us return to the study of the interplay between growth and mixing: Next we explore the influence of the rate of mixing on the growth of \({\widehat{\Gamma}}_{n}(\phi)\). We shall need the following definitions.
+
+**Definition 1.6****.**: Let \(\{a_{n}\}_{n\in{\mathbb{N}}}\) be a sequence of positive numbers converging to zero as \(n\to\infty\). We say that a diffeomorphism \(\phi\in G\)_mixes_ a function \(f\in E\) at the rate \(\{a_{n}\}\) if
+
+\[|(f\circ\phi^{n},f)_{L_{2}}|\leq a_{n}\;\forall n\in{\mathbb{N}}.\]
+
+Given a positive sequence \(a_{n}\to 0\), we call a positive integer sequence \(\{v(n)\}\)_adjoint_ to \(\{a_{n}\}\) if the following conditions hold:
+
+(4) \[\sum_{i:0<iv(n)\leq n}a_{iv(n)}\leq\frac{1}{4}\;,\]
+
+and
+
+(5) \[\frac{n}{v(n)}\to\infty\;\;\text{as}\;\;n\to\infty\;.\]
+
+**Lemma 1.7****.**: _Every positive sequence \(a_{n}\to 0\) admits an adjoint sequence._
+
+The proof is given in Section 5.
+
+In the next theorem we assume that the metric space \((M,\rho)\) satisfies Condition 1.3.
+
+**Theorem 1.8****.**: _Assume that a bi-Lipschitz homeomorphism \(\phi\in G\) mixes a Lipschitz function \(f\in E\) with \(||f||_{L_{2}}=1\) at the rate \(\{a_{n}\}\). Then for every adjoint sequence \(\{v(n)\}\) of \(\{a_{n}\}\) we have_
+
+(6) \[{\widehat{\Gamma}}_{n}(\phi)\geq\frac{1}{2\kappa^{\frac{1}{d}}{\operatorname{Lip}}(f)}\cdot\Big{[}\frac{n}{2v(n)}\Big{]}^{1/d}\;\;\forall n\in{\mathbb{N}}\;.\]
+
+_In particular, if \(\sum a_{i}<\infty\) then_
+
+(7) \[{\widehat{\Gamma}}_{n}(\phi)\geq\text{const}\cdot n^{\frac{1}{d}}\;.\]
+
+Note that the second part of the theorem is an immediate consequence of the first part. Indeed, if \(\sum a_{i}<\infty\) then the adjoint sequence can be taken constant, \(v(n)\equiv v_{0}\) and (6) implies (7).
+
+As we shall show in Section 7 below the estimate (7) is asymptotically sharp: It is attained for the shift associated with the Rudin-Shapiro sequence.
+
+**Corollary 1.9****.**: _Suppose that \(\phi\in G\) mixes a Lipschitz function at the rate \(\{a_{n}\}\) such that \(a_{n}=O(1/n^{\nu})\), where \(0<\nu<1\). Then_
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\text{const}\cdot n^{\frac{\nu}{d}}.\]
+
+Proof.: If \(a_{n}\leq c/n^{\nu}\) for some \(\nu\in(0;1)\) then one readily checks that for \(C>0\) large enough there exists a sequence \(\{v(n)\}\) adjoint to \(\{a_{n}\}\) such that \(v(n)\leq C\cdot n^{1-\nu}\). Thus
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\frac{1}{2\kappa^{\frac{1}{d}}{\operatorname{Lip}}(f)}\cdot\Big{[}\frac{n}{2Cn^{1-\nu}}\Big{]}^{1/d}\geq\text{const}\cdot n^{\frac{\nu}{d}}.\]
+
+Organization of the paper: In Section 2 we prove the universal lower growth bound given in Theorem 1.2 for a bi-Lipschitz homeomorphism which mixes a Lipschitz function (the case of homeomorphism which mixes an \(L_{2}\)–function is also considered). Furthermore, we prove Theorem 1.5 asserting that if a bi-Lipschitz homeomorphism grows sufficiently slow, it must have strong recurrence properties and in particular must be rigid in the sense of ergodic theory. The section ends with a discussion on comparison of growth rates in finitely generated groups and in groups of homeomorphisms. In Section 3 we prove Theorem 1.8 which relates the growth rate to the rate of mixing. For the proof, we derive an auxiliary fact on "almost orthonormal" sequences of Lipschitz functions. In Section 4 we generalize the main results of the paper to the case of Hölder observables. In Section 5 we prove existence of adjoint sequences used in the formulation of Theorem 1.8.
+
+Next we pass to constructing examples. In Section 6 we present an example which emphasizes the difference between the growth rates of sequences \({\widehat{\Gamma}}_{n}(\phi)\) and \(\Gamma(\phi^{n})\): We construct a volume-preserving real-analytic diffeomorphism of the \(3\)-torus which mixes a real-analytic function and such that \(\Gamma(\phi^{n_{i}})\) grows arbitrarily slowly along a suitable subsequence \(n_{i}\to\infty\). In Section 7 we show that the bound in Theorem 1.8 is sharp: It is attained in the case of a symbolic dynamical system associated to the Rudin-Shapiro sequence.
+
+Finally, in Appendix we prove Kolmogorov-Tihomirov type estimate (2).
+
+## 2. Recurrence via Arzela-Ascoli compactness
+
+Proof of Theorem 1.2.: Suppose that the assertion of the theorem is false. Then, considering a sequence \(\alpha_{k}=1/k\), \(k\in{\mathbb{N}}\) we get a sequence \(\{n_{k}\}\) so that \(n_{k}/k\geq[\|f\|_{\infty}/0.7]+1\) and
+
+\[R_{k}:={\operatorname{Lip}}(f)\cdot{\widehat{\Gamma}}_{n_{k}}(\phi)<\tau(n_{k}/k,\|f\|_{\infty})\;.\]
+
+This yields
+
+\[D(R_{k},1.4,\|f\|_{\infty})\leq n_{k}/k<m+1,\]
+
+where \(m=[n_{k}/k]\geq 1\). Consider \(m+1\) functions
+
+\[f,f\circ\phi^{k},\ldots,f\circ\phi^{mk}\;.\]
+
+Since
+
+\[{\operatorname{Lip}}(g\circ\psi)\leq{\operatorname{Lip}}(g)\cdot\Gamma(\psi)\;\;\forall g\in E,\psi\in G\;,\]
+
+these functions lie in the subset \(E_{R_{k},\|f\|_{\infty}}\subset E\). Recall that \(E_{R_{k},\|f\|_{\infty}}\) can be covered by \(D(R_{k},1.4,\|f\|_{\infty})\leq m\) balls (in the uniform norm) of the radius \(0.7\). By the pigeonhole principle, there is a pair of functions from our collection lying in the same ball. In other words for some natural numbers \(p>q\) we have \(||f\circ\phi^{pk}-f\circ\phi^{qk}||_{\infty}\leq 1.4\). Put \(j=(p-q)k\). We have
+
+\[||f-f\circ\phi^{j}||_{L_{2}}\leq||f-f\circ\phi^{j}||_{\infty}\leq 1.4\;.\]
+
+Since
+
+\[||f||_{L_{2}}=||f\circ\phi^{j}||_{L_{2}}=1,\]
+
+we have
+
+\[(f,f\circ\phi^{j})_{L_{2}}=\frac{1}{2}(||f||_{L_{2}}^{2}+||f\circ\phi^{j}||_{L_{2}}^{2}-||f-f\circ\phi^{j}||_{L_{2}}^{2})\geq\frac{1}{2}(1+1-1.4^{2})=0.02\;.\]
+
+Note that \(j\geq k\) and thus increasing \(k\) we get the above inequality for arbitrarily large values of \(j\). This contradicts the assumption that \(\phi\) mixes \(f\).
+
+Denote by \(H\) the group of all bi-Lipschitz homeomorphisms (not necessarily measure preserving) of a compact metric space \((M,\rho)\). An argument similar to the one used in the proof above shows that if the growth rate of \({\widehat{\Gamma}}_{n}(\phi)\) is sufficiently slow, the cyclic subgroup \(\{\phi^{n}\}\) generated by \(\phi\) has the identity map as its limit point with respect to \(C^{0}\)-topology (cf. a discussion in D’Ambra-Gromov [2, 7.10.C,D]). Here is a precise statement. Denote by \(\Lambda\) the space of Lipschitz self-maps of \(M\). For \(\phi\in\Lambda\) write \({\operatorname{Lip}}(\psi)\) for the Lipschitz constant of \(\psi\). Equip \(\Lambda\) with the \(C^{0}\)-distance
+
+\[\text{dist}(\phi,\psi)=\sup_{x\in M}\rho(\phi(x),\psi(x))\;.\]
+
+Denote by \(\Lambda_{R}\) the subset consisting of all maps \(\psi\) from \(\Lambda\) with \({\operatorname{Lip}}(\psi)\leq R\). This subset is compact with respect to the metric dist by the Arzela-Ascoli theorem. Denote by \(\Delta(R,\epsilon)\) the minimal number of \(\epsilon/2\)-balls required to cover \(\Lambda_{R}\). For \(t\geq\Delta(0,\epsilon)=\mathcal{N}_{\epsilon}(M)\) put
+
+\[\theta(t,\epsilon)=\sup\{R\geq 0\;:\;\Delta(R,\epsilon)\leq t\}\;.\]
+
+**Theorem 2.1****.**: _Let \(\phi:M\to M\) be a bi-Lipschitz homeomorphism. Assume that the identity map is not a limit point with respect to \(C^{0}\)-topology for the cyclic subgroup \(\{\phi^{n}\}\). Then for every sequence \(\epsilon_{n}\to 0\) there exists \(\alpha>0\) so that_
+
+\[\widehat{\Gamma}_{n}(\phi)\geq\theta(\alpha n,\epsilon_{n})\]
+
+_for all sufficiently large \(n\)._
+
+Proof.: Suppose that the assertion of the theorem is false. For every \(\alpha=1/k\), \(k\in{\mathbb{N}}\) we can choose \(n_{k}>\max(\mathcal{N}_{\epsilon_{k}}(M),k)\) so that
+
+\[\widehat{\Gamma}_{n_{k}}(\phi)<\theta(n_{k}/k,\epsilon_{k}).\]
+
+Put \(m_{k}=[n_{k}/k]\) and \(R_{k}={\widehat{\Gamma}}_{n_{k}}(\phi)\). Since \(R_{k}<\theta(n_{k}/k,\epsilon_{k})\), we obtain
+
+\[\Delta(R_{k},\epsilon_{k})\leq n_{k}/k<m_{k}+1.\]
+
+Consider \(m_{k}+1\) maps \({\text{{\bf 1}}},\phi^{k},\ldots,\phi^{km_{k}}\). They lie in \(\Lambda_{R_{k}}\). Since \(\Delta(R_{k},\epsilon_{k})\leq m_{k}\), it follows that at least two of these maps lie in the same \(\epsilon_{k}/2\)-ball covering of \(\Lambda_{R_{k}}\). Therefore there exist \(p>q\) so that
+
+\[\text{dist}(\phi^{pk},\phi^{qk})\leq\epsilon_{k}\;.\]
+
+Put \(l_{k}=(p-q)k\), and note that \(\text{dist}(\phi^{pk},\phi^{qk})=\text{dist}({\text{{\bf 1}}},\phi^{l_{k}})\). Thus \(\text{dist}({\text{{\bf 1}}},\phi^{l_{k}})\leq\epsilon_{k}\), and since \(k\) divides \(l_{k}\) we have \(l_{k}\to\infty\). We conclude that \(\phi^{l_{k}}\to{\text{{\bf 1}}}\;,\) which contradicts the fact that the identity map is not a limit point (with respect to \(C^{0}\)-topology) for the sequence \(\{\phi^{n}\}\).
+
+**Remark 2.2****.**: Assume that the metric space \((M,\rho)\) satisfies Condition 1.3 with exponent \(d>0\). Since \(\Lambda_{R}=\mathcal{D}^{R}_{M}(M)\), by (2), we have
+
+\[\Delta(R,\epsilon)\leq\mathcal{N}_{\epsilon/4}(M)^{\mathcal{N}_{\epsilon/(4R)}(M)}\leq(\kappa(\epsilon/4)^{-d})^{\kappa(\epsilon/(4R))^{-d}}.\]
+
+Thus
+
+\[\theta(t,\epsilon)\geq\text{const}\frac{\epsilon\cdot\log^{1/d}t}{\log^{1/d}1/\epsilon}\;.\]
+
+**Corollary 2.3****.**: _Let \(\phi:M\to M\) be a bi-Lipschitz homeomorphism, where \(M\) satisfies Condition 1.3. Assume that the identity map is not a limit point with respect to \(C^{0}\)-topology for the cyclic group \(\{\phi^{n}\}\). Let \(\{\eta(n)\}\) be a sequence of positive numbers such that \(\eta(n)\to+\infty\) as \(n\to+\infty\) and \(\eta(n)=o(\log n)\). Then \({\widehat{\Gamma}}_{n}(\phi)\geq\eta(n)^{1/d}\) for all sufficiently large \(n\)._
+
+Proof.: An application of Theorem 2.1 for \(\epsilon_{n}=(\eta(n)/\log n)^{\frac{1}{2d}}\) gives the existence of \(\alpha>0\) for which
+
+\[{\widehat{\Gamma}}_{n}(\phi) \geq \theta(\alpha n,\epsilon_{n})\geq\text{const}\frac{\epsilon_{n}\cdot\log^{1/d}\alpha n}{\log^{1/d}1/\epsilon_{n}}\geq\text{const}\frac{\left(\frac{\eta(n)}{\log n}\right)^{\frac{1}{2d}}\cdot\log^{1/d}n}{\log^{1/d}\frac{\log n}{\eta(n)}}\]
+\[= \text{const}\frac{\left(\frac{\log n}{\eta(n)}\right)^{\frac{1}{2d}}}{\log^{1/d}\frac{\log n}{\eta(n)}}\cdot\eta(n)^{1/d}\geq\eta(n)^{1/d}\]
+
+for all sufficiently large \(n\).
+
+Theorem 1.5 is an immediate consequence of Corollary 2.3.
+
+**Remark 2.4****.**: Consider _any_ group \(H\) equipped with a pseudo-norm \(\ell\): \(\ell(h)\geq 0\) all \(h\in H\), \(\ell(h^{-1})=\ell(h)\) and \(\ell(hg)\leq\ell(h)+\ell(g)\). For an element \(h\in G\) put
+
+\[{\widehat{\ell}}_{n}(h)=\max_{i=1,\ldots,n}\ell(h^{n})\;.\]
+
+It is instructive to compare possible growth rates of cyclic subgroups in the following two cases:
+
+* (i)\(H\) is a finitely generated group, \(\ell\) is the word norm;
+* (ii)\(H\) is the group of all bi-Lipschitz homeomorphisms equipped with the pseudo-norm \(\ell=\log\Gamma\).
+
+We claim that in the first case, condition
+
+(8) \[\liminf_{n\to\infty}\frac{{\widehat{\ell}}_{n}(\phi)}{\log n}=0\]
+
+is equivalent to the fact that \(\phi\) is of finite order. Indeed, assume that \(\phi\) satisfies (8). Denote by \(H_{R}\subset H\) the ball of radius \(R\) centred at \(\phi\) in the word norm. Denote by \(K\) the number of elements in the generating set of \(H\). Then the cardinality of \(H_{R}\) does not exceed \(K^{R+1}\). Condition (8) guarantees that there exists \(n>0\) arbitrarily large so that \({\widehat{\ell}}_{n}(\phi)\leq\log n/(2\log K)\). Consider \(n+1\) elements \({\text{{\bf 1}}},\phi,\ldots,\phi^{n}\). All of them lie in the set \(H_{R}\) with \(R={\widehat{\ell}}_{n}(\phi)\). This set contains at most \(K^{R+1}\leq K\sqrt{n}\) elements. Since \(K\sqrt{n}<n+1\) for large \(n\) we get that among \({\text{{\bf 1}}},\phi,\ldots,\phi^{n}\) there are at least two equal elements, hence \(\phi\) is of finite order. The claim follows.
+
+In contrast to this, in the case (ii), the group of bi-Lipschitz homeomorphisms may have elements of infinite order which satisfy (8), see [3, 9]. These elements are "exotic" from the algebraic viewpoint: they cannot be included into any finitely generated subgroup \(H^{\prime}\) of \(H\) so that the inclusion
+
+\[(H^{\prime},\text{word\;norm})\hookrightarrow(H,\log\Gamma)\]
+
+is quasi-isometric. It would be interesting to explore more thoroughly the dynamics of these exotic elements.
+
+Corollary 2.3 shows that if such an exotic element is of a "very slow" growth then it has strong recurrence properties. The argument based on the Arzela compactness, which was used in its proof, imitates the argument showing that condition (8) characterizes elements of finite order in finitely generated groups. Let us compare these results for bi-Lipschitz homeomorphisms of \(d\)-dimensional spaces. Consider such a homeomorphism, say, \(\phi\) with \({\widehat{\ell}}_{n}(\phi)=o(\log n)\), which means that it is algebraically exotic in the sense of the discussion above. If \(\phi\) satisfies a stronger inequality \({\widehat{\ell}}_{n}(\phi)\leq(\frac{1}{d}-\epsilon)\log\log n\), it is strongly recurrent by Corollary 2.3 above. We conclude this discussion with the following open problem: explore dynamical properties of those bi-Lipschitz homeomorphisms of \(d\)-dimensional spaces whose growth sequence \({\widehat{\ell}}_{n}(\phi)\) falls into the gap between \(\frac{1}{d}\log\log n\) and \(o(\log n)\).
+
+## 3. Almost orthonormal systems of Lipschitz functions
+
+In this section we prove Theorem 1.8. We start with the following general result on "almost orthonormal" systems of functions:
+
+**Theorem 3.1****.**: _Let \(\{f_{i}\}\) be a sequence of linear independent Lipschitz functions from \(E\) with \(||f_{i}||_{L_{2}}=1\) with the following property: There exists a sequence of positive real numbers \(a_{n}\to 0\) so that \(|(f_{i},f_{j})_{L_{2}}|\leq a_{i-j}\) for all \(j<i\). Let \(\{v(n)\}\) be an adjoint sequence of \(\{a_{n}\}\). Then_
+
+(9) \[\max_{i=1,\ldots,n}{\operatorname{Lip}}(f_{i})\geq\frac{1}{2\kappa^{\frac{1}{d}}}\cdot\Big{[}\frac{n}{2v(n)}\Big{]}^{1/d}\;\;\forall n\in{\mathbb{N}}\;.\]
+
+**Lemma 3.2****.**: _Let \(f_{i}\in L_{2}(M)\), \(i=1,\ldots,N\) be a sequence of functions with \(||f_{i}||_{L_{2}}=1\) for all \(i\) and \(|(f_{i},f_{j})_{L_{2}}|\leq\alpha_{i-j}\) for \(j<i\), where \(\sum_{i=1}^{N}\alpha_{i}\leq 1/4\). Then for every real numbers \(c_{1},\ldots,c_{N}\) we have_
+
+\[||\sum_{i=1}^{N}c_{i}f_{i}||_{L_{2}}^{2}\geq\frac{1}{2}\sum_{i=1}^{N}c_{i}^{2}\;.\]
+
+Proof.: Put
+
+\[h=\sum_{i=1}^{N}c_{i}f_{i}\;\text{ and }\;C=\sqrt{\sum_{i=1}^{N}c_{i}^{2}}\;.\]
+
+Then
+
+\[||h||_{L_{2}}^{2}=C^{2}+I\;,\]
+
+where \(I=2\sum_{j<i}c_{i}c_{j}(f_{i},f_{j})\). By the Cauchy-Schwarz inequality,
+
+\[|I|\leq 2\sum_{p=1}^{N}\sum_{j=1}^{N-p}|c_{j}|\cdot|c_{j+p}|\cdot\alpha_{p}\leq 2\cdot\frac{1}{4}\cdot C^{2}=C^{2}/2\;.\]
+
+Thus
+
+\[||h||_{L_{2}}^{2}\geq C^{2}-C^{2}/2=C^{2}/2\]
+
+as required.
+
+Proof of Theorem 3.1.: We shall assume that \(2v(n)\leq n\), otherwise the inequality (9) holds by trivial reasons. Put \(q(n)=[n/(2v(n))]\) and \(\delta=(\kappa/q(n))^{1/d}\). By the definition of \(\kappa\) and \(d\), there exists a \(\delta\)-net on \(M\) consisting of \(p\leq q(n)\) points. Denote by \(E^{\prime}\subset E\) the codimension \(p\) subspace consisting of all those functions which vanish at the points of the net.
+
+Let \(V\) be the linear span of the functions \(f_{iv(n)}\), \(i=1,\ldots,2q(n)\). Then the dimension of \(W:=V\cap E^{\prime}\) is \(\geq 2q(n)-p\geq q(n)\). It is well known [4, p.103] that there exists \(h\in W\) with
+
+(10) \[||h||_{\infty}\geq\sqrt{\dim W}||h||_{L_{2}}\;.\]
+
+Write \(h=\sum_{i=1}^{2q(n)}c_{i}f_{iv(n)}.\) Note that \(|(f_{iv(n)},f_{jv(n)})_{L_{2}}|\leq a_{(i-j)v(n)}\) for \(i<j\). Put \(\alpha_{i}=a_{iv(n)}\). By the definition of \(v(n)\), we have
+
+\[\sum_{i=1}^{2q(n)}\alpha_{i}\leq\frac{1}{4}\;,\]
+
+and hence by Lemma 3.2
+
+\[||h||_{L_{2}}^{2}\geq C^{2}/2\;,\;\text{with}\;\;C=\sqrt{\sum_{i=1}^{2q(n)}c_{i}^{2}}\;.\]
+
+We conclude from (10) that
+
+\[||h||_{\infty}\geq\frac{1}{\sqrt{2}}\cdot\sqrt{q(n)}\cdot C\;.\]
+
+Recall now that \(h\) vanishes at all the points of the \(\delta\)-net. Thus
+
+(11) \[{\operatorname{Lip}}(h)\geq||h||_{\infty}/\delta\geq\frac{1}{\sqrt{2}}\cdot\sqrt{q(n)}\cdot C\cdot(\kappa/q(n))^{-1/d}\;.\]
+
+Next, let us estimate \({\operatorname{Lip}}(h)\) from above. Put
+
+\[\Pi_{n}:=\max_{i=1,\ldots,n}{\operatorname{Lip}}(f_{i})\;.\]
+
+We have
+
+\[{\operatorname{Lip}}(h)={\operatorname{Lip}}\big{(}\sum_{i=1}^{2q(n)}c_{i}f_{iv(n)}\big{)}\leq\Pi_{n}\cdot\sqrt{2q(n)}\cdot C\;.\]
+
+Combining this inequality with lower bound (11) we get
+
+\[\Pi_{n}\geq\frac{1}{2\kappa^{\frac{1}{d}}}\cdot q(n)^{\frac{1}{d}}\;,\]
+
+as required.
+
+Reduction of Theorem 1.8 to Theorem 3.1: We start with the following auxiliary lemma.
+
+**Lemma 3.3****.**: _Assume that \(\phi\in G\) mixes a function \(f\in E\). Then for every \(m>0\) the functions \(f,f\circ\phi,\ldots,f\circ\phi^{m}\) are linearly independent elements of \(E\)._
+
+Proof.: Assume that \(\|f\|_{L_{2}}=1\) and on the contrary that for some \(m\) these functions are linearly dependent. Then for some \(p\in{\mathbb{N}}\)
+
+\[f\circ\phi^{p}\in V:=\text{Span}(f,f\circ\phi,\ldots,f\circ\phi^{p-1})\]
+
+which implies that _every_ function of the form \(f\circ\phi^{n},n\in{\mathbb{Z}}\) belongs to \(V\). The space \(V\) is finite-dimensional and every element of the sequence \(\{f\circ\phi^{n}\},n\in{\mathbb{Z}}\) has unit \(L_{2}\)-norm. Thus this sequence has a subsequence converging to an element \(g\in V\) of unit \(L_{2}\)-norm. Since \(\phi\) mixes \(f\), we have \((g,f\circ\phi^{n})_{L_{2}}=0\) for every \(n\in{\mathbb{Z}}\). It follows that \(g=0\), contrary to \(\|g\|_{L_{2}}=1\). This completes the proof.
+
+Proof of Theorem 1.8.: Put \(f_{i}=f\circ\phi^{i},i\in{\mathbb{N}}\). Since \(\phi\) mixes \(f\) at the rate \(\{a_{i}\}\) we have \(|(f_{i},f_{j})_{L_{2}}|\leq a_{i-j}\) for all \(j<i\). The functions \(\{f_{i}\}\) are linearly independent by Lemma 3.3. Thus all the assumptions of Theorem 3.1 hold. Theorem 1.8 readily follows from Theorem 3.1 combined with the inequality
+
+\[\max_{i=1,\ldots,n}{\operatorname{Lip}}(f_{i})\leq{\widehat{\Gamma}}_{n}(\phi)\cdot{\operatorname{Lip}}(f)\;.\]
+
+**Remark 3.4****.**: Assume that \(\{f_{i}\}\) is an orthonormal system (in the \(L_{2}\)-sense) of Lipschitz functions with zero mean. Put
+
+\[\Pi_{n}:=\max_{i=1,\ldots,n}{\operatorname{Lip}}(f_{i})\;.\]
+
+It follows from Theorem 3.1 that
+
+\[\Pi_{n}\geq\text{const}\cdot n^{\frac{1}{d}}\;.\]
+
+For an illustration, consider the Euclidean torus \({\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}\). Let \(\lambda_{1}\leq\lambda_{2}\leq\ldots\) be the sequence of the eigenvalues (taken with their multiplicities) of the Laplace operator. Each \(\lambda_{n}\) has the form \(4\pi^{2}|v|^{2}\), where \(v\) runs over \({\mathbb{Z}}^{d}\setminus\{0\}\). Choose the sequence of eigenfunctions \(f_{n}\) corresponding to \(\lambda_{i}\) so that the eigenfunctions corresponding to \(4\pi^{2}|v|^{2}\) are \(\sqrt{2}\sin 2\pi(x,v)\) and \(\sqrt{2}\cos 2\pi(x,v)\). It follows that
+
+\[{\operatorname{Lip}}(f_{n})\approx|v|\approx\lambda_{n}^{1/2}\approx n^{1/d}\;,\]
+
+where the last asymptotic (up to a multiplicative constant) is just the Weyl law. It follows that the exponent of the power-law in the right hand side of the inequality (9) is sharp.
+
+## 4. From Lipschitz to Hölder observables
+
+Assume that the metric space \((M,\rho)\) satisfies Condition 1.3 with exponent \(d>0\). Let \(\phi:(M,\rho,\mu)\to(M,\rho,\mu)\) be a bi-Lipschitz homeomorphism. Suppose that \(f:M\to{\mathbb{R}}\) is a Hölder continuous function with exponent \(\beta\in(0;1]\) which is mixed by \(\phi\). Let \(\rho_{\beta}\) stand for the metric on \(M\) given by \(\rho_{\beta}(x,y)=\rho(x,y)^{\beta}\). Under the new metric \(f\) becomes a Lipschitz function and \(\phi\) remains bi-Lipschitz with \(\Gamma_{\rho_{\beta}}(\phi)=\Gamma(\phi)^{\beta}\). Moreover the metric space \((M,\rho_{\beta})\) satisfies Condition 1.3 with exponent \(d/\beta\). By Corollary 1.4, we have
+
+\[{\widehat{\Gamma}}_{n}(\phi)^{\beta}=\widehat{(\Gamma_{\rho_{\beta}})}_{n}(\phi)\geq{\rm const}\cdot\log^{\frac{\beta}{d}}n\]
+
+which yields the following:
+
+**Corollary 4.1****.**: _If \(\phi\in G\) mixes a Hölder continuous function then there exists \(\lambda>0\) so that_
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\lambda\cdot\log^{\frac{1}{d}}n\]
+
+_for all natural \(n\)._
+
+In the same manner an application of Theorem 1.8 and Corollary 1.9 gives the following:
+
+**Corollary 4.2****.**: _Suppose that \(\phi\in G\) mixes a Hölder continuous function at the rate \(\{a_{n}\}\) such that \(\sum a_{n}<\infty\). Then there exists \(\lambda>0\) so that_
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\lambda\cdot n^{\frac{1}{d}}\]
+
+_for all natural \(n\). If \(a_{n}=O(1/n^{\nu})\), where \(0<\nu<1\) then there exists \(\lambda>0\) so that_
+
+\[{\widehat{\Gamma}}_{n}(\phi)\geq\lambda\cdot n^{\frac{\nu}{d}}\]
+
+_for all natural \(n\)._
+
+## 5. Existence of an adjoint sequence
+
+Proof of Lemma 1.7.: Making a rescaling if necessary assume that \(a_{n}\leq 1\) for all \(n\). Choose \(N_{k}\nearrow\infty,k\in{\mathbb{N}}\) so that \(N_{1}=1\) and \(a_{i}\leq 1/k\) for all \(i\geq N_{k}\). Put \(b_{n}:=1/k\) for \(n\in[N_{k};N_{k+1})\). Thus \(\{b_{n}\}\) a non-increasing positive sequence which majorates \(\{a_{n}\}\) and converges to zero.
+
+Define \(v(n)\) as _the minimal_ integer \(k\) with
+
+\[\frac{b_{k}}{k}<\frac{1}{4n}\;.\]
+
+Note that \(v(n)\to\infty\) as \(n\to\infty\). By definition
+
+\[\frac{b_{v(n)-1}}{v(n)-1}\geq\frac{1}{4n}\;.\]
+
+Thus we get that
+
+\[\frac{v(n)}{4n}\leq b_{v(n)-1}+\frac{1}{4n}\;,\]
+
+and hence \(v(n)/n\to 0\) which yields assumption (5). Furthermore, using monotonicity of \(b_{n}\) and inequality \(b_{v(n)}/v(n)<1/(4n)\) which follows from the definition of \(v(n)\) we estimate
+
+\[\sum_{i:0<iv(n)\leq n}b_{iv(n)}\leq\frac{n}{v(n)}\cdot b_{v(n)}\leq\frac{1}{4}\,\]
+
+and we get assumption (4).
+
+## 6. Slowly growing diffeomorphisms
+
+As we have shown above, if a bi-Lipschitz homeomorphism \(\phi\) of a \(d\)-dimensional compact metric space mixes a Lipschitz function, the growth rate of the sequence \(\widehat{\Gamma}_{n}(\phi)\) is at least \(\sim\log^{1/d}n\) (see Corollary 1.4). Furthermore, \(\widehat{\Gamma}_{n}(\phi)\geq\text{const}\cdot n^{\nu/d}\) provided the mixing rate is \(\sim n^{-\nu}\) for some \(\nu\in(0;1)\) (see Corollary 1.9 ). In this section we work out an example which shows that the behavior of the sequence \(\Gamma(\phi^{n})\) is essentially different from the one of \(\widehat{\Gamma}_{n}(\phi)\) even in real-analytic category. In addition, this example gives us an opportunity to test our lower bounds on \(\widehat{\Gamma}_{n}(\phi)\) in terms of the rate of mixing.
+
+Consider the three dimensional torus \({\mathbb{T}}^{3}={\mathbb{R}}^{3}/{\mathbb{Z}}^{3}\) equipped with the Euclidean metric and the Lebesgue measure. Fix any concave increasing function \(u:[0;+\infty)\to[0;+\infty)\) such that
+
+\[\lim_{x\to+\infty}u(x)=+\infty,\;u(1)\geq 1\text{ and }u(x)\leq x^{3/4}.\]
+
+**Theorem 6.1****.**: _There exists a real-analytic measure-preserving diffeomorphism \(\phi:{\mathbb{T}}^{3}\to{\mathbb{T}}^{3}\) with the following properties:_
+
+* (i)\(\phi\) _mixes a nonzero real-analytic function at the rate_ \(\{\log u(n)/u(n)^{1/3}\}\)_;_
+* (ii)_There exists a positive constant_ \(c_{1}>0\) _such that_ \(\Gamma(\phi^{n})\leq c_{1}u(n)\) _for infinitely many_ \(n\in{\mathbb{N}}\)_;_
+* (iii)_There exist positive constants_ \(c_{2},c_{3}\) _such that_
+
+\[c_{2}\frac{\sqrt{n}}{\log u(n)}\leq{\widehat{\Gamma}}_{n}(\phi)\leq c_{3}u(\sqrt{n})\sqrt{n},\]
+
+_where the left hand side inequality holds for every natural_ \(n\) _and the right hand side holds for infinitely many_ \(n\)_._
+
+In particular, this theorem shows that \(\Gamma(\phi^{n})\) can grow arbitrarily slowly along a subsequence even when \(\phi\) mixes a real-analytic function.
+
+**Remark 6.2****.**: Taking \(u(x)=x^{3\nu}\), for \(0<\nu<1/4\), we get a diffeomorphism \(\phi\) which mixes a real-analytic function at the rate \(1/n^{\nu-\epsilon}\) (for arbitrary small \(\epsilon>0\)) and such that \({\widehat{\Gamma}}_{n}(\phi)\geq\text{const}\cdot n^{1/2-\epsilon}\). Notice that applying Corollary 1.9 we get \({\widehat{\Gamma}}_{n}(\phi)\geq\text{const}\cdot n^{\nu/3-\epsilon}\). Thus Corollary 1.9 gives a correct prediction of the appearance of a power law in the lower bound for \({\widehat{\Gamma}}_{n}(\phi)\), though with a non-optimal exponent. It is an interesting open problem to find the sharp value of the exponent in Corollary 1.9.
+
+Our construction of a diffeomorphism \(\phi\) in Theorem 6.1 and the estimate of the rate of mixing follows the work of Fayad [6] (see also [7]). The main additional difficulty in our situation is due to the fact that we have to keep track of the growth of the differential.
+
+Preliminaries: We denote by \({\mathbb{T}}\) the circle group \({\mathbb{R}}/{\mathbb{Z}}\) which we will constantly identify with the interval \([0;1)\) with addition mod \(1\). For a real number \(t\) denote by \(\|t\|\) its distance to the nearest integer number. For an irrational \(\alpha\in{\mathbb{T}}\) denote by \(\{q_{n}\}\) its sequence of denominators, i.e.
+
+\[q_{0}=1,\;\;q_{1}=a_{1},\;\;q_{n+1}=a_{n+1}q_{n}+q_{n-1},\]
+
+where \([0;a_{1},a_{2},\dots]\) is the continued fraction expansion of \(\alpha\). Then
+
+(12) \[\frac{1}{2q_{n+1}}<\|q_{n}\alpha\|<\frac{1}{q_{n+1}}\;\;\;\;\text{ for each natural }n.\]
+
+Let \(T:{\mathbb{T}}\to{\mathbb{T}}\) stand for the corresponding ergodic rotation \(Tx=x+\alpha\). Every measurable function \(\varphi:{\mathbb{T}}\rightarrow{\mathbb{R}}\) determines the measurable cocycle over the rotation \(T\) given by
+
+\[\varphi^{(n)}(x)=\left\{\begin{array}[]{ccc}\varphi(x)+\varphi(Tx)+\ldots+\varphi(T^{n-1}x)&\mbox{if}&n>0\\ 0&\mbox{if}&n=0\\ -\left(\varphi(T^{n}x)+\ldots+\varphi(T^{-1}x)\right)&\mbox{if}&n<0.\end{array}\right.\]
+
+If \(\varphi:{\mathbb{T}}\to{\mathbb{R}}\) is a continuous function then
+
+\[\|\varphi^{(m+n)}\|_{\infty}\leq\|\varphi^{(m)}\|_{\infty}+\|\varphi^{(n)}\|_{\infty}\text{ and }\|\varphi^{(-n)}\|_{\infty}=\|\varphi^{(n)}\|_{\infty}\]
+
+for all integer \(m,n\).
+
+Recall that
+
+(13) \[4\|x\|\leq|e^{2\pi ix}-1|\leq 2\pi\|x\|\;\;\;\;\text{ for each real }x.\]
+
+The construction: Let us consider a pair of irrational numbers \((\alpha,\alpha^{\prime})\) such that the sequences of denominators \(\{q_{n}\}\), \(\{q_{n}^{\prime}\}\) of convergents for their continued fraction expansion satisfy
+
+(14) \[2u^{-1}(e^{q^{\prime}_{n-1}})\leq\frac{q_{n}}{u(q_{n})}\leq 3u^{-1}(e^{q^{\prime}_{n-1}}),\;\;2u^{-1}(e^{q_{n}})\leq\frac{q^{\prime}_{n}}{u(q^{\prime}_{n})}\leq 3u^{-1}(e^{q_{n}})\]
+
+for any \(n\geq n_{0}(\alpha,\alpha^{\prime})\). Here \(n_{0}\) is a sufficiently large positive integer which will be chosen in the course of the proof. For a given pair we consider real analytic functions \(\varphi,\psi\) on \({\mathbb{T}}\) given by
+
+(15) \[\varphi(x)=\sum_{n=n_{0}}^{\infty}\frac{\cos 2\pi q_{n}x}{2\pi q_{n}u^{-1}(e^{q_{n}})},\;\;\psi(y)=\sum_{n=n_{0}}^{\infty}\frac{\cos 2\pi q^{\prime}_{n}y}{2\pi q^{\prime}_{n}u^{-1}(e^{q^{\prime}_{n}})}.\]
+
+Let us consider the volume–preserving diffeomorphism \(\phi:{\mathbb{T}}^{3}\to{\mathbb{T}}^{3}\) given by
+
+\[\phi(x,y,z)=(x+\alpha,y+\alpha^{\prime},z+\varphi(x)+\psi(y)).\]
+
+We claim that \(\phi\) has all the properties listed in Theorem 6.1.
+
+Starting growth estimates: Then for each integer \(n\) we have
+
+\[\phi^{n}(x,y,z)=(x+n\alpha,y+n\alpha^{\prime},z+\varphi^{(n)}(x)+\psi^{(n)}(y))\]
+
+and hence \(\Gamma(\phi^{n})\sim\max(\|\varphi^{\prime(n)}\|_{\infty},\|\psi^{\prime(n)}\|_{\infty})\).
+
+**Lemma 6.3****.**: _For every \(x,y\in{\mathbb{T}}\) and \(k\in{\mathbb{N}}\) we have_
+
+\[|\varphi^{\prime(q_{k})}(x)|\leq\frac{6q_{k}}{u^{-1}(e^{q_{k}})},\;\;|\varphi^{\prime\prime(q_{k})}(x)|\leq\frac{6q^{2}_{k}}{u^{-1}(e^{q_{k}})},\]
+
+\[|\psi^{\prime(q^{\prime}_{k})}(y)|\leq\frac{48q^{\prime}_{k}}{u^{-1}(e^{q^{\prime}_{k}})},\;\;|\psi^{\prime\prime(q^{\prime}_{k})}(y)|\leq\frac{48q^{\prime 2}_{k}}{u^{-1}(e^{q^{\prime}_{k}})}.\]
+
+Proof.: Since
+
+\[\varphi^{(m)}(x)=\sum_{n=n_{0}}^{\infty}\frac{1}{2\pi q_{n}u^{-1}(e^{q_{n}})}\operatorname{Re}e^{2\pi iq_{n}x}\frac{e^{2\pi imq_{n}\alpha}-1}{e^{2\pi iq_{n}\alpha}-1},\]
+
+we obtain
+
+(16) \[\varphi^{\prime(m)}(x)=\sum_{n=n_{0}}^{\infty}\frac{1}{u^{-1}(e^{q_{n}})}\operatorname{Im}e^{2\pi iq_{n}x}\frac{e^{2\pi imq_{n}\alpha}-1}{e^{2\pi iq_{n}\alpha}-1},\]
+
+hence
+
+\[|\varphi^{\prime(q_{k})}(x)|\leq\sum_{n=n_{0}}^{\infty}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi iq_{k}q_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}.\]
+
+In the next chain of inequalities we use that by increasing \(n_{0}\) we can assume that \(\sum_{n=n_{0}}^{\infty}{q_{n}}/{u^{-1}(e^{q_{n}})}<1/4\). We have
+
+\[\sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi iq_{k}q_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}\]
+\[\leq \sum_{n=n_{0}}^{k-1}\frac{2}{u^{-1}(e^{q_{n}})}\frac{\|q_{k}q_{n}\alpha\|}{\|q_{n}\alpha\|}\leq\sum_{n=n_{0}}^{k-1}\frac{2}{u^{-1}(e^{q_{n}})}\frac{q_{n}\|q_{k}\alpha\|}{\|q_{n}\alpha\|}\]
+\[\leq \sum_{n=n_{0}}^{k-1}\frac{4}{u^{-1}(e^{q_{n}})}\frac{q_{n}q_{n+1}}{q_{k+1}}\leq\frac{4q_{k}}{q_{k+1}}\sum_{n=n_{0}}^{k-1}\frac{q_{n}}{u^{-1}(e^{q_{n}})}\leq\frac{q_{k}}{q_{k+1}}\leq\frac{q_{k}}{q^{\prime}_{k}}\;.\]
+
+In view of (14),
+
+\[\frac{q_{k}}{q^{\prime}_{k}}\leq\frac{1}{2u(q^{\prime}_{k})}\frac{q_{k}}{u^{-1}(e^{q_{k}})}\leq\frac{q_{k}}{u^{-1}(e^{q_{k}})}\;.\]
+
+It follows that
+
+\[\sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi iq_{k}q_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}\leq\frac{q_{k}}{u^{-1}(e^{q_{k}})}\;.\]
+
+Furthermore,
+
+\[\sum_{n=k}^{\infty}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi iq_{k}q_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}\leq\sum_{n=k}^{\infty}\frac{2q_{k}}{u^{-1}(e^{q_{n}})}\leq\frac{4q_{k}}{u^{-1}(e^{q_{k}})},\]
+
+and the required upper bound for \(|\varphi^{\prime(q_{k})}(x)|\) follows.
+
+Since
+
+\[|\varphi^{\prime\prime(q_{k})}(x)|\leq\sum_{n=n_{0}}^{\infty}\frac{2\pi q_{n}}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi iq_{k}q_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|},\]
+
+similar arguments to those above show that \(|\varphi^{\prime\prime(q_{k})}(x)|\leq 48q_{n}^{2}/u^{-1}(e^{q_{k}})\).
+
+The remaining inequalities are proved similarly.
+
+**Lemma 6.4****.**: _For every natural \(m\) and \(k\) we have_
+
+\[\|\varphi^{\prime(m)}\|_{\infty}\leq\frac{6m}{u^{-1}(e^{q_{k}})}+q_{k},\;\;\|\varphi^{\prime\prime(m)}\|_{\infty}\leq\frac{48mq_{k}}{u^{-1}(e^{q_{k}})}+q_{k},\]
+
+\[\|\psi^{\prime(m)}\|_{\infty}\leq\frac{6m}{u^{-1}(e^{q^{\prime}_{k}})}+q^{\prime}_{k},\;\;\|\psi^{\prime\prime(m)}\|_{\infty}\leq\frac{48mq^{\prime}_{k}}{u^{-1}(e^{q^{\prime}_{k}})}+q^{\prime}_{k}.\]
+
+Proof.: Write \(m\) as \(m=pq_{k}+r\), where \(p=[m/q_{k}]\) and \(0\leq r<q_{k}\). Then
+
+\[\|\varphi^{\prime(m)}\|_{\infty}\leq p\|\varphi^{\prime(q_{k})}\|_{\infty}+\|\varphi^{\prime(r)}\|_{\infty}\leq\frac{m}{q_{k}}\frac{6q_{k}}{u^{-1}(e^{q_{k}})}+r\|\varphi^{\prime}\|_{\infty}\leq\frac{6m}{u^{-1}(e^{q_{k}})}+q_{k}.\]
+
+The remaining inequalities are proved similarly.
+
+A van der Corput like Lemma:¹ For estimating the rate of mixing, we shall need the following version of the van der Corput Lemma:
+
+Footnote 1: It is known also as a stationary phase argument.
+
+**Lemma 6.5****.**: _Let \(f:{\mathbb{T}}\rightarrow{\mathbb{R}}\) be a \(C^{1}\) function. Suppose there exist a family \(\{(a_{j};b_{j})\subset{\mathbb{T}}:j=1,\ldots,s\}\) of pairwise disjoint intervals and a real positive number \(a\) such that \(|f^{\prime}(x)|\geq a>0\) for all \(x\in{\mathbb{T}}\setminus\bigcup_{j=1}^{s}(a_{j};b_{j})\). Then_
+
+(17) \[\left|\int_{\mathbb{T}}e^{2\pi if(x)}dx\right|\leq\frac{1}{2\pi}\frac{\|f^{\prime\prime}\|_{\infty}}{a^{2}}+\frac{s}{\pi a}+\sum_{j=1}^{s}(b_{j}-a_{j}).\]
+
+Proof.: Without loss of generality we can assume that \(a_{1}<b_{1}<\ldots<a_{s}<b_{s}<a_{1}\). Put \(D=\bigcup_{j=1}^{s}(a_{j};b_{j})\) and \(a_{s+1}=a_{1}\). Then
+
+\[\left|\int_{\mathbb{T}}e^{2\pi if(x)}\,dx\right| \leq \left|\int_{{\mathbb{T}}\setminus D}e^{2\pi if(x)}\,dx\right|+\sum_{j=1}^{s}(b_{j}-a_{j})\]
+\[= \left|\int_{{\mathbb{T}}\setminus D}\frac{1}{2\pi if^{\prime}(x)}\,de^{2\pi if(x)}\right|+\sum_{j=1}^{s}(b_{j}-a_{j}).\]
+
+Integrating by parts gives
+
+\[\left|\int_{{\mathbb{T}}\setminus D}\frac{1}{2\pi if^{\prime}(x)}\,de^{2\pi if(x)}\right|\]
+\[= \left|\sum_{j=1}^{s}\left(\frac{e^{2\pi if(a_{j+1})}}{2\pi f^{\prime}(a_{j+1})}-\frac{e^{2\pi if(b_{j})}}{2\pi f^{\prime}(b_{j})}-\frac{1}{2\pi}\int_{b_{j}}^{a_{j+1}}e^{2\pi if(x)}\,d\left(\frac{1}{f^{\prime}(x)}\right)\right)\right|\]
+\[= \left|\sum_{j=1}^{s}\left(\frac{e^{2\pi if(a_{j+1})}}{2\pi f^{\prime}(a_{j+1})}-\frac{e^{2\pi if(b_{j})}}{2\pi f^{\prime}(b_{j})}+\frac{1}{2\pi}\int_{b_{j}}^{a_{j+1}}e^{2\pi if(x)}\frac{f^{\prime\prime}(x)}{(f^{\prime}(x))^{2}}\,dx\right)\right|\]
+\[\leq \frac{1}{2\pi}\sum_{j=1}^{s}\left[\left(\frac{1}{|f^{\prime}(a_{j})|}+\frac{1}{|f^{\prime}(b_{j})|}\right)+\sum_{j=1}^{s}|a_{j+1}-b_{j}|\frac{\|f^{\prime\prime}\|_{\infty}}{a^{2}}\right]\]
+\[\leq \frac{1}{2\pi}\frac{\|f^{\prime\prime}\|_{\infty}}{a^{2}}+\frac{s}{\pi a}.\]
+
+**Lemma 6.6****.**: _There exists \(C>0\) such that_
+
+\[I_{m}:=\left|\int_{{\mathbb{T}}^{2}}e^{2\pi i(\varphi^{(m)}(x)+\psi^{(m)}(y))}\,dxdy\right|\leq C\frac{\log u(m)}{u(m)^{1/3}}.\]
+
+Proof.: For each \(m\) large enough there exists a natural number \(k\geq n_{0}\) such that
+
+\[u^{-1}(e^{q_{k}})\leq\frac{m}{u(m)}\leq u^{-1}(e^{q^{\prime}_{k}})\text{ or }u^{-1}(e^{q^{\prime}_{k}})\leq\frac{m}{u(m)}\leq u^{-1}(e^{q_{k+1}}).\]
+
+Suppose that \(m/u(m)\in[u^{-1}(e^{q_{k}});u^{-1}(e^{q^{\prime}_{k}})]\). Then
+
+\[\frac{m}{u(m)}\leq u^{-1}(e^{q^{\prime}_{k}})\leq\frac{q_{k+1}}{2u(q_{k+1})}\leq\frac{q_{k+1}/2}{u(q_{k+1}/2)}\]
+
+and hence \(m\leq q_{k+1}/2\) because of the concavity of \(u\).
+
+Put
+
+\[a_{j}=\frac{1}{2q_{k}}\left(j-\frac{1}{u(m)^{1/3}}\right)-\frac{(m-1)\alpha}{2},\;\;b_{j}=\frac{1}{2q_{k}}\left(j+\frac{1}{u(m)^{1/3}}\right)-\frac{(m-1)\alpha}{2}\]
+
+for \(j=1,\ldots,2q_{k}\). If \(x\in{\mathbb{T}}\setminus\bigcup_{j=1}^{2q_{k}}(a_{j};b_{j})\), then
+
+\[1/u(m)^{1/3}\leq\|2q_{k}(x+(m-1)\alpha/2)\|\leq|\sin 2\pi q_{k}(x+(m-1)\alpha/2)|.\]
+
+By (16),
+
+\[|\varphi^{\prime(m)}(x)| \geq \frac{1}{u^{-1}(e^{q_{k}})}\left|\operatorname{Im}e^{2\pi iq_{k}x}\frac{e^{2\pi imq_{k}\alpha}-1}{e^{2\pi iq_{k}\alpha}-1}\right|\]
+\[-\sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi imq_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}-\sum_{n=k+1}^{\infty}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi imq_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}.\]
+
+Note that
+
+\[\left|\operatorname{Im}e^{2\pi iq_{k}x}\frac{e^{2\pi imq_{k}\alpha}-1}{e^{2\pi iq_{k}\alpha}-1}\right|\]
+\[= \left|\frac{1}{2i}\left(e^{2\pi iq_{k}x}\frac{e^{2\pi iq_{k}m\alpha}-1}{e^{2\pi iq_{k}\alpha}-1}-e^{-2\pi iq_{k}x}\frac{e^{-2\pi iq_{k}m\alpha}-1}{e^{-2\pi iq_{k}\alpha}-1}\right)\right|\]
+\[= \left|\frac{1}{2i}\frac{e^{2\pi iq_{k}m\alpha}-1}{e^{2\pi iq_{k}\alpha}-1}\left(e^{2\pi iq_{k}x}-e^{-2\pi iq_{k}(x+(m-1)\alpha)}\right)\right|\]
+\[= \frac{|e^{2\pi iq_{k}m\alpha}-1|}{|e^{2\pi iq_{k}\alpha}-1|}|\sin 2\pi q_{k}(x+(m-1)\alpha/2)|.\]
+
+Since \(m\leq q_{k+1}/2\) and \(\|q_{k}\alpha\|<1/q_{k+1}\), we have
+
+\[\|mq_{k}\alpha\|\leq m\|q_{k}\alpha\|\leq\frac{1}{2}q_{k+1}\|q_{k}\alpha\|<\frac{1}{2},\]
+
+hence \(\|mq_{k}\alpha\|=m\|q_{k}\alpha\|\). It follows that
+
+\[\frac{|e^{2\pi iq_{k}m\alpha}-1|}{|e^{2\pi iq_{k}\alpha}-1|}\geq\frac{\|q_{k}m\alpha\|}{2\|q_{k}\alpha\|}=\frac{m}{2}.\]
+
+Thus
+
+\[{\left|\operatorname{Im}e^{2\pi iq_{k}x}\frac{e^{2\pi imq_{k}\alpha}-1}{e^{2\pi iq_{k}\alpha}-1}\right|}\geq\frac{m}{2u(m)^{1/3}u^{-1}(e^{q_{k}})}\;.\]
+
+Since \(\|q_{n}\alpha\|>1/(2q_{n+1})\), we have
+
+\[\sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi imq_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|} \leq \sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}\frac{1}{2\|q_{n}\alpha\|}\leq\sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}q_{n+1}\]
+\[\leq q_{k}\sum_{n=n_{0}}^{k-1}\frac{1}{u^{-1}(e^{q_{n}})}\leq q_{k}.\]
+
+Moreover
+
+\[\sum_{n=k+1}^{\infty}\frac{1}{u^{-1}(e^{q_{n}})}\frac{|e^{2\pi imq_{n}\alpha}-1|}{|e^{2\pi iq_{n}\alpha}-1|}\leq m\sum_{n=k+1}^{\infty}\frac{2}{u^{-1}(e^{q_{n}})}\leq\frac{4m}{u^{-1}(e^{q_{k+1}})}.\]
+
+Therefore, if \(x\in{\mathbb{T}}\setminus\bigcup_{j=1}^{2q_{k}}(a_{j};b_{j})\), then
+
+\[|\varphi^{\prime(m)}(x)|\geq\frac{m}{2u(m)^{1/3}u^{-1}(e^{q_{k}})}-q_{k}-\frac{4m}{u^{-1}(e^{q_{k+1}})}.\]
+
+Since \(u^{-1}(e^{q_{k}})\leq m/u(m)\leq m\), we have
+
+\[q_{k}\leq\log u(m)\leq\frac{\log u(m)}{u(m)^{2/3}}\frac{m}{u(m)^{1/3}u^{-1}(e^{q_{k}})}.\]
+
+Moreover, since
+
+\[u^{-1}(e^{q_{k}})\leq\frac{m}{u(m)}\leq q_{k+1}\text{ and }u(m)\leq m^{3/4},\]
+
+we have
+
+\[\frac{m}{u^{-1}(e^{q_{k+1}})} \leq \frac{m}{u(m)^{1/3}u^{-1}(e^{q_{k}})}\frac{m/(u(m))^{2/3}}{u^{-1}(e^{m/u(m)})}\]
+\[\leq \frac{m}{u(m)^{1/3}u^{-1}(e^{q_{k}})}\frac{(m/u(m))^{2}}{u^{-1}(e^{m/u(m)})}.\]
+
+Therefore, for \(m\) large enough,
+
+(18) \[|\varphi^{\prime(m)}(x)|\geq\frac{m}{4u(m)^{1/3}u^{-1}(e^{q_{k}})}\text{ for all }x\in{\mathbb{T}}\setminus\bigcup_{j=1}^{2q_{k}}(a_{j};b_{j}).\]
+
+On the other hand, by Lemma 6.4,
+
+\[|\varphi^{\prime\prime(m)}(x)|\leq\frac{48mq_{k}}{u^{-1}(e^{q_{k}})}+q_{k}\leq\frac{50mq_{k}}{u^{-1}(e^{q_{k}})}.\]
+
+An application of Lemma 6.5 for the function \(\varphi^{(m)}\) and the family of intervals \((a_{i};b_{i})\), \(i=1,\ldots,2q_{k}\) gives
+
+\[\left|\int_{{\mathbb{T}}}e^{2\pi i\varphi^{(m)}(x)}\,dx\right|\]
+\[\leq \frac{1}{2\pi}\frac{\frac{50mq_{k}}{u^{-1}(e^{q_{k}})}}{\left(\frac{m}{4u(m)^{1/3}u^{-1}(e^{q_{k}})}\right)^{2}}+\frac{2q_{k}}{\frac{\pi m}{4u(m)^{1/3}u^{-1}(e^{q_{k}})}}+\frac{2}{u(m)^{1/3}}\]
+\[= \frac{400q_{k}u^{-1}(e^{q_{k}})u(m)^{2/3}}{\pi m}+\frac{4q_{k}u^{-1}(e^{q_{k}})u(m)^{1/3}}{\pi m}+\frac{2}{u(m)^{1/3}}\]
+\[\leq \frac{200q_{k}u^{-1}(e^{q_{k}})u(m)^{2/3}}{m}+\frac{2}{u(m)^{1/3}}.\]
+
+Since \(u^{-1}(e^{q_{k}})\leq m/u(m)\), we have \(q_{k}\leq\log u(m)\) and
+
+\[\frac{q_{k}u^{-1}(e^{q_{k}})u(m)^{2/3}}{m}\leq\frac{\log u(m)}{u(m)^{1/3}}.\]
+
+Consequently
+
+\[\left|\int_{{\mathbb{T}}}e^{2\pi i\varphi^{(m)}(x)}\,dx\right|\leq 202\frac{\log u(m)}{u(m)^{1/3}}.\]
+
+When \(m/u(m)\in[u^{-1}(e^{q^{\prime}_{k}});u^{-1}(e^{q_{k+1}})]\), proceeding in the same way we obtain
+
+\[\left|\int_{{\mathbb{T}}}e^{2\pi i\psi^{(m)}(y)}\,dy\right|\leq 202\frac{\log u(m)}{u(m)^{1/3}}.\]
+
+Therefore for each natural \(m\) we have
+
+\[I_{m}=\left|\int_{{\mathbb{T}}}e^{2\pi i\varphi^{(m)}(x)}dx\right|\left|\int_{{\mathbb{T}}}e^{2\pi i\psi^{(m)}(y)}dy\right|\leq 202\frac{\log u(m)}{u(m)^{1/3}}.\]
+
+Proof of Theorem 6.1.:  
+
+**(i):** Take \(f:{\mathbb{T}}^{3}\to{\mathbb{R}}\) given by \(f(x,y,z)=\sin 2\pi z\). Then in view of Lemma 6.6 we obtain
+
+\[|(f\circ\phi^{n},f)| = \frac{1}{2}\left|\operatorname{Im}\int_{{\mathbb{T}}^{2}}e^{2\pi i(\varphi^{(n)}(x)+\psi^{(n)}(y))}\,dxdy\right|\]
+\[\leq \frac{1}{2}\left|\int_{{\mathbb{T}}^{2}}e^{2\pi i(\varphi^{(n)}(x)+\psi^{(n)}(y))}\,dxdy\right|\leq{\rm const}\cdot\frac{\log u(n)}{u(n)^{1/3}}\]
+
+for all \(n\in{\mathbb{N}}\) large enough.
+
+**(ii):** Since
+
+\[\phi^{n}(x,y,z)=(x+n\alpha,y+n\alpha^{\prime},z+\varphi^{(n)}(x)+\psi^{(n)}(y))\]
+
+it suffices to show that \(\max(\|\varphi^{\prime(n)}\|_{\infty},\|\psi^{\prime(n)}\|_{\infty})\leq c_{1}u(n)\) for infinitely many \(n\in{\mathbb{N}}\). By Lemma 6.3 and Lemma 6.4,
+
+\[\|\varphi^{\prime(q_{k})}\|_{\infty}\leq 1\text{ and }\|\psi^{\prime(q_{k})}\|_{\infty}\leq\frac{6q_{k}}{u^{-1}(e^{q^{\prime}_{k-1}})}+q^{\prime}_{k-1}.\]
+
+From (14) we have
+
+\[q^{\prime}_{k-1}\leq\log u(q_{k})\text{ and }u^{-1}(e^{q^{\prime}_{k-1}})\geq\frac{q_{k}}{3u(q_{k})}.\]
+
+It follows that
+
+\[\|\psi^{\prime(q_{k})}\|_{\infty}\leq 18u(q_{k})+\log u(q_{k})\leq 20u(q_{k})\]
+
+for all \(k\) large enough.
+
+**(iii):** Set
+
+\[g_{m}:=\max(\|\varphi^{\prime(m)}\|_{\infty},\|\psi^{\prime(m)}\|_{\infty})\text{ and }\widehat{g}_{m}=\max_{0\leq i\leq m}g_{i}.\]
+
+It suffices to show that
+
+(19) \[c_{2}\frac{\sqrt{m}}{\log u(m)}\leq\widehat{g}_{m}\leq c_{3}u(\sqrt{m})\sqrt{m}\;,\]
+
+where the left hand side inequality holds for every natural \(m\) and the right hand side holds for infinitely many \(m\).
+
+By Lemma 6.4,
+
+(20) \[\widehat{g}_{m}\leq\max\left(\frac{6m}{u^{-1}(e^{q_{k}})}+q_{k},\frac{6m}{u^{-1}(e^{q^{\prime}_{k}})}+q^{\prime}_{k}\right)\]
+
+for every natural \(m\) and \(k\). Choose \(x\) and \(y\) so that
+
+\[\sin(2\pi q_{k}(x+(m-1)\alpha/2))=\sin(2\pi q^{\prime}_{k}(y+(m-1)\alpha^{\prime}/2))=1\;.\]
+
+Proceeding along the same lines as in the proof of Lemma 6.6 one readily shows that
+
+(21) \[u^{-1}(e^{q_{k}})\leq\frac{m}{u(m)}\leq u^{-1}(e^{q^{\prime}_{k}}) \Longrightarrow g_{m}\geq|\varphi^{\prime(m)}(x)|\geq\frac{m}{4u^{-1}(e^{q_{k}})},\]
+(22) \[u^{-1}(e^{q^{\prime}_{k}})\leq\frac{m}{u(m)}\leq u^{-1}(e^{q_{k+1}}) \Longrightarrow g_{m}\geq|\psi^{\prime(m)}(y)|\geq\frac{m}{4u^{-1}(e^{q^{\prime}_{k}})}.\]
+
+To prove the lower bound on \(\widehat{g}_{m}\) suppose that \(u^{-1}(e^{q_{k}})\leq m/u(m)\leq u^{-1}(e^{q^{\prime}_{k}})\) (the case of \(u^{-1}(e^{q^{\prime}_{k}})\leq m/u(m)\leq u^{-1}(e^{q_{k+1}})\) is treated similarly).
+
+_Case 1._ Suppose that \(m\leq(u^{-1}(e^{q_{k}}))^{2}\). Set \(m_{0}:=[u^{-1}(e^{q_{k}})]\). Then
+
+\[u^{-1}(e^{q_{k}})/2\leq m_{0}\leq u^{-1}(e^{q_{k}})\leq m\]
+
+and
+
+\[u^{-1}(e^{q^{\prime}_{k-1}})\leq q_{k}\leq\frac{e^{q_{k}/3}}{2}\leq\left(\frac{u^{-1}(e^{q_{k}})}{2}\right)^{1/4}\leq m_{0}^{1/4}\leq\frac{m_{0}}{u(m_{0})}\leq u^{-1}(e^{q_{k}}).\]
+
+Therefore in view of (22), we obtain
+
+\[\widehat{g}_{m}\geq g_{m_{0}}\geq\frac{m_{0}}{4u^{-1}(e^{q^{\prime}_{k-1}})}\geq\frac{u^{-1}(e^{q_{k}})}{8u^{-1}(e^{q^{\prime}_{k-1}})}\geq\frac{u^{-1}(e^{q_{k}})}{4q_{k}}\geq\frac{\sqrt{m}}{4\log u(m)}.\]
+
+_Case 2._ Suppose that \(m\geq(u^{-1}(e^{q_{k}}))^{2}\). Then in view of (21), we obtain
+
+\[\widehat{g}_{m}\geq g_{m}\geq\frac{m}{4u^{-1}(e^{q_{k}})}\geq\sqrt{m}/4.\]
+
+The desired lower bound on \(\widehat{g}_{m}\) follows.
+
+To prove the upper bound on \(\widehat{g}_{m}\) in formula (19) we take \(m=(q^{\prime}_{k})^{2}\). Then
+
+\[\frac{6m}{u^{-1}(e^{q_{k}})}+q_{k}=\frac{6(q^{\prime}_{k})^{2}}{u^{-1}(e^{q_{k}})}+q_{k}\leq 18u(q^{\prime}_{k})q^{\prime}_{k}+q_{k}\leq 20u(q^{\prime}_{k})q^{\prime}_{k}\leq 20u(\sqrt{m})\sqrt{m}.\]
+
+Moreover
+
+\[\frac{6m}{u^{-1}(e^{q^{\prime}_{k}})}+q^{\prime}_{k}=\frac{6(q^{\prime}_{k})^{2}}{u^{-1}(e^{q^{\prime}_{k}})}+q^{\prime}_{k}\leq 2q^{\prime}_{k}=2\sqrt{m}.\]
+
+Finally, from (20) we have
+
+\[\widehat{g}_{m}\leq 20u(\sqrt{m})\sqrt{m}.\]
+
+This completes the proof.
+
+## 7. Growth of the Rudin-Shapiro shift
+
+In the present section we prove the following result.
+
+**Theorem 7.1****.**: _Fix \(d>0\). There exists a bi-Lipschitz homeomorphism \(\phi\) of a compact measure metric space \((X,\rho,\mu)\) with the following properties:_
+
+* (i)_The upper box dimension (see formula (_1_) above) of_ \((X,\rho)\) _equals_ \(d\)_. Furthermore, for every_ \(\delta>0\) _there exists a_ \(\delta\)_-net in_ \(X\) _containing at most_ \(\text{const}\cdot\delta^{-d}\) _points (see Condition_ 1.3 _above);_
+* (ii)_The homeomorphism_ \(\phi\) _mixes a nonzero Lipschitz function_ \(f:X\to{\mathbb{R}}\) _with zero mean at the speediest possible rate, i.e._ \((f\circ\phi^{k},f)_{L_{2}(X,\mu)}=0\) _for all_ \(k\neq 0\)_;_
+* (iii)_There exist_ \(c_{1},c_{2}>0\) _so that the growth rate of_ \(\phi\) _satisfies_
+
+\[c_{1}\cdot n^{1/d}\leq{\widehat{\Gamma}}_{n}(\phi)\leq c_{2}\cdot n^{1/d}\]
+
+_for all_ \(n\in{\mathbb{N}}\)_._
+
+Thus we confirm that the lower bound (7) in Theorem 1.8 is sharp. As we shall explain below, the homeomorphism \(\phi\) can be chosen as the shift associated to the Rudin-Shapiro sequence.
+
+In what follows we work in the framework of the theory of symbolic dynamical systems associated to substitutions (see [12, 8]). Let us consider a finite alphabet \(\mathcal{A}\). Denote by \(\mathcal{A}^{*}=\bigcup_{n\geq 1}\mathcal{A}^{n}\) the set of all finite words over the alphabet \(\mathcal{A}\). A _substitution_ on \(\mathcal{A}\) is a mapping \(\zeta:\mathcal{A}\to\mathcal{A}^{*}\). Any substitution \(\zeta\) induces two maps, also denoted by \(\zeta\), one from \(\mathcal{A}^{*}\) to \(\mathcal{A}^{*}\) and another from \(\mathcal{A}^{\mathbb{N}}\) to \(\mathcal{A}^{\mathbb{N}}\) by putting
+
+\[\zeta(a_{0}a_{1}\ldots a_{n})=\zeta(a_{0})\zeta(a_{1})\ldots\zeta(a_{n})\text{ for every }a_{0}a_{1}\ldots a_{n}\in\mathcal{A}^{*},\]
+
+\[\zeta(a_{0}a_{1}\ldots a_{n}\ldots)=\zeta(a_{0})\zeta(a_{1})\ldots\zeta(a_{n})\ldots\text{ for every }a_{0}a_{1}\ldots a_{n}\ldots\in\mathcal{A}^{\mathbb{N}}.\]
+
+If there exists a letter \(a\in\mathcal{A}\) so that \(\zeta(a)\) consists of at least two letters and starts with \(a\), the word \(\zeta^{n}(a)\) starts with \(\zeta^{n-1}(a)\) and is strictly longer than \(\zeta^{n-1}(a)\). Thus \(\zeta^{n}(a)\) converges in the obvious sense as \(n\to\infty\) to an infinite word \(v\in\mathcal{A}^{\mathbb{N}}\) such that \(\zeta(v)=v\).
+
+We can associate to the sequence \(v\) a topological dynamical system as follows. Let \(\mathcal{L}(v)\) denote the language of the sequence \(v\), i.e. the set of all finite words (over the alphabet \(\mathcal{A}\)) which occur in \(v\). Let \(X_{v}\subset\mathcal{A}^{\mathbb{Z}}\) stand for the set of all sequences \(x=\{x_{n}\}_{n\in{\mathbb{Z}}}\in\mathcal{A}^{\mathbb{Z}}\) such that \(x_{n}x_{n+1}\ldots x_{n+k-1}\in\mathcal{L}(v)\) for all \(n\in{\mathbb{Z}}\) and \(k\in{\mathbb{N}}\). Obviously, \(X_{v}\) is a compact subset of \(\mathcal{A}^{\mathbb{Z}}\) with the product topology and \(X_{v}\) is invariant under the two-sided Bernoulli shift \(\phi:\mathcal{A}^{\mathbb{Z}}\to\mathcal{A}^{\mathbb{Z}}\), \([\phi(\{x_{k}\}_{k\in{\mathbb{Z}}})]_{n}=x_{n+1}\). Therefore we can consider \(\phi\) as a homeomorphism of \(X_{v}\).
+
+A substitution \(\zeta\) is called _primitive_ if there exists \(k\geq 1\) such that \(\zeta^{k}(a)\) contains \(b\) for every \(a,b\in\mathcal{A}\). If \(\zeta\) is primitive, the space \(X=X_{v}\) does not depend on the choice of \(v\). Furthermore, the corresponding homeomorphism \(\phi:X\to X\) is minimal and uniquely ergodic. Unique ergodicity of \(\phi\) can be deduced from the analogous result in [12, Chapter V] for the one-sided shift: Given two words \(z,w\in\mathcal{L}(v)\), denote by \(\Omega_{z}(w)\) the number of appearances of \(z\) as a sub-word in \(w\). Unique ergodicity of the one-sided shift yields (see [12, Corollary IV.14]) existence of a positive function \(\omega:\mathcal{L}(v)\to(0;1]\) so that for every \(z\)
+
+(23) \[\frac{\Omega_{z}(w)}{\text{length}(w)}\to\omega(z)\;\;\text{uniformly in}\;\;w\;\;\text{as}\;\;\text{length}(w)\to\infty\;.\]
+
+This in turn yields, exactly as in [12, Corollary IV.14], unique ergodicity of the two-sided shift \(\phi\).
+
+Let us consider the Rudin–Shapiro sequence \(v=\{v_{n}\}_{n\geq 0}\) over the alphabet \(\mathcal{A}=\{-1,+1\}\) which is defined by the relation
+
+\[v_{0}=1,\;\;v_{2n}=v_{n},\;\;v_{2n+1}=(-1)^{n}v_{n}\text{ for any }n\geq 0.\]
+
+It arises from the fixed point \(ABACABDB\ldots\) of the primitive substitution \(A\mapsto AB,B\mapsto AC,C\mapsto DB,D\mapsto DC\) after replacing \(A,B\) by \(+1\) and \(C,D\) by \(-1\). As above, we associate to the sequence \(v\) the topological space \(X\subset\mathcal{A}^{{\mathbb{Z}}}\) and the two-sided shift \(\phi:X\to X\). Notice that \(\phi\) is uniquely ergodic as a factor of the corresponding uniquely ergodic substitution system.
+
+Proof of Theorem 7.1.: Let \(\mu\) be the unique \(\phi\)-invariant Borel probability measure on \(X\). We shall show that after a suitable choice of a metric on \(X\), the shift \(\phi:X\to X\) possesses properties (i)-(iii) stated in the theorem.
+
+Choosing the metric: Fix a concave increasing function \(u:[0;+\infty)\to[0;+\infty)\) such that \(u(0)=0\) and \(u(t)\to+\infty\) as \(t\to+\infty\). Then define a metric \(\rho\) on \(X\) by putting \(\rho(x,y)=e^{-u(k(x,y))}\), where \(k(x,y)=\min\{|k|:x_{k}\neq y_{k},k\in{\mathbb{Z}}\}\) for two distinct sequences \(x,y\in X\). Of course, \(\phi\) is a bi-Lipschitz homeomorphism with \({\widehat{\Gamma}}_{n}(\phi)\leq e^{u(n)}\).
+
+Denote by \(\{p_{n}(v)\}\) the complexity of the sequence \(v\), that is \(p_{n}(v)\) is the number of different words of length \(n\) occurring in \(v\). As it was shown in [1], \(p_{n}(v)=8(n-1)\) for every \(n\geq 2\) (in fact, a simpler estimate \(n\leq p_{n}(v)\leq\text{const}\cdot n\) is sufficient for our purposes, see Propositions 1.1.1 and 5.4.6 in [8]).
+
+Suppose that \(u(t)=d^{-1}\log t\) for all \(t\) large enough. Given \(k>2\), put \(p=p_{2k-1}(v)\) and consider all possible words \(w^{(1)},\ldots,w^{(p)}\) from \(\mathcal{L}(v)\) of length \(2k-1\). Fix arbitrary elements \(x^{(i)}\in X\), \(i=1,\ldots,p\) so that \(x^{(i)}_{-k+1}x^{(i)}_{-k+2}\ldots x^{(i)}_{k-2}x^{(i)}_{k-1}=w^{(i)}\). Note that the points \(x^{(i)}\) lie at the distance \(\geq e^{-u(k-1)}=(k-1)^{-1/d}\) one from the other. Furthermore, every point of \(X\) lies at the distance \(\leq e^{-u(k)}=k^{-1/d}\) from \(x^{(i)}\) for some \(i=1,\ldots,p\). Recalling that \(p=16k-16\) we conclude that the upper box dimension of \(X\) equals \(d\), and moreover for every \(\delta>0\) there exists a \(\delta\)-net in \(X\) containing at most \(\text{const}\cdot\delta^{-d}\) points. Thus we get property (i) in Theorem 7.1.
+
+Mixing: Consider a function \(f:X\to{\mathbb{R}}\), \(f(x)=x_{0}\). Clearly, \(f\) is Lipschitz with respect to \(\rho\). Let us check that \((f\circ\phi^{k},f)_{L_{2}(X,\mu)}=0\) for all \(k\neq 0\). We prove this property by combining the unique ergodicity of \(\phi\) with the following fact, see [8, Proposition 2.2.5]:
+
+\[\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}v_{n}v_{n+k}=0\;\;\forall k\in{\mathbb{N}}\;.\]
+
+Indeed, there exists a sequence \(y\in X\) such that \(y_{n}=v_{n}\) for all \(n\geq 0\) (see Lemma 7.3 below). Then
+
+\[\int_{X}f(\phi^{k}x)f(x)\,d\mu(x) = \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(\phi^{k+n}y)f(\phi^{n}y)\]
+\[= \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}v_{n}v_{n+k}=0.\]
+
+This proves property (ii) in Theorem 7.1.
+
+Growth bounds: The lower bound (7) in Theorem 1.8 yields \({\widehat{\Gamma}}_{n}(\phi)\geq\text{const}\cdot n^{1/d}\). On the other hand
+
+\[{\widehat{\Gamma}}_{n}(\phi)\leq e^{u(n)}=n^{1/d},\]
+
+which yields property (iii) in Theorem 7.1.
+
+This completes the proof.
+
+**Remark 7.2****.**: Let us modify the metric \(\rho\) defined above by taking the function \(u(t)\) to be of an arbitrarily slow growth. As a result we get an example of a bi-Lipschitz homeomorphism \(\phi\) of a compact metric measure space \((M,\rho,\mu)\) of _infinite_ box dimension which mixes a Lipschitz function \(f\) at the speediest possible rate, that is \((f,f\circ\phi^{n})_{L_{2}}=0\) for all \(n\in{\mathbb{N}}\), and such that the growth rate of \({\widehat{\Gamma}}_{n}(\phi)\) is arbitrarily slow. This illustrates the significance of Condition 1.3 on the metric \(\rho\) for the validity of the statement of Theorem 1.8.
+
+We conclude this section with the following lemma which was used in the proof of Theorem 7.1 above.
+
+**Lemma 7.3****.**: _There exists a sequence \(y\in X\) so that \(y_{n}=v_{n}\) for all \(n\geq 0\)._
+
+Proof.: By (23), for every \(n\in{\mathbb{N}}\) the word \(v_{0}\ldots v_{n}\) appears infinitely many times as a subword in \(v\). Thus we can find a sequence of words of the form \(y^{(n)}=y^{(n)}_{-n}\ldots y^{(n)}_{-1}v_{0}\ldots v_{n}\), \(n\in{\mathbb{N}}\) in the language \(\mathcal{L}(v)\). Next we choose a collection \(\{\{n_{k}^{l}\}_{k\in{\mathbb{N}}}\}_{l\in{\mathbb{N}}}\) of increasing sequences of natural numbers by the following inductive procedure: Since \(\{y^{(n)}_{-1}\}_{n\in{\mathbb{N}}}\) takes only two values, we can find an increasing sequence \(\{n_{k}^{1}\}_{k\in{\mathbb{N}}}\) such that \(\{y^{(n_{k}^{1})}_{-1}\}_{k\in{\mathbb{N}}}\) is constant. Assume that the sequence \(\{n_{k}^{l}\}_{k\in{\mathbb{N}}}\) is already chosen. Choose \(\{n_{k}^{l+1}\}_{k\in{\mathbb{N}}}\) as a subsequence of \(\{n_{k}^{l}\}_{k\in{\mathbb{N}}}\) for which \(\{y^{(n_{k}^{l+1})}_{-l-1}\}_{k\in{\mathbb{N}}}\) is constant. Now we can define the desired sequence \(y=\{y_{k}\}_{k\in{\mathbb{Z}}}\in X\) by putting
+
+\[y_{-k}=y^{(n_{k}^{k})}_{-k}\text{ for }k>0\text{ and }y_{k}=v_{k}\text{ for }k\geq 0.\]
+
+## 8. Appendix: Kolmogorov-Tihomirov formula
+
+In this section we prove formula (2). Cover \(A\) by \(n=\mathcal{N}_{\epsilon/(4R)}(A)\) balls \(A_{1},\ldots,A_{n}\) of radius \(\epsilon/(8R)\) centered at \(a_{1},\ldots,a_{n}\in A\) respectively and cover \(Y\) by \(m=\mathcal{N}_{\epsilon/4}(Y)\) balls \(Y_{1},\ldots,Y_{m}\) of radius \(\epsilon/8\) centered at \(y_{1},\ldots,y_{m}\) respectively. Put \(I=\{1,\ldots,n\}\), \(J=\{1,\ldots,m\}\). For a map \(\sigma:I\to J\) set
+
+\[X_{\sigma}=\{f\in\mathcal{D}^{A}_{R}(Y)\;:\;f(a_{i})\in Y_{\sigma(i)}\;\forall i\in I\}\;.\]
+
+Obviously, \(\mathcal{D}^{A}_{R}(Y)\) is covered by \(m^{n}\) sets \(X_{\sigma}\). _Warning:_ some of these sets might be in fact empty.
+
+Assume that \(f,g\in X_{\sigma}\cap\mathcal{D}^{A}_{R}(Y)\). Take any point \(a\in A\). Choose \(a_{i}\) so that \(\rho_{1}(a,a_{i})<\epsilon/(8R)\). Then \(\rho_{2}(f(a),f(a_{i}))<\epsilon/8\) and \(\rho_{2}(g(a),g(a_{i}))<\epsilon/8\) since the Lipschitz constant of \(f\) and \(g\) is \(\leq R\). Furthermore, \(\rho_{2}(f(a_{i}),y_{\sigma(i)})<\epsilon/8\) and \(\rho_{2}(g(a_{i}),y_{\sigma(i)})<\epsilon/8\). Thus \(\rho_{2}(f(a),g(a))<\epsilon/2\). Since this is true for all points \(a\) in a compact space \(A\) we conclude that \(\text{dist}(f,g)<\epsilon/2\). It follows that the set \(X_{\sigma}\cap\mathcal{D}^{A}_{R}(Y)\) is either empty, or is fully contained in a ball of radius \(\epsilon/2\) (in the sense of metric dist) centered at any of its points.
+
+Looking at all \(\sigma\in J^{I}\), we get a covering of \(\mathcal{D}^{A}_{R}(Y)\) by at most \(m^{n}\) of metric balls of radius \(\epsilon/2\), as required.
+
+**Acknowledgements****.**: We are grateful to A. Katok for very useful comments on the first draft of this paper which have led us to an application of Theorem 1.5 to bi-Lipschitz ergodic theory. We thank G. Forni, G. Lederman and M. Sodin for useful discussions and the referee for very helpful remarks and suggestions.
+
+## References
+
+* [1]J.-P. Allouche, J.O. Shallit, , _Complexité des suites de Rudin-Shapiro généralisées_. J. Theor. Nombres Bordeaux **5** (1993), 283–302.
+* [2]G. D’Ambra, M. Gromov, _Lectures on transformation groups: geometry and dynamics._ Surveys in differential geometry (Cambridge, MA, 1990), 19–111, Lehigh Univ., Bethlehem, PA, 1991.
+* [3] A. Borichev, _Slow area-preserving diffeomorphisms of the torus,_ Israel J. Math. **141** (2004), 277–284.
+* [4] I. Chavel, _Eigenvalues in Riemannian geometry,_ Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984.
+* [5] K. Falconer, _Fractal geometry. Mathematical foundations and applications._ John Wiley & Sons, Inc., Hoboken, NJ, 2003.
+* [6] B. Fayad, _Skew products over translations on \(T^{d},d\geq 2\)_, Proc. Amer. Math. Soc. **130** (2002), 103–109.
+* [7] B. Fayad, _Analytic mixing reparametrizations of irrational flows_, Ergodic Theory Dynam. Systems **22** (2002), 437–468.
+* [8] Fogg, N. Pytheas, _Substitutions in dynamics, arithmetics and combinatorics._ Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Lecture Notes in Mathematics, 1794. Springer-Verlag, Berlin, 2002.
+* [9] R. Fuchs, _Slow diffeomorphisms of a manifold with \(\mathbb{T}^{2}\) action,_ Nonlinearity **19** (2006), no. 7, 1635–1641.
+* [10] A. Katok, J.-P. Thouvenot, _Spectral properties and combinatorial constructions in ergodic theory_, in _Handbook of dynamical systems, B. Hasselblatt and A. Katok eds., Vol. 1B, 649–743, Elsevier B. V., Amsterdam, 2006._
+* [11] A.N. Kolmogorov, V.M. Tihomirov, \(\varepsilon\)_-entropy and \(\varepsilon\)-capacity of sets in function spaces,_ Uspehi Mat. Nauk **14** (1959), 3–86 (in Russian). English translation: Amer. Math. Soc. Transl. 17, 277–364 (1961).
+* [12] M. Queffélec, _Substitution dynamical systems—spectral analysis._ Lecture Notes in Mathematics, 1294. Springer-Verlag, Berlin, 1987.
+

arxiv_ground_truth/0706.2397.md

@@ -0,0 +1,316 @@
+# Inversely Unstable Solutions of Two-Dimensional Systems on Genus-_p_ Surfaces and the Topology of Knotted Attractors
+
+Yi SONG and Stephen P. BANKS,
+Department of Automatic Control and Systems Engineering,
+University of Sheffield, Mappin Street,
+Sheffield, S1 3JD.
+e-mail: s.banks@sheffield.ac.uk
+
+###### Abstract
+
+In this paper, we will show that a periodic nonlinear ,time-varying dissipative system that is defined on a genus-\(p\) surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.
+**Keywords**: Knotted attractor, Automorphic Functions, \(C^{\infty}\) Functions, Periodic orbit, Inversely unstable solution.
+
+## 1 Introduction
+
+The general theory of dynamical systems is, of course, a subject with a long and distinguished history , (see, for example, [Smale, 1967], [Bowen, 1928] and [Manning, 1974]). In particular, the study of the dynamical behaviour of non-conservative and chaotic systems has attracted a lot of attention in the past, (see, for example, [Levinson, 1944], [Martins, 2004] and [Wiggins, 1988]). Consider a system
+
+\[\left\{\begin{array}[]{l}\dot{x}=F(x,y,t)\\ \dot{y}=G(x,y,t),\end{array}\right.\] (1)
+
+where \(F(x,y,t)\) and \(G(x,y,t)\) are both periodic in t. It will be called dissipative or non-conservative if there is a locally proper invariant set on the corresponding \(2\)-manifold on which the system is defined. Most real systems are of this kind. Up to the present a great deal of interest has been paid to the study of the topology of this invariant set (e.g. [Levinson, 1944]).
+
+Recently, in [Martins, 2004], it is shown that a system
+
+\[\ddot{x}+h(x)\dot{x}+g(t,x)=0,\] (2)
+
+where \(h\) and \(g\) are smooth functions, periodic on both \(x\) and \(t\), which is essentially a periodic nonlinear 2-dimensional, time-varying oscillator with appropriate damping, contains an invariant set which is not homeomorphic to a circle if there exists an inversely unstable solution.
+
+In this paper we are interested in generalizing this result and we will show that instead of just one invariant set, several attractors can coexist and even be linked in a higher genus surface on which the system is defined. We will also study the topology of these attractors in a similar way to [Martins, 2004] and show the existence of an inversely unstable solution implies that the specific invariant set is not homeomorphic to a circle.
+
+Moreover, in [Banks, 2002], a way to express a system situated on a genus-\(p\) surface in terms of a spherical one is presented. This is achieved by opening each handle, i.e., cutting along a fundamental circuit which contains no equilibria and adding appropriate fixed points on the resulting sphere (as shown in fig (1)). In this paper, we will also study the relation between dissipative systems on a \(p\)-hole surface and those sitting on a sphere.
+
+Figure 1: Express a Genus-1 System onto a Sphere
+
+In order to motivate the ideas, we reformulate Martins’ result in the following way.
+
+The system given by (1) can be written in the form
+
+\[\left\{\begin{array}[]{l}\dot{y_{1}}=y_{2}-H(y_{1})\\ \dot{y_{2}}=-g(t,y_{1})\end{array}\right.\] (3)
+
+where \(H(x)={\int}_{0}^{x}h(s)ds\), and \(g\) is \(T\)-periodic in \(t\). The \(Poincar\acute{e}\) map is defined as \(P(y_{0})=y(T;0,y_{0})\). Since the vector field \((y_{1},y_{2})\to\big{(}y_{2}-H(y_{1}),-g(t,y_{1})\big{)}\) is periodic with period \(R=\big{(}1,h(1)\big{)}\), the solutions \(y\) and \(y+kR\)\((k\in\mathbb{Z})\) are equivalent and so the system may be defined on a cylinder, as in fig (2).
+
+Figure 2: The Invariant Set Defined on a Cylinder
+
+Here \(\mathcal{A}\) is the invariant set
+
+\[\mathcal{A}=\bigcap_{n\in\mathbb{N}}\overline{P}^{n}(\overline{B}_{\rho_{0}})\]
+
+where \(B_{\rho_{0}}\) is some bounded set (which exists because the system is dissipative, as implied by the arrows in fig (2)). In [Martins, 2004], he shows that \(\mathcal{A}\) is not homeomorphic to a circle if there is an inversely unstable periodic orbit somewhere; we can think of the problem as sitting on a torus with one unstable cycle, as in fig (3). It is in this form that we shall generalize the result to higher genus surfaces.
+
+Figure 3: Invariant Set in the Torus Case
+
+## 2 Systems On Genus-p Surfaces
+
+In [Banks & Song, 2006] we have shown how to write down analytic (or meromorphic) systems on genus-\(p\) surfaces by the use of automorphic functions. These systems are not general enough, however, to include systems with knots, chaotic annuli, etc.. So we must consider vector fields which are \(C^{\infty}\) but which are invariant under certain linear, fractional transforms. This will be the analogue of systems which are periodic in [Martins, 2004] and have inversely unstable periodic motions – the latter now becoming knots on the genus-\(p\) surface.
+
+In order to generate the most general \(C^{\infty}\) systems on genus-\(p\) surfaces, consider a fundamental domain \(F\) in the upper-half plane model of the hyperbolic plane for the surface as in fig (4).
+
+Figure 4: Fundamental region of a genus-\(p\) surface
+
+Let \(\Gamma\) be a \(Fuchsian\) group (see [Ford, 1929] and [Banks & Song, 2006]) with a subset \(\Gamma_{1}=\{T_{i}\}\) that contains maps which pair the sides of \(F\). Each map \(T_{i}\) is of the form:
+
+\[T_{i}(z)=\frac{az+b}{cz+d}\] (4)
+
+and we shall consider them in real form:
+
+\[T_{i}(x,y)=\big{(}\tau_{ix}(x,y),\tau_{iy}(x,y)\big{)}\] (5)
+
+where \(z=x+iy\).
+
+Because of the need to generate \(C^{\infty}\) systems defined on a genus-\(p\) surface, we must ensure that, if \(T_{i}\) pairs the sides \(C_{i_{1}}\) and \(C_{i_{2}}\), as in fig (4), then the vector field \(v\) in the hyperbolic plane at corresponding points \(q\) satisfies
+
+\[v\big{(}T_{i}(q)\big{)}=(T_{i})_{\ast}\big{(}v(q)\big{)}\] (6)
+
+where \((T_{i})_{\ast}\) is the tangent map of \(T_{i}\).
+
+**Lemma 2.1**: _There exists a map from F onto a rectangle R which is one-to-one on the interior and \(C^{\infty}\)\(apart\) from at the cusp points._
+
+**Proof.** We shall construct the map explicitly so that the required properties will be clear.
+
+Figure 5: Map the fundamental region _F_ onto a rectangle _R_
+
+An elementary calculation shows that
+
+\[\left\{\begin{array}[]{l}x={\phi}_{1}(r,\theta)\\ y={\phi}_{2}(r,\theta)\end{array}\right.\]
+
+where
+
+\[{\phi}_{1}(r,\theta) = \frac{\pi}{\pi-2\alpha}\cdot(\theta-\alpha)\]
+\[{\phi}_{2}(r,\theta) = r\] (7)
+
+where \(\alpha\) is the value of the starting angle corresponding to the curve within the fundamental region in the \((r,\theta)-plane\)(as shown in fig (5)).   \(\Box\)
+
+We shall call \(R\) the modified fundamental region, and write this map as
+
+\[\phi:(r,\theta)\to(x,y).\] (8)
+
+Let
+
+\[D_{i}={\phi}(C_{i})\] (9)
+
+be the edges of the curves \(C_{i}\) on the boundary of \(F\). From the above remarks we see that a vector field _w_ on \(R\) which is associated with one on the original surface must satisfy
+
+\[w\Big{(}{\phi}\big{(}T_{i}(q)\big{)}\Big{)}={\phi}_{\ast}\Big{(}(T_{i})_{\ast}\big{(}w(q)\big{)}\Big{)},\quad q\in D_{i_{1}}\] (10)
+
+where \(T_{i}\) pairs the segments \(D_{i_{1}}\) and \(D_{i_{2}}\). Let
+
+\[m_{1},m_{2},\cdots,m_{4p}\in R,\]
+
+denote the points
+
+\[m_{i}=\phi(i)\]
+
+(i.e., the \(\phi\)-image of the cusp points on \(F\)). Then we have
+
+**Lemma 2.2**: _Any vector field w which is \(C^{\infty}\) on the interior of R and satisfies (10) where \(\phi\) is given by (2) and such that_
+
+\[w(m_{i})=0\]
+
+_defines a unique vector field on a genus-p \((p>1)\) surface._
+
+**Proof.** The only part left to prove is the converse. This follows from the above remarks and the \({Poincar\acute{e}}\) index theorem—any dynamical system on a surface of genus \(p>1\) must have at least one equilibrium point. We can choose this as the cusp points of \(F\).   \(\Box\)
+
+Figure 6: Closed curves on a surface
+
+We next consider the existence of periodic knotted trajectories on the surface. By the above results we can restrict attention to a rectangle \(R\) as shown in fig (5). Any closed curve on the surface is given by a set of non-intersecting curves which ‘match’ in the sense of (10) at identified boundaries. For example, the set of curves shown in fig (6.a) form a closed curve on the corresponding surface; moreover, fig (6.b) stands for a _trefoil knot_ on a \(2\)-hole torus if we identify the sides properly.
+
+Of course, the knot type of this closed curve depends on the embedding of the surface in \(\mathbb{R}^{3}\) (or \(\mathrm{S}^{3}\)). For instance, the surface in fig (6.a) could be embedded as in fig (7), which also gives a knot diagram from which one can calculate a knot invariant (such as the Kauffman Polynomial).
+
+Figure 7: A \(Trefoil\) Knot On a Surface
+
+Let \(\psi_{i}(x,y,t)\) denote the curve \(C_{i}\) within the modified fundamental region \(R\), \(f_{i}(x,y)\) and \(g_{i}(x,y)\) be any \(C^{\infty}\) functions that guarantee the matching of vector fields at the identified boundaries via (10). Hence we have
+
+**Lemma 2.3**: _If there are \(C_{i}\) (\(1\leqslant i\leqslant k\)) curves within the modified fundamental region that stand for a periodic trajectory of a dynamical system on a genus-p surface in \(\mathbb{R}^{3}\), then this system can be defined by_
+
+\[\dot{x} = \sum_{i=1}^{k}\Bigg{(}\bigg{(}\frac{\partial\psi_{i}}{\partial y}+\psi_{i}f_{i}\bigg{)}\cdot\prod_{j\neq i}\psi_{j}\Bigg{)}\]
+\[\dot{y} = \sum_{i=1}^{k}\Bigg{(}\bigg{(}-\frac{\partial\psi_{i}}{\partial x}+\psi_{i}g_{i}\bigg{)}\cdot\prod_{j\neq i}\psi_{j}\Bigg{)}\] (11)
+
+**Proof.** Since \(\psi_{i}\) defines the curve \(C_{i}\) in region \(R\), we get
+
+\[\psi_{i}(x,y,t_{i}) = 0\] (12)
+\[\qquad\frac{\partial\psi_{i}}{\partial x}\cdot\dot{x}+\frac{\partial\psi_{i}}{\partial y}\cdot\dot{y} = 0\] (13)
+
+For each curve \(C_{i}\), \(\psi_{i}\) switches off all terms in (11) except the \(i\)th one. Substitute (11) into (13) and we have
+
+\[{\frac{\partial\psi_{i}}{\partial x}}\bigg{(}{\frac{\partial\psi_{i}}{\partial y}}+\psi_{i}f_{i}\bigg{)}{\prod_{j\neq i}{\psi_{j}}}+{\frac{\partial\psi_{i}}{\partial y}}\bigg{(}-{\frac{\partial\psi_{i}}{\partial x}}+\psi_{i}g_{i}\bigg{)}{\prod_{j\neq i}{\psi_{j}}}\]
+\[={\prod_{j\neq i}}{\psi_{j}}\bigg{(}{\frac{\partial\psi_{i}}{\partial x}}\psi_{i}f_{i}+{\frac{\partial\psi_{i}}{\partial y}}\psi_{i}g_{i}\bigg{)}=0,\] (14)
+
+so the lemma follows.   \(\Box\)
+
+## 3 The \(\mathbf{Poincar\acute{e}}\) Map and Knotted Attractors
+
+Equation (11) now can be regarded as a general form of dynamical systems in the hyperbolic upper-half plane, which can be situated on a genus-\(p\) surface after identification of the corresponding sides.
+
+Again Consider the \({Poincar\acute{e}}\) map \(P:\mathbb{R}^{2}\to\mathbb{R}^{2}\) given by
+
+\[P(x_{0},y_{0})=Y\big{(}T;0,(x_{0},y_{0})\big{)},\]
+
+where \(Y(t)=Y\big{(}t;0,(x_{0},y_{0})\big{)}\) is the solution of (11) starting from point \((x_{0},y_{0})\). Because of the ‘periodicity’ from the automorphic form which is defined by the \(Fuchsian\) group \(\Gamma\), we have
+
+\[P\big{(}\Gamma_{i}(x_{0},y_{0})\big{)}=\Gamma_{i}\big{(}P(x_{0},y_{0})\big{)}\] (15)
+
+where \(\Gamma_{i}\) is the transformation from one fundamental region to another one next to it. Moreover, if \((x,y)\) is a solution of (11), so is \(\Gamma_{i}^{n}(x,y)\)\((n\in\mathbb{Z})\). Restricting our attention onto one fundamental region \(F\)(as shown in fig (4)), we only need to consider the dynamics within it, and obviously the \(Poincar\acute{e}\) map is well defined on \(F\).
+
+If the system given by (11) is dissipative, then there exists an unstable periodic orbit, which means all the trajectories are pointing outward along this closed curve. Assume that it is represented by \(\{\psi_{i}\}\)\((1\leq i\leq k)\), we are mainly interested in what the dynamics will look like on the rest of the surface.
+
+To begin, we need to perform some surgery to the \(2\)-manifold. By cutting along this closed orbit, the genus-\(p\) surface will effectively turn into a \((p-1)\)-hole torus with two boundary circles being introduced.
+
+_Remark._ To make this statement much clearer, we now look at an example of cutting along a _trefoil knot_ that sits on a torus.
+
+Figure 8: Cut a \(Trefoil\) Knot On a Torus
+
+As shown in fig (8), by cutting the torus along this _trefoil knot_ and identifying the corresponding parts on both sides, ‘m’ and ‘n’, we get a cylinder with the two ends being the original _trefoil knot_. This surgery can always be performed on the genus-\(p\) surface such that the knot along which is cut will generate one pair of the sides in the fundamental domain, and this results in the constructed 2-manifold being a \((g-1)\)-hole torus with two boundary circles (as shown in fig (9)).
+
+Figure 9: Cut a \(g\)-hole Torus Open Along a Knot
+
+In [Martins, 2004], he studied the torus case, (i.e., \(\textrm{genus}=1\)), and showed that if there exists a trivial unstable periodic orbit, then an invariant set \(\mathcal{A}\), a band around the tube, which may or may not be homeomorphic to a circle, must exist (see fig (3) for an illustration). \(\mathcal{A}\) is a compact, non-empty, connected set, and it acts as an attractor towards which all the dynamics converge. It is given by the iterations of the \(Poinca\acute{e}\) map within the fundamental region to a well-defined bounded set.
+
+In the case of genus-\(2\) surfaces, the same argument applies for the existence of the invariant set as in [Martins, 2004], while the exact number of the invariant sets may vary. More specifically, if we cut a \(2\)-hole torus along a knotted trajectory, topologically the surface will turn into a torus but with 2 boundary circles, as illustrated in fig (10).
+
+Figure 10: Cut a \(2\)-hole Torus
+
+Suppose that this knot is unstable; after the surgery, all the dynamics are pointing outward from the two resulting boundary circles. Since fig (10) is essentially a cylinder with a torus attached in the middle, from [Martins, 2004], we know that there exists some invariant set \(\mathcal{A}\). However, the number of invariant sets differs from that of the genus-\(1\) case. There can be at most three invariant sets, individually as shown in fig (11).
+
+Figure 11: Possible Invariant Sets in a Genus-\(2\) Surface
+
+In this figure, \(\mathcal{A}\) denotes the invariant set, while \(m\) and \(m^{\prime}\) stand for the saddle type equilibrium points which have \(-1\) index respectively. This makes sense because \(Index(m)+Index(m^{\prime})=-2\), which accounts for the correct \(Poincar\acute{e}\) characteristic for a genus-\(2\) surface. Note that the actual invariant set may be a combination of two or all of these three individual ones (see fig (12) for the illustration).
+
+Figure 12: Possible Combination of the Invariant Sets
+
+As the genus of a surface increases, the number of the invariant sets will increase accordingly, but they are all based on the three basic types shown in fig (11).
+
+**Lemma 3.1**: _For a system situated on the genus-\(p\) surface, if only one unstable periodic orbit exists, then there can be at most \((2p-1)\) attractors that might be knotted themselves and linked together. Moreover, a surgery can always be performed to make them look like a combination of the basic invariant sets shown in fig (11)._
+
+**Proof.** We prove it by induction.
+
+In the torus case, it is known that the attractor is a band as shown in fig (3) ([Martins, 2004]); and on a \(2\)-hole torus, from the discussion above, there can be at most \(3\) attractors.
+
+Assume it is true for the genus-\(p\) surface such that it has \((2p-1)\) invariant sets at most, then by adding the genus by \(1\), we essentially introduce another hole to the manifold which will give two more attractors at most, this proves the lemma.   \(\Box\)
+
+## 4 Inversely Unstable Solutions and the Topology of Knotted Attractors
+
+Inversely unstable solutions to a dynamical system has been studied for a long time. To be precise, we restate the main idea, which can be found in [Levinson, 1944], for example.
+
+Denote (11) by
+
+\[\left\{\begin{array}[]{l}\dot{x}=F(x,y,t)\\ \dot{y}=G(x,y,t)\end{array}\right.\] (16)
+
+where \(F\) and \(G\) are both \(T\)-periodic in \(t\).
+
+**Definition 4.1**: _Suppose \((a,b)\in\mathbb{Z}\times\mathbb{N}\), \(b\geqslant 1\), we shall say that a solution \(z=(x,y)\) of (16) is \((a,b)\)-periodic iff_
+
+\[z(t+bT)=\Gamma_{i}^{a}\big{(}z(t)\big{)}\]
+
+_where \(\Gamma_{i}\) is the map between one fundamental region and the other one next to it._
+
+Note that these solutions correspond to the trajectories that ‘wind around’ one of the tubes in the genus-\(p\) surface \(a\) times within a \(bT\) time interval before closing. If \(\big{(}x(t),y(t)\big{)}\) is a \((a,b)\)-periodic solution, then the initial point \(A\), \(\big{(}x(t_{0}),y(t_{0})\big{)}\), is a fixed point of \(M=P^{b}-\Big{(}\Gamma^{a}\big{(}z(0)\big{)}-z(0)\Big{)}\). Assume \(A\) is an isolated fixed point, and let \(A_{0}\) denote the point \(\big{(}x(t_{0})+u_{0},y(t_{0})+v_{0}\big{)}\) near \(A\) in the hyperbolic upper-half plane. Applying the \(Poincar\acute{e}\) map once we have
+
+\[A_{1}=P(A_{0})\]
+
+and \(A_{1}\) is denoted by \(\big{(}x(t_{0})+u_{1},y(t_{0})+v_{1}\big{)}\). By using a power series in \(u_{0}\) and \(v_{0}\) with coefficients functions in \(t\), we can express the solution trajectory of \(\big{(}x(t),y(t)\big{)}\) starting at \(A_{0}\) by
+
+\[X(t) = x(t)+c_{1}(t)u_{0}+c_{2}(t)v_{0}+c_{3}(t)u_{0}^{2}+c_{4}(t)u_{0}v_{0}+\cdots\]
+\[Y(t) = y(t)+d_{1}(t)u_{0}+d_{2}(t)v_{0}+d_{3}(t)u_{0}^{2}+d_{4}(t)u_{0}v_{0}+\cdots\] (17)
+
+In particular, by setting \(t=t_{0}+T\), we have
+
+\[u_{1} = au_{0}+bv_{0}+a_{1}u_{0}^{2}+b_{1}u_{0}v_{0}+\cdots\]
+\[v_{1} = cu_{0}+dv_{0}+c_{1}u_{0}^{2}+d_{1}u_{0}v_{0}+\cdots\] (18)
+
+If we denote \(\big{(}x(t_{0})+u_{0},y(t_{0})+v_{0}\big{)}\) and \(\big{(}x(t_{0})+u_{1},y(t_{0})+v_{1}\big{)}\) by \((x_{0},y_{0})\) and \((x_{1},y_{1})\) respectively, then
+
+\[J(\frac{x_{1},y_{1}}{x_{0},y_{0}})=J(\frac{u_{1},v_{1}}{u_{0},v_{0}})\] (19)
+
+where \(J\) is the Jacobian of the \(Poincar\acute{e}\) map for the point \((x_{0},y_{0})\). For very small values \(u_{0}\) and \(v_{0}\), (4) is determined by its linear terms. So the characteristic multiplier can be determined by
+
+\[(a-\lambda)(d-\lambda)-bc=0.\]
+
+Using the notations above, we have
+
+**Definition 4.2**: _Given an \((a,b)\)-periodic solution \(\big{(}x(t),y(t)\big{)}\) of (16) such that \(\big{(}x(t_{0}),y(t_{0})\big{)}\) is an isolated fixed point of \(M\), we shall say the solution \(\big{(}x(t),y(t)\big{)}\) is inversely unstable iff \(\lambda_{2}<-1<\lambda_{1}<0\)._
+
+In [Martins, 2004], it is shown that in the torus case, the invariant set \(\mathcal{A}\) may not be homeomorphic to a circle. We shall now extend the ideas to higher genus surfaces. To do this we need
+
+**Definition 4.3**: _A system defined on a surface S is dissipative relative to a knot K if there is a neighbourhood, say N, of K in S such that on \(\partial(S/N)\), the vector field is pointing into N._
+
+Then we have
+
+**Theorem 4.1**: _Given a system defined by (11) on a genus-\(p\) surface, which is dissipative relative to a knot K situated on this surface as well, if there exists an inversely unstable solution \((x_{I},y_{I})\) within the (knotted) attractor \(\mathcal{A}_{I}\), then \(\mathcal{A}_{I}\) is not homeomorphic to the circle \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\)._
+
+**Proof.** We shall prove this theorem in a geometrical way. Due to the dissipative nature, there exist one or more unstable periodic orbits, and each of them is equivalent to a knot on the surface that the system is defined on respectively. By cutting the surface along one of these knots, we can reduce the surface genus by \(1\) while introducing two boundary circles (as shown in fig (10)). Now gluing the two circles will produce a tube containing an attractor \(\mathcal{A}\). Assume that there exists an inversely unstable solution in \(\mathcal{A}\). Let \(A\) be a fixed point of the associated \(Poincar\acute{e}\) map. Choose a neighbourhood \(U\) where \(A\) is the only fixed point within \(U\). Suppose \(A_{0}\) is a point in \(U\) close to \(A\) (see fig (13) for illustration). If we apply the \(Poincar\acute{e}\) map to point \(A_{0}\), with the dynamics being determined by the characteristic multipliers, which are \(\lambda_{2}<-1<\lambda_{1}<0\), \(A_{0}\) will move to \(A_{1}\), a point lies in the other half plane with respect to \(y\)-axis and is much closer to the fixed point \(A\). Now apply the \(Poincar\acute{e}\) map to point \(A_{1}\), and this time the characteristic multipliers will become \(0<\lambda_{1}^{2}<1<\lambda_{2}^{2}\) under the action of \(P^{2}\), which gives a directly unstable solution that moves \(A_{1}\) to \(A_{2}\), a point further away in the left-half plane. With the iteration of \(Poincar\acute{e}\) map, the corresponding characteristic multipliers will be alternatively positive and negative. However, all neighbouring dynamics tend towards the knotted attractor by dissipativity. In other words, within the invariant set near the inversely unstable solution, the dynamics tend either to get close to this trajectory or escape from it, while at the boundary, they are pushed back by the external dissipative condition. This is why chaotic behaviour can happen which means that \(\mathcal{A}\) is not homeomorphic to a circle. The same idea follows when there are more than one attractor contain separate inversely unstable solutions.   \(\Box\)
+
+Figure 13: How an Inversely Unstable Solution will Affect the Dynamics
+
+So generally speaking, a dissipative system given by (4) that situated on a genus-\(p\) surface can have at most \(p\) topologically distinct knotted attractors; whether they are homeomorphic to a circle individually depends on the existence of inversely unstable solution within themselves.
+
+It is known that any dynamical system sitting on a \(2\)-manifold with \(p\) genus can be represented on a sphere by cutting each handle along a fundamental circuit which contains no equilibrium point and filling in the dynamics within the resulting region bounded by these curves (see [Banks, 2002]).
+
+Conversely, we can get higher genus surface systems by performing surgery on certain spherical ones. Specifically, given a spherical system, irrespective of the rest of the dynamics, as long as it contains \(2\) stable equilibria, we can choose a small neighbourhood \(M_{i}\)\((i=1,2)\) around each of them such that they are the only equilibrium points within each region. Glue in a dissipative region with attractor \(\mathcal{A}\) as in fig (14), cut this attractor open, twist it and identify the two ends together in the appropriate way, we then obtain the desired knot. If the attractor contains an inversely unstable solution, then it is not homeomorphic to a circle, which means chaotic behaviour will occur within this invariant set.
+
+Figure 14: Construct a Torus System from a Spherical One
+
+Hence we have proved
+
+**Theorem 4.2**: _Any dynamical system on a genus-\(p\) surface that contains a set of k \((k\geq 1,k\in\mathbb{N})\) (knotted) dissipative attractors each containing an inversely unstable orbit can be represented by a system with at least 2k stable equilibrium points on a sphere. Conversely, starting from a spherical system that contains at least 2k stable equilibria, we can construct a system on a genus-\(p\)\(2\)-manifold that contains k knotted attractors each with chaotic behaviour._
+
+_Remark._ An important consequence of this theorem is that we can determine the general structure of a system with \(k\) ‘chaotic’ dissipative attractors by studying systems with \(2k\) stable equilibrium points on the sphere. Of course, such a system must have other equilibrium points so that the total index is 2, by the \(Poincar\acute{e}\) index theorem. Thus the remaining equilibrium points must have index \(2-2k\). This implies the existence of some hyperbolic points.
+
+## 5 Examples
+
+In this section we show that we can obtain systems with dissipative chaotic behaviour by choosing stable and unstable knotted orbits, and the unstable orbit acts as the dissipative ‘repeller’.
+
+Figure 15: A Surface of Genus Two Carrying Two Distinct Knot Types
+
+In [Banks, 2002], it is shown that for a dynamical system on a surface of genus \(p\), it can carry at most \(p\) distinct types of (homotopically nontrivial) knots. For example, fig (15) shows the two distinct knot types that a system can have on a genus-\(2\) surface.
+
+Assume that these two knots act as two attractors, (the existence of chaotic behaviour will depend on whether there is an inversely unstable solution within each attractor,) then there must exist one or more unstable orbits due to which these two invariant sets are generated. To find it out explicitly, we first represent the system onto a sphere with four holes, which is achieved by cutting along two fundamental circuits to open the handles out, as shown in fig (16.a).
+
+Figure 16: Spherical Representation For the Attractors and The Possible Dynamics Elsewhere
+
+The unstable orbits therefore should bound each part of the attractors presented on the sphere such that they can push the dynamics toward the invariant sets and introduce possible chaotic behaviour. Moreover, there must exist some equilibrium points to give the correct index of a genus-\(2\) surface, which is \(-2\). Fig (16.b) shows a possible solution trajectories of two unstable orbits which satisfy the criteria discussed above. Please note that the solution trajectories may not be unique.
+
+Figure 17: Genus-\(2\) Surface Containing 2 Knotted Attractors and the Corresponding Dynamics
+
+Recover the original 2-manifold by gluing the corresponding boundary circles, we eventually get a system on a genus-\(2\) surface. It has two unstable periodic cycles, which generate two knotted attractors with distinct types, and two saddle equilibrium points which give the correct index of \(-2\)(See fig (17) for an illustration).
+
+Moreover, as in fig (18), if each invariant set contains an inversely unstable orbit, then around each knot there exists a band within which chaotic behaviour will occur.
+
+Figure 18: Genus-\(2\) Surface Containing 2 Invariant Sets With Inversely Unstable Orbit In
+
+Now if reduce the number of invariant sets by one and assume the existence of only one unstable orbit, following the same algorithm as above, we get one possible solution for the dynamics as in fig (19). Note that again there are two saddle equilibria to count for the correct index.
+
+Figure 19: Genus-\(2\) Surface Containing 1 Knotted Attractors and the Corresponding Dynamics
+
+Under the existence of inversely unstable orbit, chaotic behaviour will occur within the invariant set (see fig (20)).
+
+Figure 20: Genus-\(2\) Surface Containing 1 Knotted Attractor With an Inversely Unstable Orbit
+
+## 6 Conclusion
+
+We have studied dynamical systems on a genus-\(p\) surface and extend the _generalized automorphic functions_ (see [Banks & Song, 2006]) to define a general form for these systems (both analytic and non-analytic). Also we look at the topology of knotted attractors under the existence of unstable periodic orbits and prove that for a genus-\(p\) surface with only one unstable cycle, the number of invariant sets may vary while a maximum of \((2p-1)\) must not be exceeded. Moreover, we extend the result in [Martins, 2004] and show that for a higher genus \((genus>1)\) surface, the individual attractor is not homeomorphic to a circle if there exists an inversely unstable solution within itself. This is purely because of the property of inversely unstable solution which can generate a local behaviour to make the dynamics fight against the effect of global unstable orbit.
+
+In the future paper, we will consider _automorphic functions_ in 3-dimension which will give us systems naturally defined on genus-\(p\) solid 3-manifolds.
+
+## References
+
+* [1] Banks, S. P. “Three-dimensional stratifications, knots and bifurcations of two-dimensional dynamical systems”, Int. J. of Bifurcation and Chaos, Vol. 12, No. 1 (2002) 1-21.
+* [2] Banks, S. P. and Song, Y. “Elliptic and automorphic dynamical systems on surfaces”, Int. J. of Bifurcation and Chaos, Vol. 16, No. 4 (2006) 911-923.
+* [3] Bowen, R. “On axiom A diffeomorphisms”, AMS, Providence, RT., 1978.
+* [4] Ford, L. R. “Automorphic functions”, McGraw–Hill, 1929
+* [5] Levinson, N. “Transformation theory of non-linear differential equations of the second order”, The Annals of Mathematics, 2nd Ser., Vol. 45, No. 4. (Oct., 1944), 723-737.
+* [6] Manning, A. “There are no new Anosov diffeomeophisms on tori”, Amer. J. Math., 96 (1974), 422.
+* [7] Martins, R. “The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction”, J. of Differential Equations, Vol. 212, Issue 2, 15 May 2005, 351-365.
+* [8] Richeson, D. and Wiseman, J. “Bounded homeomorphisms of the open annulus”, New York J. Math. 9 (2003) 55-68.
+* [9] Smale, S. “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747.
+* [10] Wiggins, S. “Global bifurcations and chaos: Analytic methods”, Applied math. Sciences, (1988), New York: Springer–Verlag.
+

arxiv_ground_truth/0707.0311.md

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+# On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs¹
+
+¹
+
+Footnote 1: footnotetext: This research was supported by a Grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.
+
+ Rom Pinchasi
+
+ Mathematics Dept., Technion—Israel Institute of Technology, Haifa 32000, Israel. room@math.technion.ac.il.
+
+Günter Rote
+
+ Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195 Berlin, Germany. rote@inf.fu-berlin.de
+
+###### Abstract
+
+We answer a question raised by Walter Morris, and independently by Alon Efrat, about the maximum cardinality of an anti-chain composed of intersections of a given set of \(n\) points in the plane with half-planes. We approach this problem by establishing the equivalence with the problem of the maximum monotone path in an arrangement of \(n\) lines. A related problem on convex pseudo-discs is also discussed in the paper.
+
+## 1 Introduction
+
+Let \(P\) be a set of \(n\) points in the plane, no three of which are collinear. A subset of \(P\) is called _linearly separable_ if it is the intersection of \(P\) with a closed half-plane. A \(k\)-set of \(P\) is a subset of \(k\) points from \(P\) which is linearly separable. Let \(\mathcal{A}_{k}=\mathcal{A}_{k}(P)\) denote the collection of all \(k\)-sets of \(P\). It is a well-known open problem to determine \(f(k)\), the maximum possible cardinality of \(\mathcal{A}_{k}\), where \(P\) varies over all possible sets of \(n\) points in general position in the plane. The current records are \(f(k)=O(nk^{1/3})\) by Dey ([2]) and \(f(\lfloor n/2\rfloor)\geq ne^{\Omega(\sqrt{\log n})}\) by Tóth ([6]).
+
+Let \(\mathcal{A}=\mathcal{A}(P)=\cup_{k=0}^{n}\mathcal{A}_{k}\) be the family of all linearly separable subsets of \(P\). The family \(\mathcal{A}\) is partially ordered by inclusion. Clearly, each \(\mathcal{A}_{k}\) is an anti-chain in \(\mathcal{A}\). The following problem was raised by Walter Morris in 2003 in relation with the _convex dimension_ of a point set (see [3]) and, as it turns out, it was independently raised by Alon Efrat 10 years before, in 1993:
+
+**Problem 1****.**: What is the maximum possible cardinality \(g(n)\) of an anti-chain in the poset \(\mathcal{A}\), over all sets \(P\) with \(n\) points?
+
+In Section 2 we show that in fact \(g(n)\) can be very large, and in particular much larger than \(f(n)\).
+
+**Theorem 1****.**: \(g(n)=\Omega(n^{2-\frac{d}{\sqrt{\log n}}})\)_, for some absolute constant \(d>0\)._
+
+In an attempt to bound from above the function \(g(n)\) one can view linearly separable sets as a special case of a slightly more general concept:
+
+**Definition 1****.**: Let \(P\) be a set of \(n\) points in general position in the plane. A Family \(F\) of subsets of \(P\) is called a family of _convex pseudo-discs_ if the following two conditions are satisfied:
+
+1. 1.Every set in \(F\) is the intersection of \(P\) with a convex set.
+2. 2.If \(A\) and \(B\) are two different sets in \(F\), then both sets \({\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\) and \({\mbox{conv}(B)}\setminus{\mbox{conv}(A)}\) are connected (or empty).
+
+One natural example for a family of convex pseudo-discs is the family \(\mathcal{A}(P)\), where \(P\) is a set of \(n\) points in general position in the plane. To see this, observe that every linearly separable set is the intersection of \(P\) with a convex set, namely, a half-plane. It is therefore left to verify that if \(A=P\cap H_{A}\) and \(B=P\cap H_{B}\), where \(H_{A}\) and \(H_{B}\) are two half-planes, then both \({\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\) and \({\mbox{conv}(B)}\setminus{\mbox{conv}(A)}\) are connected. Let \(A^{\prime}=A\setminus H_{B}=A\setminus B=A\setminus{\mbox{conv}(B)}\). Since \({\mbox{conv}(A^{\prime})}\cap{\mbox{conv}(B)}=\emptyset\), we have \({\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\supset{\mbox{conv}(A^{\prime})}\). For any \(x\in{\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\), we claim that there is a point \(a^{\prime}\in A^{\prime}\) such that the line segment \([x,a^{\prime}]\) is fully contained in \({\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\). This will clearly show that \({\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\) is connected. Let \(a_{1},a_{2},a_{3}\) be three points in \(A\) such that \(x\) is contained in the triangle \(a_{1}a_{2}a_{3}\). If each line segment \([x,a_{i}]\), for \(i=1,2,3\), contains a point of \({\mbox{conv}(B)}\), it follows that \(x\in{\mbox{conv}(B)}\), contrary to our assumption. Thus there must be a line segment \([x,a_{i}]\) that is contained in \(A^{\prime}=A\setminus{\mbox{conv}(B)}\), and we are done.
+
+In Section 3 we bound from above the maximum size of a family of convex pseudo-discs of a set \(P\) of \(n\) points in the plane, assuming that this family of subsets of \(P\) is by itself an anti-chain with respect to inclusion:
+
+**Theorem 2****.**: _Let \(F\) be a family of convex pseudo-discs of a set \(P\) of \(n\) points in general position in the plane. If no member of \(F\) is contained in another, then \(F\) consists of at most \(4\binom{n}{2}+1\) members._
+
+Clearly, in view of Theorem 1, the result in Theorem 2 is nearly best possible. We show by a simple construction that Theorem 2 is in fact tight, apart from the constant multiplicative factor of \(n^{2}\).
+
+## 2 Large anti-chains of linearly separable sets
+
+Instead of considering Problem 1 directly, we will consider a related problem.
+
+**Definition 2****.**: For a pair \(x,y\) of points and a pair \(\ell_{1},\ell_{2}\) of non-vertical lines, we say that \(x,y\)_strongly separate_\(\ell_{1},\ell_{2}\) if \(x\) lies strictly above \(\ell_{1}\) and strictly below \(\ell_{2}\), and \(y\) lies strictly above \(\ell_{2}\) and strictly below \(\ell_{1}\).
+
+We will also take the dual viewpoint and say that \(\ell_{1},\ell_{2}\) strongly separate \(x,y\). (In fact, this relation is invariant under the standard point-line duality.)
+
+If we have a set \(L\) of lines, we say that the point pair \(x,y\) is _strongly separated_ by \(L\), if \(L\) contains two lines \(\ell_{1},\ell_{2}\) that strongly separate \(x,y\).
+
+A pair of lines \(\ell_{1},\ell_{2}\) is said to be strongly separated by a set \(P\) of points if there are two points \(x,y\in P\) that strongly separate \(\ell_{1}\) and \(\ell_{2}\).
+
+Using the above terminology one can reduce Problem 1 to the following problem:
+
+**Problem 2****.**: Let \(P\) be a set of \(n\) points in the plane. What is the maximum possible cardinality \(h(n)\) (taken over all possible sets \(P\) of \(n\) points) of a set of lines \(L\) in the plane such that for every two lines \(\ell_{1},\ell_{2}\in L\), \(P\) strongly separates \(\ell_{1}\) and \(\ell_{2}\).
+
+Figure 1: Problem 2.
+
+To see the equivalence of Problem 1 and Problem 2, let \(P\) be a set of \(n\) points and \(L\) be a set of \(h(n)\) lines that answer Problem 2. We can assume that none of the points lie on a line of \(L\). Then with each of the lines \(\ell\in L\) we associate the subset of \(P\) which is the intersection of \(P\) with the half-plane below \(\ell\). We thus obtain \(h(n)\) subsets of \(P\) each of which is a linearly separable subset of \(P\). Because of the condition on \(L\) and \(P\), none of these linearly separable sets may contain another. Therefore we obtain \(h(n)\) elements from \(\mathcal{A}(P)\) that form an anti-chain, hence \(g(n)\geq h(n)\).
+
+Conversely, assume we have an anti-chain of size \(g(n)\) in \(\mathcal{A}(P)\) for a set \(P\) of \(n\) points. Each linearly separable set is the intersection of \(P\) with a half-plane, which is bounded by some line \(\ell\). We can assume without loss of generality that none of these lines is vertical, and at least half of the half-spaces lie below their bounding lines. These lines form a set \(L\) of at least \(\lceil g(n)/2\rceil\) lines, and each pair of lines is separated by two points from the \(n\)-point set \(P\). Thus, \(h(n)\geq\lceil g(n)/2\rceil\).
+
+Before reducing Problem 2 to another problem, we need the following simple lemma.
+
+**Lemma 1****.**: _Let \(\ell_{1},\ldots,\ell_{n}\) be \(n\) non-vertical lines arranged in increasing order of slopes. Let \(P\) be a set of points. Assume that for every \(1\leq i<n\), \(P\) strongly separates \(\ell_{i}\) and \(\ell_{i+1}\). Then for every \(1\leq i<j\leq n\), \(P\) strongly separates \(\ell_{i}\) and \(\ell_{j}\)._
+
+Proof.: We prove the lemma by induction on \(j-i\). For \(j=i+1\) there is nothing to prove. Assume \(j-i\geq 2\). We first show the existence of a point \(x\in P\) that lies above \(\ell_{i}\) and below \(\ell_{j}\). Let \(B\) denote the intersection point of \(\ell_{i}\) and \(\ell_{j}\). Let \(r_{i}\) denote the ray whose apex is \(B\), included in \(\ell_{i}\), and points to the right. Similarly, let \(r_{j}\) denote the ray whose apex is \(B\), included in \(\ell_{j}\), and points to the right.
+
+Since the slope of \(\ell_{i+1}\) is between the slope of \(\ell_{i}\) and the slope of \(\ell_{j}\), \(\ell_{i+1}\) must intersect either \(r_{i}\) or \(r_{j}\) (or both, in case it goes through \(B\)).
+
+**Case 1.**\(\ell_{i+1}\) intersects \(r_{i}\). Then there is a point \(x\in P\) that lies above \(\ell_{i}\) and below \(\ell_{i+1}\). This point \(x\) must also lie below \(\ell_{j}\).
+
+**Case 2.**\(\ell_{i+1}\) intersects \(r_{j}\). Then, by the induction hypothesis, there is a point \(x\in P\) that lies above \(\ell_{i+1}\) and below \(\ell_{j}\). This point \(x\) must also lie above \(\ell_{i}\).
+
+The existence of a point \(y\) that lies above \(\ell_{j}\) and below \(\ell_{i}\) is symmetric. ∎
+
+By Lemma 1, Problem 2 is equivalent to following problem.
+
+**Problem 3****.**: What is the maximum cardinality \(h(n)\) of a collection of lines \(L=\{\ell_{1},\ldots,\ell_{h(n)}\}\) in the plane, indexed so that the slope of \(\ell_{i}\) is smaller than the slope of \(\ell_{j}\) whenever \(i<j\), such that there exists a set \(P\) of \(n\) points that strongly separates \(\ell_{i}\) and \(\ell_{i+1}\), for every \(1\leq i<h(n)\)?
+
+We will consider the dual problem of Problem 3:
+
+**Problem 4****.**: What is the maximum cardinality \(h(n)\) of a set of points \(P=\{p_{1},\ldots,p_{h(n)}\}\) in the plane, indexed so that the \(x\)-coordinate of \(p_{i}\) is smaller than the \(x\)-coordinate of \(p_{j}\), whenever \(i<j\), such that there exists a set \(L\) of \(n\) lines that strongly separates \(p_{i+1}\) and \(p_{i}\), for every \(1\leq i<h(n)\)?
+
+We will relate Problem 4 to another well-known problem: the question of the longest monotone path in an arrangement of lines.
+
+Consider an \(x\)-monotone path in a line arrangement in the plane. The _length_ of such a path is the number of different line segments that constitute the path, assuming that consecutive line segments on the path belong to different lines in the arrangement. (In other words, if the path passes through a vertex of the arrangement without making a turn, this does not count as a new edge.)
+
+**Problem 5****.**: What is the maximum possible length \(\lambda(n)\) of an \(x\)-monotone path in an arrangement of \(n\) lines?
+
+A construction of [1] gives a simple line arrangement in the plane which consists of \(n\) lines and which contains an \(x\)-monotone path of length \(\Omega(n^{2-\frac{d}{\sqrt{\log n}}})\) for some absolute constant \(d>0\). No upper bound that is asymptotically better than the trivial bound of \(O(n^{2})\) is known.
+
+Problem 5 is closely related to Problem 4, and hence also to the other problems:
+
+**Proposition 1****.**:
+\[h(n)\geq\left\lceil\frac{\lambda(n)+1}{2}\right\rceil,\] (1)
+
+\[\lambda(n)\geq h(n)-2\] (2)
+
+Proof.: We first prove \(h(n)\geq\lceil(\lambda(n)+1)/2\rceil\). Let \(L\) be a simple arrangement of \(n\) lines that admits an \(x\)-monotone path of length \(m=\lambda(n)\). Denote by \(x_{0},x_{1},\ldots,x_{m}\) the vertices of a monotone path arranged in increasing order of \(x\)-coordinates. In this notation \(x_{1},\ldots,x_{m-1}\) are vertices of the line arrangement \(L\), while \(x_{0}\) and \(x_{m}\) are chosen arbitrarily on the corresponding two rays which constitute the first and last edges, respectively, of the path. For each \(1\leq i<m\) let \(s_{i}\) denote the line that contains the segment \(x_{i-1}x_{i}\), and let \(r_{i}\) denote the line through the segment \(x_{i}x_{i+1}\).
+
+For \(1\leq i<m\), we say that the path bends downward at the vertex \(x_{i}\) if the slope of \(s_{i}\) is greater than the slope of \(r_{i}\), and it bends upward if the slope of \(s_{i}\) is smaller than the slope of \(r_{i}\). Without loss of generality we may assume that at least half of the vertices \(x_{1},\ldots,x_{m-1}\) of the monotone path are downward bends.
+
+Figure 2: Constructing a solution for Problem 4.
+
+Let \(i_{1}<i_{2}<\dots<i_{k}\) be all indices such that \(x_{i_{j}}\) is a downward bend, where \(k\geq(m-1)/2\). Observe that for every \(1\leq j<k\), the monotone path between \(x_{i_{j}}\) and \(x_{i_{j+1}}\) is an upward-bending convex polygonal path.
+
+We will now define \(k+1\) points \(p_{0},p_{1},\ldots,p_{k}\) such that for every \(0\leq j<k\) the \(x\)-coordinate of \(p_{j}\) is smaller than the \(x\)-coordinate of \(p_{j+1}\), and the line \(r_{i_{j}}\) lies above \(p_{j}\) and below \(p_{j+1}\) while the line \(s_{i_{j}}\) lies below \(p_{j}\) and above \(p_{j+1}\). This construction will thus show that \(h(n)\geq\lceil\frac{\lambda(n)+1}{2}\rceil\).
+
+For every \(1\leq j\leq k\) let \(U_{j}\) and \(W_{j}\) denote the left and respectively the right wedges delimited by \(r_{i_{j}}\) and \(s_{i_{j}}\). That is, \(U_{j}\) is the set of all points that lie below \(r_{i_{j}}\) and above \(s_{i_{j}}\). Similarly, \(W_{j}\) is the set of all points that lie above \(r_{i_{j}}\) and below \(s_{i_{j}}\).
+
+**Claim 1****.**: _For every \(1\leq j<k\), \(W_{j}\) and \(U_{j+1}\) have a nonempty intersection._
+
+Proof.: We consider two possible cases:
+
+**Case 1.**\(i_{j+1}=i_{j}+1\). In this case \(r_{i_{j}}=s_{i_{j+1}}\). Therefore any point above the line segment \([x_{i_{j}}x_{i_{j+1}}]\) that is close enough to that segment lies both below \(s_{i_{j}}\) and below \(r_{i_{j+1}}\) and hence \(W_{j}\cap U_{j+1}\neq\emptyset\).
+
+**Case 2.**\(i_{j+1}-i_{j}>1\). In this case, as we observed earlier, the monotone path between \(x_{i_{j}}\) and \(x_{i_{j+1}}\) is a convex polygonal path. Therefore, \(r_{i_{j}}\) and \(s_{i_{j+1}}\) are different lines that meet at a point \(B\) whose \(x\)-coordinate is between the \(x\)-coordinates of \(x_{i_{j}}\) and \(x_{i_{j+1}}\). Any point that lies vertically above \(B\) and close enough to \(B\) belongs to both \(W_{j}\) and \(U_{j+1}\). ∎
+
+Now it is very easy to construct \(p_{0},p_{1},\ldots,p_{k}\), see Figure 2. Simply take \(p_{0}\) to be any point in \(U_{1}\), and for every \(1\leq j<k\) let \(p_{j}\) be any point in \(W_{j}\cap U_{j+1}\). Finally, let \(p_{k}\) be any point in \(W_{k}\). It follows from the definition of \(U_{1},\ldots,U_{k}\) and \(W_{1},\ldots,W_{k}\) that for every \(0\leq j<k\), \(r_{i_{j+1}}\) lies above \(p_{j}\) and below \(p_{j+1}\) and the line \(s_{i_{j+1}}\) lies below \(p_{j}\) and above \(p_{j+1}\).
+
+We now prove the opposite direction: \(\lambda(n)\geq h(n)-2\).
+
+Assume we are given \(h(n)\) points \(p_{1},\ldots,p_{h(n)}\) sorted by \(x\)-coordinate and a set of \(n\) lines \(L\) such that every pair \(p_{i},p_{i+1}\) is strongly separated by \(L\). By perturbing the lines if necessary, we can assume that none of the lines goes through a point, and no three lines are concurrent. For \(1<i<h(n)\), let \(f_{i}\) be the face of the arrangement that contains \(p_{i}\), and let \(A_{i}\) and \(B_{i}\) be, respectively, the left-most and right-most vertex in this face. (The faces \(f_{i}\) are bounded, and therefore \(A_{i}\) and \(B_{i}\) are well-defined.) The monotone path will follow the upper boundary of each face \(f_{i}\) from \(A_{i}\) to \(B_{i}\).
+
+We have to show that we can connect \(B_{i}\) to \(A_{i+1}\) by a monotone path. This follows from the separation property of \(L\). Let \(s_{i},r_{i}\) be a pair of lines that strongly separates \(p_{i}\) and \(p_{i+1}\) in such a way that \(r_{i}\) lies above \(p_{i}\) and below \(p_{i+1}\) and \(s_{i}\) lies below \(p_{i}\) and above \(p_{i+1}\). Since \(B_{i}\) lies on the boundary of the face \(f_{i}\) that contains \(p_{i}\), \(B_{i}\) lies also between \(r_{i}\) and \(s_{i}\), including the possibility of lying on these lines. We can thus walk on the arrangement from \(B_{i}\) to the right until we hit \(r_{i}\) or \(s_{i}\), and from there we proceed straight to the intersection point \(Q_{i}\) of \(r_{i}\) and \(s_{i}\). Similarly, there is a path in the arrangement from \(A_{i+1}\) to the left that reaches \(Q_{i}\). and these two paths together link \(B_{i}\) with \(A_{i+1}\).
+
+To count the number of edges of this path, we claim that there must be at least one bend between \(B_{i}\) and \(A_{i+1}\) (including the boundary points \(B_{i}\) and \(A_{i+1}\)). If there is no bend at \(Q_{i}\), the path must go straight through \(Q_{i}\), say, on \(r_{i}\). But then the path must leave \(r_{i}\) at some point when going to the right: if the path has not left \(r_{i}\) by the time it reaches \(A_{i+1}\) and \(A_{i+1}\) lies on \(r_{i}\), then the path must bend upward at this point, since it proceeds on the upper boundary of the face \(f_{i+1}\) that lies above \(r_{i}\).
+
+Thus, the path makes at least \(h(n)-3\) bends (between \(B_{i}\) and \(A_{i+1}\), for \(1<i<h(n)-1\)) and contains at least \(h(n)-2\) edges. ∎
+
+Now it is very easy to give a lower bound for \(g(n)\), and prove Theorem 1. Indeed, this follows because \(g(n)\geq h(n)\) and \(h(n)\geq\lceil\frac{\lambda(n)+1}{2}\rceil=\Omega(n^{2-\frac{d}{\sqrt{\log n}}})\),
+
+The close relation between Problems 1 and 5 comes probably as no big surprise if one considers the close connection between \(k\)-sets and _levels_ in arrangements of lines (see [4, Section 3.2]). For a given set of \(n\) points \(P\), the \(k\)-sets are in one-to-one correspondence with the faces of the dual arrangements of lines which have \(k\) lines passing below them and \(n-k\) lines passing above them (or vice versa). The lower boundaries of these cells form the \(k\)-th level in the arrangement, and the upper boundaries form the \((k+1)\)-st level.
+
+Our chain of equivalence from Problem 1 to Problem 5 extends this relation between \(k\)-sets and levels in a way that is not entirely trivial: for example, establishing that we get sets that form an antichain requires some work, whereas for \(k\)-sets this property is fulfilled automatically.
+
+## 3 Proof of Theorem 2
+
+The heart of our argument uses a linear algebra approach first applied by Tverberg [7] in his elegant proof for a theorem of Graham and Pollak [5] on decomposition of the complete graph into bipartite graphs.
+
+Let \(F\) be a collection of convex pseudo-discs of a set \(P\) of \(n\) points in general position in the plane. We wish to bound from above the size of \(F\) assuming that no set in \(F\) contains another. For every directed line \(L=\overrightarrow{xy}\) passing through two points \(x\) and \(y\) in \(P\) we denote by \(L_{x}\) the collection of all sets \(A\in F\) that lie in the closed half-plane to the left of \(L\) such that \(L\) touches \({\mbox{conv}(A)}\) at the point \(x\) only. Similarly, let \(L_{y}\) be the collection of all sets \(A\in F\) that lie in the closed half-plane to the left of \(L\) such that \(L\) touches \({\mbox{conv}(A)}\) at the point \(y\) only. Finally, let \(L_{xy}\) be those sets \(A\in F\) that lie in the closed half-plane to the left of \(L\) such that \(L\) supports \({\mbox{conv}(A)}\) at the edge \(xy\).
+
+**Definition 3****.**: Let \(A\) and \(B\) be two sets in \(F\). Let \(L\) be a directed line through two points \(x\) and \(y\) in \(P\). We say that \(L\) is a common tangent of the _first kind_ with respect the pair \((A,B)\) if \(A\in L_{x}\) and \(B\in L_{y}\).
+
+We say that \(L\) is a common tangent of the second kind with respect to \((A,B)\) if \(A\in L_{xy}\) and \(B\in L_{y}\), or if \(A\in L_{x}\) and \(B\in L_{xy}\).
+
+The crucial observation about any two sets \(A\) and \(B\) in \(F\) is stated in the following lemma.
+
+**Lemma 2****.**: _Let \(A\) and \(B\) be two sets in \(F\). Then exactly one of the following conditions is true._
+
+1. 1._There is precisely one common tangent of the first kind with respect to_ \((A,B)\) _and no common tangent of the second kind with respect to_ \((A,B)\)_, or_
+2. 2._there is no common tangent of the first kind with respect to_ \((A,B)\)_, and there are precisely two common tangents of the second kind with respect_ \((A,B)\)_._
+
+Figure 3: The two cases of common tangents in Lemma 2
+
+Proof.: The idea is that because \(A\) and \(B\) are two pseudo-discs and none of \({\mbox{conv}(A)}\) and \({\mbox{conv}(B)}\) contains the other, then as we roll a tangent around \(C={\mbox{conv}(A\cup B)}\), there is precisely one transition between \(A\) and \(B\), and this is where the situation described in the lemma occurs (see Figure 3).
+
+Formally, by our assumption on \(F\), none of \(A\) and \(B\) contains the other. Any directed line \(L\) that is a common tangent of the first or second kind with respect to \(A\) and \(B\) must be a line supporting \({\mbox{conv}(A\cup B)}\) at an edge.
+
+Let \(x_{0},\ldots,x_{k-1}\) denote the vertices of \(C={\mbox{conv}(A\cup B)}\) arranged in counterclockwise order on the boundary of \(C\). In what follows, arithmetic on indices is done modulo \(k\).
+
+There must be an index \(i\) such that \(x_{i}\in A\setminus B\), for otherwise every \(x_{i}\) belongs to \(B\) and therefore \({\mbox{conv}(B)}={\mbox{conv}(A\cup B)}\supset{\mbox{conv}(A)}\) and therefore \(B\supset A\) (because both \(A\) and \(B\) are intersections of \(P\) with convex sets) in contrast to our assumption. Similarly, there must be an index \(i\) such that \(x_{i}\in B\setminus A\).
+
+Let \(I_{A}\) be the set of all indices \(i\) such that \(x_{i}\in A\setminus B\), and let \(I_{B}\) be the set of all indices \(i\) such that \(x_{i}\in B\setminus A\).
+
+We claim that \(I_{A}\) (and similarly \(I_{B}\)) is a set of consecutive indices. To see this, assume to the contrary that there are indices \(i,j,i^{\prime},j^{\prime}\) arranged in a cyclic order modulo \(k\) such that \(x_{i},x_{i^{\prime}}\in A\setminus B\) and \(x_{j},x_{j^{\prime}}\in B\). Then it is easy to see that \({\mbox{conv}(A)}\setminus{\mbox{conv}(B)}\) is not a connected set because \(x_{i}\) and \(x_{i^{\prime}}\) are in different connected components of this set.
+
+We have therefore two disjoint intervals \(I_{A}=\{i_{A},i_{A}+1,\ldots,j_{A}\}\) and \(I_{B}=\{i_{B},i_{B}+1,\ldots,j_{B}\}\). It is possible that \(i_{A}=j_{A}\) or \(i_{B}=j_{B}\).
+
+Observe that \(x_{i_{A}},x_{j_{A}},x_{i_{B}},x_{j_{B}}\) are arranged in this counterclockwise cyclic order on the boundary of \(C\), and for every index \(i\notin I_{A}\cup I_{B}\), \(x_{i}\in A\cap B\). The only candidates for common tangents of the first kind or of the second kind with respect to \(A\) and \(B\) are of the form \(\overrightarrow{x_{i}x_{i+1}}\), that is, they must pass through two consecutive vertices of \(C\).
+
+We distinguish two possible cases:
+
+1. 1.\(i_{B}=j_{A}+1\). In this case the line through \(x_{j_{A}}\) and \(x_{i_{B}}\) is the only common tangent of the first kind with respect to \((A,B)\) and there are no common tangents of the second kind with respect to \((A,B)\).
+2. 2.\(i_{B}\neq j_{A}+1\). In this case, there is no common tangent of the first kind with respect to \((A,B)\). The line through \(x_{i_{B}-1}\) and \(x_{i_{B}}\) and the line through \(x_{j_{A}}\) and \(x_{j_{A}+1}\) are the only common tangents of the second kind with respect to \((A,B)\).
+
+This completes the proof of the lemma. ∎
+
+Let \(A_{1},\ldots,A_{m}\) be all the sets in \(F\), and for every \(1\leq i\leq m\) let \(z_{i}\) be an indeterminate associated with \(A_{i}\). For each directed line \(L=\overrightarrow{xy}\), define the following polynomial \(P_{L}\):
+
+\[P_{L}(z_{1},\ldots,z_{m})=\\ \biggl{(}\sum_{A_{i}\in L_{x}}z_{i}\biggr{)}\biggl{(}\sum_{A_{j}\in L_{y}}z_{j}\biggr{)}+\frac{1}{2}\biggl{(}\sum_{A_{i}\in L_{x}}z_{i}\biggr{)}\biggl{(}\sum_{A_{j}\in L_{xy}}z_{j}\biggr{)}+\frac{1}{2}\biggl{(}\sum_{A_{i}\in L_{y}}z_{i}\biggr{)}\biggl{(}\sum_{A_{j}\in L_{xy}}z_{j}\biggr{)}\]
+
+This polynomial contains a term \(z_{u}z_{v}\) whenever \(L\) is a tangent line for the pair \((A_{u},A_{v})\) or for the pair \((A_{v},A_{u})\) (of the first or of the second kind, and with coefficient 1 or \(\frac{1}{2}\), accordingly). If we sum this equation over all directed lines \(L\), it follows by Lemma 2 that every term \(z_{u}z_{v}\) with \(u\neq v\) appears with coefficient 2:
+
+\[\sum_{L}P_{L}(z_{1},\ldots,z_{m})=\sum_{u<v}2z_{u}z_{v}=(z_{1}+\dotsb+z_{m})^{2}-(z_{1}^{2}+\dots+z_{m}^{2})\] (3)
+
+Consider the system of linear equations \(\sum_{A_{i}\in L_{x}}z_{i}=0\) and \(\sum_{A_{i}\in L_{y}}z_{i}=0\), where \(L=\overrightarrow{xy}\) varies over all directed lines determined by \(P\). Add to this system the equation \(z_{1}+\dots+z_{m}=0\). There are \(4\binom{n}{2}+1\) equations in this system and if \(m>4{\binom{n}{2}}+1\), there must be a nontrivial solution. However, it is easily seen that a nontrivial solution \((z_{1},\ldots,z_{m})\) will result in a contradiction to (3). This is because the left-hand side of (3) vanishes, while the right-hand side equals \(-(z_{1}^{2}+\dots+z_{m}^{2})\neq 0\). We conclude that \(|F|=m\leq 4{\binom{n}{2}}+1\). ∎
+
+We now show by a simple construction that Theorem 2 is tight apart from the multiplicative constant factor of \(n^{2}\). Fix three rays \(r_{1},r_{2}\), and \(r_{3}\) emanating from the origin such that the angle between two rays is \(120\) degrees. For each \(i=1,2,3\), let \(p^{i}_{1},\ldots,p^{i}_{n}\) be \(n\) points on \(r_{i}\), indexed according to their increasing distance from the origin. Slightly perturb the points to get a set \(P\) of \(3n\) points in general position in the plane. For every \(1\leq j,k,l\leq n\) define
+
+\[F_{jkl}=\{p^{1}_{1},\ldots,p^{1}_{j}\}\cup\{p^{2}_{1},\ldots,p^{2}_{k}\}\cup\{p^{3}_{1},\ldots,p^{3}_{l}\}.\]
+
+It can easily be checked that the collection of all \(F_{jkl}\) such that \(1\leq j,k,l\leq n\) and \(j+k+l=n+2\) is an anti-chain of convex pseudo-discs of \(P\). This collection consists of \(\binom{n+1}{2}\) sets.
+
+## References
+
+* [BRSSS04] J. Balogh, O. Regev, C. Smyth, W. Steiger, and M. Szegedy, Long monotone paths in line arrangements. _Discrete Comput. Geom._**32** (2004), no. 2, 167–176.
+* [D98] T. K. Dey, Improved bounds for planar \(k\)-sets and related problems. _Discrete Comput. Geom._**19** (1998), no. 3, 373–382.
+* [ES88] P. H. Edelman and M. E. Saks, Combinatorial representation and convex dimension of convex geometries. _Order_**5** (1988), no. 1, 23–32.
+* [E87] H. Edelsbrunner, _Algorithms in Combinatorial Geometry_, EATCS Monographs on Theoret. Comput. Sci., vol. 10, Springer-Verlag, Berlin, 1987.
+* [GP72] R. L. Graham and H. O. Pollak, On embedding graphs in squashed cubes. In _Proc. Conf. Graph Theory Appl._, Western Michigan Univ., May 10–13, 1972, ed. Y. Alavi, D. R. Lick, and A. T. White, Lecture Notes in Mathematics, vol. 303, Springer-Verlag, Berlin, 1972, pp. 99–110.
+* [T01] G. Tóth, Point sets with many \(k\)-sets, _Discrete Comput. Geom._**26** (2001) no. 2, 187–194.
+* [T82] H. Tverberg, On the decomposition of \(K_{n}\) into complete bipartite graphs. _J. Graph Theory_**6** (1982), no. 4, 493–494.
+

arxiv_ground_truth/0707.1301.md

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+# Static Screening and Delocalization Effects in the Hubbard-Anderson Model
+
+Peter Henseler
+
+Johann Kroha
+
+Physikalisches Institut, Universität Bonn, Nußallee 12, D-53115 Bonn, Germany
+
+Boris Shapiro
+
+Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
+
+###### Abstract
+
+We study the suppression of electron localization due to the screening of disorder in a Hubbard-Anderson model. We focus on the change of the electron localization length at the Fermi level within a static picture, where interactions are absorbed into the redefinition of the random on-site energies. Two different approximations are presented, either one yielding a nonmonotonic dependence of the localization length on the interaction strength, with a pronounced maximum at an intermediate interaction strength. In spite of its simplicity, our approach is in good agreement with recent numerical results.
+
+pacs: 71.10.Fd, 73.20.Fz, 72.15.Rn, 71.30.+h
+
+## I Introduction
+
+Understanding the interplay between disorder and electron-electron interactions remains one of the major challenges in modern condensed matter physics, experimental as well as theoretical. The research in this field has been stimulated by the possible metallic behavior in two-dimensional disordered interacting systems.[1] A metallic phase at zero temperature in two or less dimensions would be in contrast to the prediction of the scaling theory of Anderson localization.[2] The possible existence of a metallic phase, induced by interactions, is a long standing[3] and still controversial problem, discussed by many authors during the last three decades.[4]
+One of the ideas proposed and discussed by several authors is that interactions lead to a partial screening of the random potential and, thus, reduce the effect of localization. In particular, the 2\(d\) disordered Hubbard model (the Hubbard-Anderson model) has been studied, mostly numerically, and it was demonstrated that repulsive interactions can have a delocalizing effect.[5; 6; 7; 8; 9; 10; 11]
+In this paper, we present an analytical study of the screening effect, focussing on the case of strong disorder at zero temperature, when the Hubbard-Anderson model is in the regime of an Anderson insulator. Our approach is based on an exact treatment of the atomic limit, followed by ”switching on” the intersite hopping \(t\), under the assumption that the atomic-limit occupation numbers do not change. Let us emphasize that in our static approach, the interactions only change the original on-site energies. Therefore, we are left with a single-particle Anderson Hamiltonian, and the question is how the localized single-particle states may change due to the new, renormalized probability distribution of the on-site energies. In this sense, the approach is close, although not identical, to the Hartree-Fock treatment. A comparison of both methods will be presented. Although this approach is formulated for an arbitrary filling factor, it becomes inadequate close to half filling where magnetic effects dominate[12] (such effects are not considered in our work).
+Furthermore, we will show that, for fixed disorder, the localization length \(\xi\) is a nonmonotonic function of the Hubbard interaction energy \(U\), with a maximum for some intermediate value of \(U\). This is because for strong interactions, the Mott-Hubbard physics of interaction-suppressed hopping dominates, leading to the formation of two disorder broadened Hubbard bands with a reduced average density of states at the Fermi level and, as a consequence, increasing effectively the disorder strength. In spite of the simplicity of the approach, the results are in good agreement with recent numerical studies,[7; 9; 13; 14; 10; 5; 8] in the appropriate range of parameters.
+The evaluation of the localization length within our present study is limited to one dimension. However, since the competition of screening on the one hand and Mott-Hubbard physics on the other hand operates in any dimension, the nonmonotonic dependence of \(\xi\) on \(U\) should also hold in two and three dimensions, as argued below.
+
+## II Atomic-Limit Approximation
+
+In this paper, we consider the Hubbard-Anderson model, with on-site repulsion and on-site disorder, at zero temperature. The corresponding Hamiltonian is
+
+\[H = H_{0}\:+\:H_{\textrm{kin}}\:+\:H_{\textrm{e-e}}\] (1)
+\[= \sum\limits_{i,\sigma}\left(\varepsilon_{i}^{\phantom{\dagger}}-\mu\right)c_{i\sigma}^{\dagger}c_{i\sigma}^{\phantom{\dagger}}\:-\:t\!\sum\limits_{<i,j>,\sigma}\!c_{i,\sigma}^{\dagger}c_{j\sigma}^{\phantom{\dagger}}\]
+\[\:+\:U\sum\limits_{i}n_{i\uparrow}^{\phantom{\dagger}}n_{i\downarrow}^{\phantom{\dagger}}\,.\]
+
+Figure 1: Site occupation in the atomic ground state with doubly occupied (black), singly occupied (gray) and empty states (white). (a) weak interaction, (b) strong interaction for less than half filling (i.e., \(\rho<1\)), and (c) strong interaction for more than half filling \((\rho>1)\).
+
+As usual, \(c_{i\sigma}^{\dagger}(c_{i\sigma}^{\phantom{\dagger}})\) denote fermion creation (destruction) operators of an electron at site \(i\) with spin \(\sigma\), \(n_{i\sigma}^{\phantom{\dagger}}=c_{i\sigma}^{\dagger}c_{i\sigma}^{\phantom{\dagger}}\), \(t\) is the nearest-neighbor hopping amplitude, \(U\) is the on-site repulsion, \(\mu\) is the chemical potential, and \(\{\varepsilon_{i}^{\phantom{\dagger}}\}\) are the on-site energies. The latter are assumed to be independent and uniformly distributed over the interval \([-\frac{\Delta}{2},\frac{\Delta}{2}]\), with the disorder parameter \(\Delta\). To focus on the screening effect in the case of strong localization, i.e., \(\Delta\gg t\), the interaction term will be absorbed into the on-site energies, yielding a renormalized distribution of the \(\varepsilon_{i}\). This results in an effective single-particle problem with a probability function \(p_{\textrm{A}}(\varepsilon_{i})\) which is derived as follows.
+In the atomic limit (\(t=0\)), the ground state of the system can be solved exactly for an arbitrary filling factor \(\rho=\frac{\textrm{N}_{\textrm{e}}}{\textrm{N}}\), where \(\textrm{N}_{\textrm{e}}\), N are the numbers of electrons and lattice sites, respectively.[9] The chemical potential \(\mu\) and the site-dependent occupation numbers \(\langle n_{i}\rangle_{0}\) can be expressed as functions of \(\rho\), \(\Delta\), and \(U\): All sites with on-site energies below \(\mu-U\) are doubly occupied, all sites within \([\mu-U,\mu]\) are singly occupied, and all other sites are empty (see Fig. 1). Thus, the total occupation number of site \(i\) and the chemical potential are, respectively,
+
+\[\langle n_{i}\rangle_{0} = \left\{\begin{array}[]{rcl}2&,&\varepsilon_{i}\leq\mu-U\\ 1&,&\mu-U<\varepsilon_{i}\leq\mu\\ 0&,&\varepsilon_{i}>\mu\end{array}\right.\] (5)
+
+and[15]
+
+\[\mu = \left\{\begin{array}[]{rcl}\frac{1}{2}(\Delta\rho-\Delta+U)&,&\rho<1\,,\,U<\Delta\rho\\ \Delta(\rho-\frac{1}{2})&,&\rho<1\,,\,U\geq\Delta\rho\\ \frac{1}{2}(\Delta\rho-\Delta+U)&,&\rho\geq 1\,,\,U<2\Delta-\Delta\rho\\ \Delta(\rho-\frac{3}{2})+U&,&\rho\geq 1\,,\,U\geq 2\Delta-\Delta\rho.\end{array}\right.\] (10)
+
+The renormalized site energies depend on the site occupation numbers and can be read off from the poles of the (time-ordered) single-particle propagator,
+
+\[G_{i\sigma}(\omega)\:=\:\frac{\langle n_{i,\sigma}\rangle_{0}\langle n_{i,-\sigma}\rangle_{0}}{\omega-(\varepsilon_{i}-\mu+U)-i0^{+}}\] (11)
+\[\:+\:\frac{\langle n_{i,\sigma}\rangle_{0}(1-\langle n_{i,-\sigma}\rangle_{0})}{\omega-(\varepsilon_{i}-\mu)-i0^{+}}\:+\:\frac{(1-\langle n_{i,\sigma}\rangle_{0})\langle n_{i,-\sigma}\rangle_{0}}{\omega-(\varepsilon_{i}-\mu+U)+i0^{+}}\]
+\[\:+\:\frac{(1-\langle n_{i,\sigma}\rangle_{0})(1-\langle n_{i,-\sigma}\rangle_{0})}{\omega-(\varepsilon_{i}-\mu)+i0^{+}}\quad.\]
+
+The first term in Eq. (11) corresponds to the doubly occupied sites showing that these on-site energies are shifted by \(U\). The next two terms correspond to singly occupied sites. In the absence of spin polarization, \(\langle n_{i,\sigma}\rangle_{0}=\langle n_{i,-\sigma}\rangle_{0}\), half of these on-site energies are again shifted by \(U\) whereas the other half remain unchanged. Finally, the last term corresponds to the unoccupied sites, whose energies also remain unchanged. Combining Eqs. (10) and (11), the rule for replacing the bare site energy \(\varepsilon_{i}\) by a renormalized one is
+
+\[\varepsilon_{i}^{\phantom{\dagger}} \mapsto \!\!\left\{\!\!\begin{array}[]{rcl}\varepsilon_{i}^{\phantom{\dagger}}+U&\mbox{if}&\varepsilon_{i}^{\phantom{\dagger}}\leq\mu-U\\ \begin{array}[]{r}\varepsilon_{i}^{\phantom{\dagger}}+U\\ \varepsilon_{i}^{\phantom{\dagger}}\end{array}&\mbox{if}&\mu-U<\varepsilon_{i}^{\phantom{\dagger}}\leq\mu\:\left(\begin{array}[]{l}\mbox{each with}\\ \mbox{prob. of $\frac{1}{2}$}\end{array}\right)\\ \varepsilon_{i}^{\phantom{\dagger}}&\mbox{if}&\varepsilon_{i}^{\phantom{\dagger}}>\mu.\end{array}\right.\quad\] (19)
+
+Figure 2: The renormalized on-site energy probability functions (solid line) for weak repulsion \(U\): (a) atomic-limit approximation, and (b) Hartree-Fock approximation. For comparison, also the original function is shown (dashed lines).
+
+For a weak to intermediate repulsion \(U\) these shifts lead to a raise of the lowest lying on-site energies towards the Fermi level \(\mu\) resulting in a renormalized probability function \(p_{\textrm{A}}(\varepsilon)\) with a reduced width and modified shape (see Fig. 2).
+In the atomic-limit approximation the Hamiltonian Eq. (1) is replaced by the effective single-particle Anderson Hamiltonian
+
+\[H = \sum\limits_{i\sigma}\!\left(\varepsilon_{i}^{\phantom{\dagger}}-\mu\right)c_{i\sigma}^{\dagger}c_{i\sigma}^{\phantom{\dagger}}-t\sum\limits_{<i,j>\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}^{\phantom{\dagger}}\] (20)
+
+with on-site energy probability function \(p_{\textrm{A}}(\varepsilon)\), where the two-particle interaction \(U\) enters only as a (screening) parameter in \(p_{\textrm{A}}(\varepsilon)\). Note that, due to the asymmetry of \(p_{\textrm{A}}(\varepsilon)\), the average value of the renormalized on-site energies is
+
+\[\langle\varepsilon\rangle_{\textrm{A}} = \int p_{\textrm{A}}(\varepsilon)\,\varepsilon\,d\varepsilon\:=\:\frac{1}{2}\,\rho\,U\,.\] (21)
+
+In deriving Eq. (20), it was assumed that in the case of strong disorder, the occupation numbers in the atomic ground state, Eq. (5), are close to the occupation numbers \(\langle n_{i}\rangle\) in the true ground state, with finite hopping amplitude \(t\). This assumption is based on the results from the single-particle theory of localization,[16] where it is known that for strong disorder, an electron, once located at any site \(i\), will stay at that site with high probability. More precisely, the change of the occupation number of site \(i\), \(\delta\langle n_{i}\rangle\equiv\langle n_{i}\rangle-\langle n_{i}\rangle_{0}\), is of order \(t^{2}/\Delta\). The same estimate holds if one considers hopping of a single electron on the background of the other electrons, which are assumed to be immobile. Furthermore, for strong disorder, a perturbative expansion in \(t\), around the atomic ground state of Eq. (1), is possible.[17] Thus, with the same reasoning as in the noninteracting case, the leading self-energy corrections are again only of second order.
+One measure of localization is the localization length \(\xi\) which governs the exponential decay of the single-particle wave functions \(\psi(r)\) at distances far away from its localization center. It can be calculated from the probability for a transition from site \(i\) to site \(j\) as[18]
+
+\[-\frac{1}{\xi(E)} = \lim_{|x_{i}-x_{j}|\rightarrow\infty}\frac{\log\left\langle\left|G^{\textrm{R}}_{ij}(E)\right|^{2}\right\rangle}{2|x_{i}-x_{j}|},\] (22)
+
+where \(G^{\textrm{R}}_{ij}(E)\) is the retarded propagator for a particle with energy \(E\) from site \(j\) to site \(i\). In general, it is not possible to deduce \(\xi\) from the probability distribution analytically, but in case of a one-dimensional lattice, there exists a relatively simple relation[18] between \(\xi\) and \(\langle D(\omega)\rangle\), the disorder averaged density of states,
+
+\[\xi^{-1}(E) = \int\limits_{-\infty}^{\infty}\langle D(\varepsilon)\rangle\,\log\left|E-\varepsilon\right|\:d\varepsilon.\] (23)
+
+(Here and in the following, we choose units where \(t=1\) and measure \(\xi\) in units of the lattice spacing.)
+In the strong disorder limit, \(\langle D(\varepsilon)\rangle\) can be replaced by \(p_{\textrm{A}}(\varepsilon+\mu)\), so that the inverse localization length at the Fermi level \((E=0)\) is given by
+
+\[\xi^{-1} = \int\limits_{-\infty}^{\infty}p_{\textrm{A}}(\varepsilon)\,\log\left|\varepsilon-\mu\right|\,d\varepsilon\quad.\] (24)
+
+Figure 3: (Color online) Localization length \(\xi\) at the Fermi level as a function of repulsion \(U\) and lattice filling \(\rho\) for \(\Delta=15\): (a) atomic-limit approximation and (b) Hartree-Fock approximation.
+
+A plot of \(\xi\) as a function of \(U\) and \(\rho\), for fixed disorder strength \(\Delta=15\), is shown in Fig. 3. It can be seen that for each given filling factor, the localization length exhibits a pronounced maximum. This maximum can be calculated to appear at
+
+\[U_{\xi}^{\textrm{A}} = \frac{\Delta}{3}\left(\sqrt{1+3\rho(2-\rho)}-1\right)\] (25)
+\[\approx \frac{\Delta}{2}\rho(2-\rho)\,+\,\mathcal{O}(\rho^{2}(2-\rho)^{2})\:.\]
+
+The reason for this nonmonotonic behavior is simple: A weak to intermediate repulsion \(U\) changes the on-site energy distribution from a rectangular distribution to a narrower one by shifting low on-site energies toward the Fermi level (screening), see Fig. 2. In contrast, a very strong on-site repulsion enhances the localization by a large broadening of the probability density. Therefore, in between, there will be some value \(U_{\xi}^{\textrm{A}}\) for which the screening is optimal and the localization length acquires a maximum. Such behavior is expected, since a strong repulsion effectively suppresses hopping processes and leads to an accumulation of spectral weight in the upper Hubbard band.
+In Fig. 4, the localization length \(\xi\) is shown as a function of \(U\) for \(\rho=\frac{1}{2}\) (quarter filling). A similar, nonmonotonic behavior was also found in recent quantum Monte Carlo simulations[9; 13; 14; 5] and a most recent statistical dynamical mean field theory evaluation[10] of the problem, where the conductivity and the inverse participation ratio[7] of a finite system were calculated, respectively. Identifying a maximum of conductivity with a maximum of localization length, we find in all cases a good qualitative agreement with our results. Furthermore, we even find a reasonable quantitative agreement with our results for the points of maximal delocalization, \(U_{\xi}^{\rm A}\), Eq. (25), and \(U_{\xi}^{\rm H}\), defined in Eq. (38) below. Thus, there is a strong correlation between the degree of screening and the conductivity: optimal screening corresponds to maximal conductivity. In addition, our analytical results indicate that much of this physics of the nonmonotonic behavior can already be understood on the level of a static screening approximation.
+Our result appears to be at odds with the statement[9] that screening alone cannot account for the nonmonotonic behavior of the conductivity. We will discuss this point in the next section. For that discussion, we need the variance of the renormalized on-site energies which is given by
+
+\[\!\!\!\sigma^{2}_{\textrm{A}}\:\equiv\:\langle\varepsilon^{2}\rangle_{\textrm{A}}-\langle\varepsilon\rangle^{2}_{\textrm{A}}\] (26)
+\[\!\!\!=\frac{\Delta^{2}}{12}\left\{\begin{array}[]{rcl}1-\hat{\rho}\hat{U}+\hat{\rho}\hat{U}^{2}+3\hat{U}^{3}&\textrm{,}&\rho<1\,,\,\hat{U}<\rho\\ 1-2\rho^{\prime}\hat{U}+\hat{\rho}\hat{U}^{2}&\textrm{,}&\rho<1\,,\,\hat{U}\geq\rho\\ 1-\hat{\rho}\hat{U}+\hat{\rho}\hat{U}^{2}+3\hat{U}^{3}&\textrm{,}&\rho\geq 1\,,\,\hat{U}<2-\rho\\ 1-2\tilde{\rho}\hat{U}+\hat{\rho}\hat{U}^{2}&\textrm{,}&\rho\geq 1\,,\,\hat{U}\geq 2-\rho,\end{array}\right.\] (31)
+
+with
+
+\[\hat{U}\:=\:U/\Delta\:,\quad\hat{\rho}\:=\:3\rho(2-\rho)\:,\quad\rho^{\prime}\:=\:3\rho(1-\rho)\:,\]
+\[\tilde{\rho}\:=\:3\left(\frac{1}{4}-(\rho-\frac{3}{2})^{2}\right).\] (32)
+
+Here, \(\sigma^{2}_{\textrm{A}}\) has a minimum at
+
+\[U_{\sigma}^{\textrm{A}}=\left\{\begin{array}[]{ccl}\frac{\Delta}{9}\left(\sqrt{\hat{\rho}^{2}+9\hat{\rho}}-\hat{\rho}\right)&\,,&\frac{1}{3}\lesssim\rho\lesssim\frac{5}{3}\\ \Delta\frac{|1-\rho|}{1+|1-\rho|}&\,,&\textrm{otherwise},\end{array}\right.\] (35)
+
+which can be seen in Fig. 4 for \(\rho=\frac{1}{2}\) and \(\Delta=15\).
+
+So far, our calculations were restricted to one dimension because a generalization of relation (23) to two or three dimensions is, in general, not possible. However, in the limit of strong disorder, \(G_{ij}(E)\) can be calculated in good approximation by taking into account only the direct path[19] from \(i\) to \(j\). Therefore, Eq. (22) can be considered as a one-dimensional problem, leading again to Eqs. (23) and (24), respectively. Hence, we conjecture that the general result, i.e., the nonmonotonic behavior of localization, holds also in two and three dimensions.
+
+## III Site-dependent Hartree-Fock approximation
+
+Figure 4: (a) Localization length \(\xi\) at the Fermi level, normalized by its noninteracting value, as a function of the on-site repulsion \(U\) for \(\Delta=15\) and \(\rho=\frac{1}{2}\). (b) Variance of the renormalized on-site energy distribution, also normalized by its noninteracting value. The solid curves show the results for the atomic-limit approximation and the dashed curves are the Hartree-Fock results.
+
+The second model which will be discussed in this paper is the site-dependent Hartree-Fock approximation.[9; 6] In this single-particle approximation, each site has a single renormalized energy level, given by
+
+\[\varepsilon_{i}^{\phantom{\dagger}} \mapsto \varepsilon_{i}^{\phantom{\dagger}}+\frac{U}{2}\langle n_{i}^{\phantom{\dagger}}\rangle_{0},\] (36)
+
+which in Eq. (1) corresponds to the replacement,
+
+\[\!\!\!Un_{i\uparrow}^{\phantom{\dagger}}n_{i\downarrow}^{\phantom{\dagger}}\:\rightarrow\:U\left(\langle n_{i\uparrow}^{\phantom{\dagger}}\rangle\,n_{i\downarrow}^{\phantom{\dagger}}+\langle n_{i\downarrow}^{\phantom{\dagger}}\rangle\,n_{i\uparrow}^{\phantom{\dagger}}\right)\]
+\[\!\!\!\quad\rightarrow\:\frac{U}{2}\langle n_{i}^{\phantom{\dagger}}\rangle\left(n_{i\downarrow}^{\phantom{\dagger}}+n_{i\uparrow}^{\phantom{\dagger}}\right)\:\approx\:\frac{U}{2}\langle n_{i}^{\phantom{\dagger}}\rangle_{0}\left(n_{i\downarrow}^{\phantom{\dagger}}+n_{i\uparrow}^{\phantom{\dagger}}\right).\qquad\] (37)
+
+Here, the absence of any kind of magnetization was assumed. Note that for a local, energy independent interaction \(U\), the Hartree and the Hartree-Fock approximations are identical. The average on-site occupation \(\langle n_{i}^{\phantom{\dagger}}\rangle\) was taken to be the one of the atomic ground state, \(\langle n_{i}^{\phantom{\dagger}}\rangle_{0}\), Eq. (5), according to the assumption of a stable atomic configuration.
+As in the atomic-limit approximation, the shift of the occupied on-site energies leads to a renormalized distribution, with probability function \(p_{\textrm{H}}(\varepsilon)\) (Fig. 2). The screening effect now is even more pronounced because all singly occupied states are shifted by \(\frac{U}{2}\) yielding a smaller width and a stronger increase of the probability to find a state around the Fermi level. The resulting effective single-particle Hamiltonian is again given by Eq. (20), however, with the probability function \(p_{\textrm{H}}(\varepsilon)\).
+The corresponding plots for the localization length in Hartree-Fock approximation are shown in Figs. 3 and 4, respectively. Again, a pronounced maximum of \(\xi\) arises. In this case, it is found at the value
+
+\[U_{\xi}^{\textrm{H}} = \frac{\Delta}{2}\rho(2-\rho),\] (38)
+
+which up to order \(\mathcal{O}\left(\rho^{2}\cdot(2-\rho)^{2}\right)\) coincides with the result from the atomic-limit approximation Eq. (25). The increase of the localization length is considerably more pronounced in the Hartree-Fock approximation due to its narrower probability distribution and especially its larger probability density around the Fermi level. In both approaches, the effect of screening becomes stronger with decreasing disorder. However, for small values of \(\Delta\), the stability of the atomic ground state becomes doubtful.
+The average and the variance of the renormalized on-site energy distribution are, respectively,
+
+\[\langle\varepsilon\rangle_{\textrm{H}}\!\! = \langle\varepsilon\rangle_{\textrm{A}}\:=\:\frac{1}{2}\,\rho\,U\,,\] (39)
+\[\sigma^{2}_{\textrm{H}} = \frac{\Delta^{2}}{12}\left\{\begin{array}[]{rcl}1-\hat{\rho}\hat{U}+\hat{\rho}\hat{U}^{2}&\!\!,\!&\rho<1\,,\,\hat{U}<\rho\\ 1-2\rho^{\prime}\hat{U}+\rho^{\prime}\hat{U}^{2}&\!\!,\!&\rho<1\,,\,\hat{U}\geq\rho\\ 1-\hat{\rho}\hat{U}+\hat{\rho}\hat{U}^{2}&\!\!,\!&\rho\geq 1\,,\,\hat{U}<2-\rho\\ 1-2\tilde{\rho}\hat{U}+\tilde{\rho}\hat{U}^{2}&\!\!,\!&\rho\geq 1\,,\,\hat{U}\geq 2-\rho,\end{array}\right.\qquad\!\!\] (44)
+
+with the same abbreviations as in Eq. (II). Its minimum is found at the value
+
+\[U_{\sigma}^{\textrm{H}}=\left\{\begin{array}[]{rcl}\Delta/2&\,,&\frac{2}{3}<\rho<\frac{4}{3}\\ \Delta&\,,&\textrm{else}.\end{array}\right.\] (47)
+
+Figure 5: Inverse localization length \(\xi^{-1}\) as a function of \(\Delta\) for \(\rho=\frac{1}{2}\) and (a) \(U=7\) in the atomic-limit approximation and (b) \(U=14\) in the Hartree-Fock approximation, respectively.
+
+As mentioned above, it was argued in Ref. [9] that the picture of screening would be too primitive to explain the nonmonotonic behavior and the evidence for a metallic state found in the conductivity simulations by varying the repulsion strength \(U\). The argumentation was based on the observation, which the variance was a featureless, monotonically decreasing function of \(U\) around the transition point. Our results show that this reasoning is generally not conclusive. Although the static, single-particle treatment does not allow for the occurrence of a metallic state, we find a strong enhancement and nonmonotonic behavior of the localization length \(\xi\) as function of \(U\), whereas the variance is also only a monotonically decreasing function around the point of maximal delocalization. For strong disorder and a distribution which is not characterized by a single parameter (like in the present case, cf. Fig. 2), there is no simple relation between \(\xi\) and the variance \(\sigma^{2}\). Especially for lower fillings, our results, Eqs. (25), (35) and (38), (47), respectively, show that the values of \(U\) for which the maximum of \(\xi\) and the minimum of \(\sigma^{2}\) occur, can be separated systematically. Moreover, we find that the atomic-limit approximation and the Hartree-Fock approximation do yield close values of \(\xi\), although the variances can differ strongly (see Fig. 4).
+In Refs. [8] and [13], the inverse participation ratio was calculated as a function of the disorder strength \(\Delta\), and a nonmonotonic behavior with evidence for a metallic state was found. It was argued there that the screening picture would necessarily predict a monotonic increase of the inverse participation ratio with increasing \(\Delta\), excluding screening as a possible explanation. Our model, contrary to that statement, exhibits such nonmonotonic behavior as well, as shown in Fig. 5, where \(\xi^{-1}\) is plotted as a function of the disorder strength \(\Delta\), for some fixed values of \(\rho\) and \(U\). This nonmonotonicity is caused by the crossover from the regime of disorder screening by interaction, \(\Delta\gg U\), to the regime of interaction-reduced hopping, \(\Delta\ll U\), which is controlled by the parameter \(U/\Delta\). However, the exact position of the nonmonotonicity depends also on \(t/\Delta\), \(\rho\), and the probability function.
+
+## IV Conclusion
+
+We examined the effect of static disorder screening by on-site repulsion in the one-dimensional Hubbard-Anderson model for strong disorder. We presented two different approximation schemes by absorbing the interactions into a redefinition of the single-particle on-site energies. In both approaches a renormalized probability distribution with an enhanced probability of finding site energies close to the Fermi level was obtained. We calculated the localization length at the Fermi energy for these single-particle problems and found a pronounced maximum of the localization length for some intermediate value of the repulsion strength. This can be understood as a consequence of the fact that the increase of the localization length \(\xi\) for small \(U\) (\(U<U_{\xi}\)) and the decrease of \(\xi\) for large \(U\) (\(U>U_{\xi}\)) have different physical origins, namely disorder screening and reduced hopping, respectively. Similarly, a change of the ”bare” disorder \(\Delta\), for fixed repulsion, resulted in a nontrivial, nonmonotic dependence of the localization length on the disorder strength. In contrast to the case of weak disorder, we found no significant correlation between the variance of the effective on-site energy distribution and the localization length. Our results, especially in the case of the Hartree-Fock approximation, are in qualitative and to some degree even quantitative agreement with recent numerical studies. By our analytic approach, it was possible to investigate the static screening effects separately from dynamical (inelastic) processes.
+We gave an argument that the same behavior should also be found for strong disorder in two and three dimensions.[20] In three dimensions there might be an interesting possibility of an interaction induced metal-insulator transition. Such a possibility is based on the assumption that our results, obtained in the strongly localized regime, can be extrapolated up to the mobility edge. Under this assumption, a noninteracting Anderson-localized system, whose Fermi energy is sufficiently close to the mobility edge, will become metallic upon switching on interactions by shifting the mobility edge across the Fermi level. Furthermore, when the interactions exceed a certain strength, the system would reenter the insulating phase in analogy to the nonmonotonic behavior of the localization length in one dimension.[21] (Let us emphasize that throughout this paper we do not discuss the case of half filling, with its characteristic Mott’s physics.[12; 22]) A similar, and experimentally more relevant, effect could also happen under a change of the bare disorder \(\Delta\), as suggested by Fig. 5, since the relevant dimensionless parameter is \(U/\Delta\).
+
+###### Acknowledgements.
+
+This work was supported in part by the Deutsche Forschungsgemeinschaft through SFB 608. P.H. acknowledges additional support by Deutscher Akademischer Austausch Dienst (DAAD). B.S. acknowledges the hospitality of Bonn University where the present work was initiated.
+
+## References
+
+* (1) For a review, see E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. **73**, 251 (2001).
+* (2) E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. **42**, 673 (1979).
+* (3) L. Fleishman and P. W. Anderson, Phys. Rev. B **21**, 2366 (1980).
+* (4) Recently, it has been demonstrated that interaction-induced decay of single-particle excitations (Anderson localized in the absence of interactions) can lead to the interesting phenomenon of a metal-insulator transition at finite temperatures, see, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. Lett. **95**, 206603 (2005); D.M. Basko, I.L. Aleiner, and B.L. Altshuler, Annals of Physics **321**, 1126 (2006).
+* (5) P. J. H. Denteneer, R. T. Scalettar, and N. Trivedi, Phys. Rev. Lett. **83**, 4610 (1999).
+* (6) I. F. Herbut, Phys. Rev. B **63**, 113102 (2001).
+* (7) B. Srinivasan, G. Benenti, and D. L. Shepelyansky, Phys. Rev. B **67**, 205112 (2003).
+* (8) D. Heidarian and N. Trivedi, Phys. Rev. Lett. **93**, 126401 (2004).
+* (9) P. B. Chakraborty, P. J. H. Denteneer, and R. T. Scalettar, Phys. Rev. B **75**, 125117 (2007).
+* (10) Y. Song, R. Wortis, and W. A. Atkinson, arXiv:cond-mat/0707.0791.
+* (11) Such effect has also been observed in models with long-range Coulomb interactions; see Z. A. Nemeth and J.-L. Pichard, Eur. Phys. J. **B45**, 111 (2005), and references therein.
+* (12) See, e.g., M. C. O. Aguiar, V. Dobrosavljevic, E. Abrahams, and G. Kotliar,Proceedings of the Conference on Strongly Correlated Electron Systems SCES’07 (unpublished); arXiv:0704.0450, for a recent reference on the Hubbard-Anderson model at half filling in 3d.
+* (13) N. Trivedi, P. J. H. Denteneer, D. Heidarian, and R. T. Scalettar, Pramana-J. of Physics **64**, 1051 (2005).
+* (14) P. J. H. Denteneer and R. T. Scalettar, Phys. Rev. Lett. **90**, 246401 (2003).
+* (15) At half filling and strong repulsion, the chemical potential lies in the gap of the two Hubbard bands. To avoid an unnecessary case differentiation, in that case, we defined \(\mu=U-\Delta/2\). \(\mu\) defined in this way is at the top of the gap, in contrast to the more common definition, \(\mu=U/2\), at the center of the gap. Physical quantities are of course not affected by this definition.
+* (16) P. W. Anderson, Phys. Rev. **109**, 1492 (1958).
+* (17) M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev B **40**, 546 (1989).
+* (18) I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, _Introduction to the Theory of Disordered Systems_ (Wiley, New York, 1988).
+* (19) E. Medina and M. Kardar, Phys. Rev. B **46**, 9984 (1992).
+* (20) We emphasize that throughout this paper, we only consider the case of strong disorder away from half filling. For the question of disorder screening in the opposite limit of disordered Mott insulators, see, e.g., D. Tanaskovic, V. Dobrosavljevic, E. Abrahams, and G. Kotliar, Phys. Rev. Lett. **91**, 066603 (2003), as well as Ref. [12].
+* (21) This reentry resembles the double insulator-metal-insulator transition, taking place in a disordered system under a change of the magnetic field. See B. Shapiro, Phil. Mag. B **50**, 241 (1984), and the discussion in N. Mott, _Metal-Insulator Transitions_ (Taylor and Francis, 1990), p. 160.
+* (22) See K. Byczuk, W. Hofstetter, and D. Vollhardt, Phys. Rev. Lett. **94**, 056404 (2005), where reentries, similar to those discussed above, have been found.
+

arxiv_ground_truth/0707.3690.md

@@ -0,0 +1,156 @@
+Virus Structure:
+
+From Crick and Watson to a new conjecture∗
+
+Alfredo Iorio_a,b_¹ and Siddhartha Sen_c,d_²
+
+Footnote 1: E-mail: iorio@ipnp.troja.mff.cuni.cz
+
+Footnote 2: E-mail: tcss@mahendra.iacs.res.in
+
+_a_ Institute of Particle and Nuclear Physics, Charles University of Prague
+
+V Holesovickach 2, 18200 Prague 8 - Czech Republic
+
+_b_ Department of Physics “E.R.Caianiello” University of Salerno and INFN
+
+Via Allende, 84081 Baronissi (SA) - Italy
+
+_c_ School of Mathematical Science, University College Dublin
+
+Belfield, Dublin 4 - Ireland
+
+_d_ Indian Association for the Cultivation of Science
+
+Jadavpur, Calcutta 700032 - India
+
+We conjecture that certain patterns (scars), theoretically and numerically predicted to be formed by electrons arranged on a sphere to minimize the repulsive Coulomb potential (the Thomson problem) and experimentally found in spherical crystals formed by self-assembled polystyrene beads (an instance of the _generalized_ Thomson problem), could be relevant to extend the classic Caspar and Klug construction for icosahedrally-shaped virus capsids. The main idea is that scars could be produced at an intermediate stage of the assembly of the virus capsids and the release of the bending energy present in scars into stretching energy could allow for a variety of non-spherical capsids’ shapes. The conjecture can be tested in experiments on the assembly of artificial protein-cages where these scars should appear.
+
+∗ Invited talk by A.I. at the Fourth International Summer School and Workshop on Nuclear Physics Methods and Accelerators in Biology and Medicine - Prague, 8-19 July 2007.
+
+PACS: 87.10.+e, 87.15.Kg, 61.72.-y
+
+Keywords: Virus structure, Biomembranes, Crystal Defects
+
+## 1 Virus Structure
+
+General ConsiderationsViruses are small pieces of genetic material (DNA or RNA) that can efficiently encode few small identical proteins that then assemble themselves³ to form “cages” around the genetic material [1]. These cages are called _capsids_ and their shape is the main concern of this paper.
+
+Footnote 3: Sometimes, for large viruses, the assembly is done with the help of other proteins encoded to this end by the genetic material. The environment also plays an important role.
+
+Capsids are essential for protecting the genetic material and contribute to identifying cells suitable for the duplication of the genetic material⁴. Understanding the way proteins arrange to form these very resistent capsids is important: if we could find a way to undo these constructions we would be able to destroy viruses. There are three classes of capsid’s shapes [1]: _helical_ (the proteins spiralize counter-clockwise around the genetic material), _icosahedral or simple_ (the proteins arrange in morphological units of 5 and 6 following precise geometrical and topological prescriptions, as we shall soon explain), _complex_ (sphero-cylindrical, conical, tubular or even more complicated shapes, i.e. without a precise resemblance to any particular regular polyhedron). There are also _polymorphic viruses_ that change their shape, e.g., from icosahedral to tubular and _enveloped viruses_ that, in addition to the protein-capsid, also have an outer lipid bilayer (the viral envelope) taken by the host cell membranes.
+
+Footnote 4: Viruses are not able to duplicate without the help of the host cell, that is why their living nature is debatable.
+
+Icosahedral VirusesIn 1956 Crick and Watson [2] proposed that small viruses have capsids with the identical proteins (or _subunits_ or _structural units_) arranged into morphological units called _capsomers_ with the shape of hexagons and pentagons, called _hexamers_ and _pentamers_, respectively. These capsomers form polyhedrons that go under the name of _icosadeltahedrons_, with a fixed number, 12, of pentamers and a variable number of hexamers. Following Crick and Watson’s seminal idea, Caspar and Klug (CK) [3] later extended the class of viruses to which this construction applies to what they called “simple viruses”, i.e. still roughly spherical but not necessarily small viruses. The CK model for icosadeltahedral capsids is nowadays universally accepted by virologists [1], [4].
+
+The fact that exactly 12 pentamers are necessary is easily understood if we look at this problem as the analogue problem of tiling a sphere with pentagons and hexagons and we take into account the topological properties of the sphere (Euler theorem, see, e.g., [5]). The precise number of hexamers will not be fixed by this argument and needs further assumptions that we shall discuss in the next paragraph. The argument goes like this: Suppose that the ends of proteins only join three at the time. If \(N_{p}\) is the number of \(p\)-mers used to tile a sphere of unit radius, i.e. \(N_{5}\) pentamers and \(N_{6}\) hexamers, the resulting polyhedron \(P\) has \(V_{P}=1/3\sum_{N}N_{p}p\) vertices, \(E_{P}=1/2\sum_{N}N_{p}p\) edges, and \(F_{P}=\sum_{N}N_{p}\) faces, giving for the Euler characteristic \(\chi=V_{P}-E_{P}+F_{P}\), the following expression
+
+\[\sum_{N}(6-p)N_{p}=6\chi=12\;,\] (1)
+
+since for a sphere \(\chi=2\). Explicitly Eq.(1) reads
+
+\[(6-5)\;N_{5}+(6-6)\;N_{6}=12\] (2)
+
+hence to tile a sphere \(N_{5}=12\) is required, but \(N_{6}\) can be arbitrary. As said, for virus capsids \(N_{6}\) is not arbitrary but must be a specific number that we shall soon obtain. For the mathematical problem of the tiling of the sphere one might also imagine to use _heptagons_. In that case the Euler formula (1) gives
+
+\[N_{5}-N_{7}=12\;.\] (3)
+
+Thus, starting from the tiling of the sphere with exactly 12 pentagons (and an arbitrary number of hexagons) one can add _pairs_ pentagon-heptagon, but not a pentagon or a heptagon separately. Note that at this point this is only a mathematical consideration and its relevance for virus structure is all to be proved.
+
+Figure 1: Planar hexagonal lattice of identical rigid proteins. The vector \(\vec{A}=h\vec{a}+k\vec{b}\) corresponds to \(h=1\) and \(k=3\).
+
+The geometric interpretation of the Euler formula (1) is that a sphere of unit radius has curvature \(R_{\rm sphere}=+1\) and each polygon contributes to this curvature with \(R_{p}=(6-p)/12\): a hexagon with \(R_{6}=0\), a pentagon with \(R_{5}=+1/12\), a heptagon with \(R_{7}=-1/12\). This can be understood by constructing hexagons, pentagons and heptagons out of equilateral triangles of paper. A hexagon is obtained by joining together 6 triangles: they all stay in a plane. Take one triangle out and join what is left to make a pentagon and the resulting figure will bend outwards, while adding one triangle to the hexagon to make a heptagon results into an inward bending. This also tells us that a certain amount of bending energy \(E_{b}\) is necessary to convert a hexagon into a pentagon or into a heptagon. How big is \(E_{b}\) depends on the elastic properties of the material used. Let us now describe in more detail the CK construction.
+
+The CK constructionSuppose that the proteins are arranged on a plane to form the hexagonal lattice of Fig.1. Each side of the lattice represents a real protein. The basic vectors \(\vec{a}\) and \(\vec{b}\), with \(|\vec{a}|=|\vec{b}|\), join the center of the hexagon taken as the origin of the lattice with the centers of the nearest hexagons as in figure. The angle is evidently \(\varphi=60^{o}\). The 3-dimensional polyhedron these proteins will eventually form is obtained by imagining the 20 equilateral triangles with side \(|\vec{A}|=A\) - where \(\vec{A}=h\vec{a}+k\vec{b}\), and \(h,k=0,1,2,...\) - represented in Fig.2 folded to obtain the icosahedron, the platonic solid with 12 vertices, 20 faces and 30 edges. Each triangle face of the icosahedron, contains a fixed number of hexamers that are the real proteins. At each of the 12 vertices the hexamers must turn into pentamers for the topological and geometrical reasons described above. Say \(|\vec{a}|=a\), then one has \(A^{2}=a^{2}(h^{2}+k^{2}+2hk\cos\varphi)=a^{2}(h^{2}+k^{2}+hk)\equiv a^{2}T(h,k)\), with \(T(h,k)=1,3,4,7,...\).
+
+Figure 2: The equilateral triangles template and the icosadeltahedron. The 10 circled points on the planar template correspond to the 10 inner vertices of the solid, while all the outer vertices of the 5 upper triangles correspond to the north pole vertex of the solid and all the outer vertices of the 5 lower triangles correspond to the south pole vertex. At these locations the hexamers turn into pentamers. Each triangular face of the icosadeltahedron is made of \([T/2]\) hexamers (6 for the example of Fig.1).
+
+Being the area of the triangle given by \(\alpha_{A}=(\sqrt{3}/4)a^{2}T(h,k)\) and the area of one hexagon \(\alpha_{6}=(\sqrt{3}/2)a^{2}\), the number of hexagons per triangle is \(n_{6}=\alpha_{A}/\alpha_{6}=[T/2]\). The total number of subunits is obtained by counting the total number of hexagons used for the _planar_ lattice of Fig.1, which is \(N_{6}=20(T/2)=10T\), then multiplying by 6 (the number of edges of the hexagon): \(N_{\rm proteins}=60T\). On the real 3-dimensional solid (that one one might think of obtaining by folding the planar template) the \(60T\) proteins are arranged as: i) \(60\) form 12 pentamers; ii) \(60(T-1)\) form \(10(T-1)\) hexamers, for a total number of morphological units of \(N=10T+2\). The figures obtained are icosadeltahedrons characterized by the pair of integers \((h,k)\) which not only are related to the total number of proteins, but also give the “chirality” of the polyhedron. Viruses belonging to this class follow these prescriptions with great accuracy and they are classified according to the values of \(T\) (see Table 1 for some examples and Ref.[4] for an exhaustive database on icosahedral virus structures).
+
+\begin{table}
+\begin{tabular}{l r r}
+\hline
+ & \(N_{\rm Proteins}\) & \(T\) \\
+\hline
+Feline Panleukopenia Virus & 60 & 1 \\
+Human Hepatitis B & 240 & 4 \\
+Infectious Bursal Disease Virus (IBDV) & 780 & 13 \\
+_General_ & 60 T & \(h^{2}+k^{2}+hk\) \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Examples of viruses that follow the CK classification taken from Ref. [4].
+
+Recently there have been various attempts to generalize the CK model to include also certain complex viruses. One of those attempts is the model proposed in Ref.[6] - based on the continuum elastic theory of large spherical viruses of Ref.[7] - where the authors address the problem of understanding the formation of spherocylindrical and conical virus capsids. Later we shall show that, if a change in the texture of the arrangement of proteins (scar) takes place, those and many more shapes could be obtained.
+
+## 2 Lessons from the Thomson Problem
+
+Thomson ProblemLet us now turn our attention to a different but geometrically related physical set-up from which we would like to gain some insights for the generalization of the CK construction we are looking for: the Thomson problem [8]. It consists of determining the minimum energy configuration for a collection of electrons constrained to move on the surface of a sphere and interacting via the Coulomb potential. This old (and largely unsolved) problem has many generalizations for more general repulsive potentials as well as for topological defects rather than unit electric charges [9], [10]. The fact that the two problems (virus capsids construction and equilibrium configurations for charges on a sphere) are intimately related can be seen from the numerical results for the Thomson problem that have been obtained over the years. In Ref. [11] the authors proposed as solution of the problem an arrangement of \(N\) electrons on the sphere into a triangular lattice where each electron has 6 nearest neighbors sitting at the vertices of an hexagon, with the exception of 12 locations where the nearest neighbors are only 5 sitting at the vertices of a pentagon and \(N=10T+2\), with \(T=h^{2}+k^{2}+hk\): that is the icosadeltahedron. Note that in this case the electrons are constrained to be on the surface of the sphere, e.g. imagining the sphere as a metal, while the proteins have not such constraint. Furthermore, the polygons here are “imaginary”, in the sense that only the vertices are real particles, while the edges are not.
+
+Scars (and Pentagonal Buttons)Further studies [12] have shown that, even for \(N=10T+2\) electrons, when \(T\) is large enough (of the order of \(10^{2}\)), configurations which differ from the icosadeltahedron have lower energy than the corresponding icosadeltahedron. That is, when near one of the 12 pentagons two hexagons (let us call this a 5-6-6 structure) are replaced by a pair heptagon-pentagon (let us call this a 5-7-5 structure) to form a linear pattern called _scar_, the energy is lower than that of a configuration of 12 pentagons and all the rest hexagons. This indeed happens in numerical simulations for higher and higher number of electrons, where the scars become longer (e.g. 5-7-5-7-5, etc.), always respect the topological/geometrical constraint of Eq. (3), can spiralize or might even form exotic patterns like two nested pentagonal structures with five pentagons placed at the vertices of the outer pentagonal structure, five heptagons at the vertices of the inner pentagonal structure, and a pentagon in the common center (the vertex of the icosadeltahedron) (see, e.g., [9] and references therein). The latter patterns are called _pentagonal buttons_ and an explanation of their topological origin can be found in Ref.[5]. Apparently, even more complicated structures can appear in numerical simulations [9]. Scars have been experimentally found to be formed in spherical crystals of mutually repelling polystyrene beads self-assembled on water droplets in oil [13]. The repulsive potential there is not the Coulomb potential, hence that is a particular instance of the generalized Thomson problem. These experimental findings confirm that, at least in the case of scars, things go along the lines of the above outlined analysis.
+
+The lesson we learn from the Thomson problem is that under certain conditions it is energetically favorable to convert a pair 6-6, with zero total and local curvature (\(0=0+0\)) and zero bending energy, into a pair 5-7, again with zero total curvature but with nonzero local curvature (\(0=+1/12-1/12\)) hence with nonzero bending energy given by \(2E_{b}\), where \(E_{b}\) is necessary to convert a 6 into a 5 or into a 7.
+
+## 3 Scars and Virus Structure
+
+Our ConjectureWhat we propose here is that, due to the interaction with the environment (and/or with the genetic material), the formation of scars of pentamers and heptamers can take place in virus capsids during the process of assembly of the proteins. The way we believe this happens is as follows: i) At first the proteins assemble to make an icosadeltahedron following the CK prescription. ii) At an intermediate stage, due to the interaction with the environment they form scars near the location of one or more of the 12 pentamers at the vertices of the icosadeltahedron. This interaction is necessary because the needed extra bending energy (\(2E_{b}\) in the case of the formation of what we might call a “simple” scar: 5-7-5) can only come from the environment. iii) Eventually, the capsids change shape, from spherical to non-spherical via the release of the bending energy into stretching energy at the location of the scar with the consequent “annihilation” of the 5-7 pair into a 6-6 pair. The resulting capsid has the usual morphological units, pentamers and hexamers, but not the spherical shape. Thus it is to be expected that in real viruses scars should not be visible in the final stage but they should drive a change in shape from spherical to non-spherical. It is plausible, though, that i) in experiments where artificial virus capsids are synthesized, scars could be actually seen at an intermediate stage of the assembly when the “would-be-capsid” is frozen at a suitable point in time; ii) not all scars are annihilated, hence some of them could be visible on the final capsid. Note that in the presence of scars, the total number of proteins needed is the same as for the icosadeltahedral case without scars (this follows from 6 + 6 = 5 + 7) while the number and type of morphological units changes (for one simple scar: 13 pentamers, 1 heptamer, \(10T-12\) hexamers, etc.).
+
+As said earlier, there is a strong interest today in trying to generalize the CK construction to include non-spherical viruses, important examples being the retroviruses that have spherical, spherocylindrical and conical capsids (see, e.g., Ref.[14] and references therein). In the work of Ref.[6] the proposal that spontaneous curvature of the proteins in the capsids can drive a change in shape from spherical to spherocylindrical or conical shapes is extensively studied and the geometric construction of certain capsids (spherocilyndrical and conical) is carried out. The application of that approach to the case of retroviruses is then performed in Ref.[14], where the importance of the environment for the assembly of retrovirus capsids is pointed out. What we conjecture here is that the basis of these phenomena is the formation of scars. Our belief is based on the following observations: i) Scars appear in the geometrically related (generalized) Thomson problem; ii) Their formation/annihilation mechanism here seems to us a natural way to convert the energy given by the environment into bending energy (formation) and subsequently into stretching energy (annihilation); iii) This way a mechanism for producing a great variety of shapes (not only the spherocilyndrical or conical) is in place: the formation/annihilation of scars (simple or complex) in different locations on the intermediate icosadeltahedron (we suppose that this has to happen near the vertices).
+
+Other authors have speculated that scars should occur in virus capsids [13]. They expect scars to be formed only on large viruses and that means that they are expecting scars to be seen on the final capsid. This is an instance that we do not exclude but that is not essential for us as our main proposal is to ascribe the shape change to the scars formation/annihilation mechanism.
+
+Figure 3: The intermediate spherical (icosadeltahedral) capsid with the \(C_{5}\)-symmetric distribution of simple scars.
+
+Variety of shapesIt is easy to convince oneself that indeed a great variety of shapes could be obtained via the scar formation/annihilation mechanism: At the site on the intermediate icosadeltahedron where the scar is formed and then annihilated the sphere gets stretched. The amount of stretching depends on the complexity of the scar⁵. The scars could be formed symmetrically (as we shall see in the next paragraph, for a particular symmetry of formation of scars we shall naturally obtain the spherocylindrical shape) or asymmetrically hence giving rise to regular or irregular shapes. Of all these very large number of shapes only a subset will describe real virus capsids because not all the shapes will be stable or energetically favored. A systematic study can be carried on using this method and case by case it could be seen whether it fits with the elastic properties of the virus capsids and with the constraints coming from the environment [14]. What we shall do now is to construct, within our framework, one particular shape, the spherocylindrical. This will give us the opportunity to show how the method of construction works in a case that it is known to correspond to real virus capsids, like, e.g., certain bacteriophages.
+
+Footnote 5: Complex scars might not be that rare as the same amount of energy is needed for the formation of, say, one next-to-simple scar (5-7-5-7-5) and two simple scars, i.e. \(4E_{b}\).
+
+Spherocylindrical CapsidsSuppose that the intermediate icosadeltahedron is formed. We can then refer to the hexagonal lattice and to the template of Fig.1 and Fig.2. Let us imagine that the scars, e.g. all simple, are created only near the 10 inner vertices via a mechanism that respects the \(C_{5}\) rotation symmetry⁶ around the north pole-south pole axis⁷. In Fig.3 the vertices where the scars are formed are indicated with \(\bullet\), while the other two are indicated with \(\circ\). Take a pair of the equilateral triangles of that template: any one from one of the outer layers of five triangles (e.g. the layer of triangles that correspond to the north pole) and the one from the inner layer that shares an edge with it. In Fig.4 of such pairs is shown and the different nature of the vertices is represented like in Fig.3. The scars are distributed in a way that is asymmetric with respect to the two triangles, hence the net effect of their formation/annihilation mechanism will deform them differently. Depending on the actual orientation of the scar around the given vertex the deformation will be different. To obtain the spherocylindrical capsid the three scars should make the lower triangle thinner and longer (they stretch the area and make it bigger) and this has the effect of shrinking the upper triangle by making the common edge shorter. Due to the symmetry of the location of scars the two edges of the new lower triangle have to be the same. If this mechanism takes place in the same fashion for all the ten pairs⁸ of triangles of the template of Fig.1 the resulting new template is the one given in Fig.4.
+
+Footnote 6: \(C_{n}\) is the finite group of rotations of angles \(2\pi/n\), with \(n=1,2,3,...\). \(C_{5}\) is one of the subgroups of the icosahedral group, the group of all possible symmetries of the icosahedron. Its relevance for the Thomson problem has been understood in [5] where a mechanism of spontaneous symmetry breaking is seen as the responsible for some of the patterns found in numerical simulations. Here our introduction of the \(C_{5}\) symmetry is motivated solely by the need to build up the spherocylinder.
+
+Footnote 7: Of course the axis is completely arbitrary as long as it encompasses two opposite vertices.
+
+Footnote 8: Five north pole triangles paired with their common-edge inner triangles and five south pole triangles paired with their common-edge inner triangles.
+
+Figure 4: The generalized CK construction of the template driven by the scars formation-annihilation mechanism.
+
+We require that this mechanism is area preserving, i.e. that the total number of proteins needed is the same as the one needed for the icosadeltahedron, they are only rearranged. This is obtained by requiring that \(2\alpha_{A}=\alpha_{1}+\alpha_{2}\), where \(\alpha_{1}\) is the area of the upper new triangle and \(\alpha_{2}\) the area of the lower new triangle in Fig.4. This means that the three quantities must be related as
+
+\[\sqrt{3}A^{2}=B\left(\sqrt{A^{2}-\frac{1}{4}B^{2}}+\sqrt{{C}^{2}-\frac{1}{4}B^{2}}\right)\;,\] (4)
+
+with \(B<A\) and \(C>A\). Recall that, for \(a=1\), \(A^{2}=T=h^{2}+k^{2}+hk\), hence the final capsid, obtained by folding the new template of Fig.4 (see Fig.5), will have (12 pentamers and) the \(10(T-1)\) hexamers distributed differently with respect to the intermediate icosadeltahedron.
+
+Notice that this spherocylinder is slightly different from the one obtained in [6] as the upper and lower half-icosadeltahedrons are not obtained by folding five equilateral triangles but five isosceles triangles (in this sense they are no longer proper half-icosadeltahedrons but a deformation of them). This is an instance that could be experimentally tested.
+
+From this construction it is clear that a variety of shapes could be obtained this way. For instance, if the orientation of the scars in the previous setting is such that \(C\) shrinks, hence \(B\) becomes longer, then a disk-like shape is obtained. Let us stress here again that for this to correspond to real virus capsids one needs more detailed information on the elastic properties of the proteins and of the interaction with the environment.
+
+Figure 5: The spherocylindrical capsid.
+
+## 4 Conclusions
+
+In this paper we propose a mechanism of formation and subsequent annihilation of scars of pentamers-heptamers at an intermediate stage of the assembly of the virus capsid as the responsible for a great variety of non-spherical virus shapes. Our conjecture is based on the fact that scars are found in the (generalized) Thomson problem, in experiments and in numerical simulations, and on the observation that this mechanism would give a simple and plausible explanation of how the energy provided by the environment is converted into a change of capsid’s shape. The conjecture can be tested, for instance, in experiments where artificial capsids are synthesized. Scars should appear on what we called here the intermediate icosadeltahedron, then should drive the change in shape. Capsids that could perhaps be used to this end are those relative to viruses that are known to have non-spherical final shape but still pentamers and hexamers as morphological units, like for instance certain bacteriophages. This conjecture, if experimentally confirmed, would extend the classic Caspar and Klug construction for icosahedral viruses to include viruses that still have pentamers and hexamers as morphological units but no longer are icosadeltahedrons.
+
+Let us conclude by making the remark that a better understanding of the way virus capsids are formed might suggest ways of destroying a virus by, for example, making the capsid unstable.
+
+## Acknowledgments
+
+A.I. thanks Paul Voorheis of Trinity College Dublin, Daniel Grumiller of MIT Boston, for enjoyable discussions and for providing some difficult-to-find references and Arianna Calistri of the University of Padua for advice with virology. S.S. acknowledges the kind hospitality of the Institute of Particle and Nuclear Physics of Charles University Prague.
+
+## References
+
+* [1] W. Chiu, R. M. Burnett, R. L. Garcea Eds., Structural Biology of Viruses, Oxford University Press (New York) 1997.
+* [2] F. Crick, J. D. Watson, Nature **177** (1956) 473.
+* [3] D. Caspar, A. Klug, Cold Spring Harb Symp. Quant Biol. **27** (1962) 1.
+* [4] Virus Particle ExploreR (VIPER), V. Reddy,P. Natarajan,B. Okerberg,K. Li,K. Damodaran, R. Morton,C. Brooks III, J. Johnson, J. of Vir. **75** (2001) 11943 (http://viperdb.scripps.edu/)
+* [5] A. Iorio, S. Sen, Phys. Rev. B **74** (2006) 052102; ibidem **75** (2007) 099901 (E).
+* [6] T. T. Nguyen, R. F. Bruinsma, W. M. Gelbart, Phys. Rev. E **72** (2005) 051923.
+* [7] J. Lidmar, L. Mirny, D. R. Nelson, Phys. Rev. E **68** (2003) 051910.
+* [8] J. J. Thomson, Phil. Mag. **7** (1904) 237.
+* [9] M. J. Bowick, D. R. Nelson, A. Travesset, Phys. Rev. B **62** (2000) 8738.
+* [10] M. J. Bowick, A. Cacciuto, D. R. Nelson, A. Travesset, Phys. Rev. Lett. **89** (2002) 185502 and Phys. Rev. B **73** (2006) 024115.
+* [11] E. L. Altschuler, T. J. Williams, E. R. Ratner, R. Tipton, R. Stong, F. Dowla, F. Wooten, Prhys. Rev. Lett. **78** (1997) 2681.
+* [12] A. Perez-Garrido, M. J. W. Dodgson, M A. Moore, M. Ortuño, A. Diaz-Sanchez, Phys. Rev. Lett. **79** (1997) 1417.
+* [13] A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G. Nikolaides, A. Travesset, D. A. Weitz, Science **299** (2003) 1716.
+* [14] T. T. Nguyen, R. F. Bruinsma, W. .M. Gelbart, Phys. Rev. Lett. **96** (2006) 078102.
+

arxiv_ground_truth/0707.4042.md

@@ -0,0 +1,389 @@
+**The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof**
+
+James Allen Fill¹
+
+Footnote 1: Research supported by NSF grant DMS–0406104, and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.
+
+Department of Applied Mathematics and Statistics
+
+The Johns Hopkins University
+
+jimfill@jhu.edu and http://www.ams.jhu.edu/~fill/
+
+ABSTRACT
+
+A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any \(d\), the passage time from state \(0\) to state \(d\) is distributed as a sum of \(d\) independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray–Knight theorem.
+
+Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time \(T\) for an ergodic birth-and-death chain on \(\{0,\dots,d\}\) in continuous time with generator \(G\), started in state \(0\), is distributed as a sum of \(d\) independent exponential random variables whose rate parameters are the nonzero eigenvalues of \(-G\). Our approach yields the first (sample-path) construction of such a \(T\) for which individual such exponentials summing to \(T\) can be explicitly identified.
+
+_AMS_ 2000 _subject classifications._ Primary 60J25; secondary 60J35, 60J10, 60G40.
+
+_Key words and phrases._ Markov chains, birth-and-death chains, passage time, strong stationary duality, anti-dual, eigenvalues, stochastic monotonicity, Ray–Knight theorem.
+
+_Date._ Revised May 18, 2009.
+
+## 1 Introduction and summary
+
+A well-known theorem usually attributed to Keilson [12] (Theorem 5.1A, together with Remark 5.1B; see also Section 1 of [11]), but which—as pointed out by Laurent Saloff-Coste via Diaconis and Miclo [5]—can be traced back at least as far as Karlin and McGregor [10, equation (45)], states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any \(d\), the passage time from state \(0\) to state \(d\) is distributed as a sum of \(d\) independent exponential random variables with distinct rate parameters. Keilson, like Karlin and McGregor, proves this result by analytical (non-probabilistic) means.
+
+Modulo the distinctness of the rates, and with additional information (see, e.g., [2]) relating the exponential rates to spectral information about the chain, the theorem can be recast as follows.
+
+**Theorem 1.1****.**: _Consider a continuous-time birth-and-death chain with generator \(G^{*}\) on the state space \(\{0,\dots,d\}\) started at \(0\), suppose that \(d\) is an absorbing state, and suppose that the other birth rates \(\lambda^{*}_{i}\), \(0\leq i\leq d-1\), and death rates \(\mu^{*}_{i}\), \(1\leq i\leq d-1\), are positive. Then the absorption time in state \(d\) is distributed as the sum of \(d\) independent exponential random variables whose rate parameters are the \(d\) nonzero eigenvalues of \(-G^{*}\)._
+
+There is an analogue for discrete time:
+
+**Theorem 1.2****.**: _Consider a discrete-time birth-and-death chain with transition kernel \(P^{*}\) on the state space \(\{0,\dots,d\}\) started at \(0\), suppose that \(d\) is an absorbing state, and suppose that the other birth probabilities \(p^{*}_{i}\), \(0\leq i\leq d-1\), and death probabilities \(q^{*}_{i}\), \(1\leq i\leq d-1\), are positive. Then the absorption time in state \(d\) has probability generating function_
+
+\[u\mapsto\prod_{j=0}^{d-1}\left[\frac{(1-\theta_{j})u}{1-\theta_{j}u}\right],\]
+
+_where \(-1\leq\theta_{j}<1\) are the \(d\) non-unit eigenvalues of \(P^{*}\)._
+
+In this paper we will give a stochastic proof of Theorem 1.2 under the additional hypothesis that all eigenvalues of \(P^{*}\) are (strictly) positive; as we shall see later (Lemma 2.4), this implies another condition key to our development, namely, that
+
+\[p^{*}_{i-1}+q^{*}_{i}<1,\qquad 1\leq i\leq d.\] (1.1)
+
+Whenever \(P^{*}\) has nonnegative eigenvalues, the conclusion of Theorem 1.2 simplifies:
+
+_The absorption time in state \(d\) is distributed as the sum of \(d\) independent geometric random variables whose failure probabilities are the non-unit eigenvalues of \(P^{*}\)._
+
+The special-case of Theorem 1.2 for positive eigenvalues establishes the theorem in general by the following argument (which is unusual, in that it is not often easy to relate characteristics of a chain to a “lazy” modification). Choose any \(\varepsilon\in(0,1/2)\) and apply the special case of Theorem 1.2 to the “lazy” kernel \(P^{*}(\varepsilon):=(1-\varepsilon)I+\varepsilon P^{*}\). Let \(T^{*}\) and \(T^{*}(\varepsilon)\) denote the respective absorption times for \(P^{*}\) and \(P^{*}(\varepsilon)\). Then \(T^{*}(\varepsilon)\) has probability generating function (pgf)
+
+\[{\bf E}\,s^{T^{*}(\varepsilon)}=\prod_{j=0}^{d-1}\left[\frac{\varepsilon(1-\theta_{j})s}{1-(1-\varepsilon(1-\theta_{j}))s}\right].\] (1.2)
+
+But the conditional distribution of \(T^{*}(\varepsilon)\) given \(T^{*}\) is negative binomial with parameters \(T^{*}\) and \(\varepsilon\), so the pgf of \(T^{*}(\varepsilon)\) can also be computed as
+
+\[{\bf E}\,s^{T^{*}(\varepsilon)}={\bf E}\,{\bf E}\left(\left.s^{T^{*}(\varepsilon)}\right|T^{*}\right)={\bf E}\left(\frac{\varepsilon s}{1-(1-\varepsilon)s}\right)^{T^{*}}.\] (1.3)
+
+Equating (1.2) and (1.3) and letting \(u:=\varepsilon s/[1-(1-\varepsilon)s]\), we find, as desired,
+
+\[{\bf E}\,u^{T^{*}}=\left[\frac{(1-\theta_{j})u}{1-\theta_{j}u}\right].\]
+
+Later in this section we explain how Theorem 1.1 follows from Theorem 1.2, but in Section 5 we will also outline a direct stochastic proof of Theorem 1.1.
+
+**Remark 1.3****.**: (a) Theorems 1.1 and 1.2 are the starting point of an in-depth consideration of separation cut-off for birth-and-death chains in [6].
+
+(b) By a simple perturbation argument, Theorems 1.1 and 1.2 extend to all birth-and-death chains for which the birth rates \(\lambda^{*}_{i}\) (respectively, birth probabilities \(p^{*}_{i}\)), \(0\leq i\leq d-1\), are positive.
+
+(c) There is a stochastic interpretation of the pgf in Theorem 1.2 even when some of the eigenvalues are negative (see (4.23) in [4]), but we do not know a stochastic proof (i.e., a proof that proceeds by constructing random variables) in that case.
+
+(d) The condition (1.1) is closely related to the notion of (stochastic) monotonicity. All continuous-time, but not all discrete-time, birth-and-death chains are monotone. In discrete time, monotonicity for a general chain is the requirement that the distributions \(P^{*}(i,\cdot)\) in the successive rows of \(P^{*}\) be stochastically nondecreasing, i.e., that \(\sum_{k>j}P^{*}(i,k)\) be nondecreasing in \(i\) for each \(j\). As noted in [3], for a discrete-time birth-and-death chain \(P^{*}\), monotonicity is equivalent to the condition
+
+\[p^{*}_{i-1}+q^{*}_{i}\leq 1,\qquad 1\leq i\leq d.\]
+
+We need only prove the discrete-time Theorem 1.2 (or even just the special case where \(P^{*}\) has positive eigenvalues), for then given a continuous-time birth-and-death generator \(G^{*}\) we can consider the discrete-time birth-and-death kernels \(P^{*}(\varepsilon):=I+\varepsilon G^{*}\), where \(I\) denotes the identity matrix and \(\varepsilon>0\) is chosen sufficiently small that \(P^{*}(\varepsilon)\) is nonnegative and has positive eigenvalues. Let \(T(\varepsilon)\) and \(T\) denote the absorption times for \(P^{*}(\varepsilon)\) and \(G^{*}\), respectively. Then it is simple to check that \(\varepsilon T(\varepsilon)\) converges in law to \(T\); indeed, for any \(0<t<\infty\) we have
+
+\[{\bf P}(\varepsilon T(\varepsilon)\leq t)=(P^{*}(\varepsilon))^{\left\lfloor t/\varepsilon\right\rfloor}(0,d)\to(e^{tG^{*}})(0,d)=P(T\leq t).\]
+
+But the eigenvalues of \(P^{*}(\varepsilon)\) and of \(-G^{*}\) are simply related, and suitably scaled geometric random variables converge in law to exponentials, so Theorem 1.1 follows immediately.
+
+The idea of our proof of Theorem 1.2 is simple: We show that the absorption time (call it \(T^{*}\)) of \(P^{*}\) has the same distribution as \(\widehat{T}\), where \(\widehat{T}\) is the absorption time of a certain pure-birth chain \(\widehat{P}\) whose holding probabilities are precisely the non-unit eigenvalues of \(P^{*}\).
+
+We do this by reviewing (in Section 2) and then employing the Diaconis and Fill [4] theory of strong stationary duality in discrete time. In brief, a given absorbing birth-and-death chain \(P^{*}\) satisfying (1.1) is the classical set-valued strong stationary dual (SSD) of some monotone birth-and-death chain \(P\) with the same eigenvalues; naturally enough, we will call \(P\) an “anti-dual” of \(P^{*}\). But, if also the eigenvalues of \(P^{*}\) are nonnegative, then we show that this \(P\) (and indeed any ergodic birth-and-death chain with nonnegative eigenvalues) in turn also has a pure-birth SSD \(\widehat{P}\) whose holding probabilities are precisely the non-unit eigenvalues of \(P\). Since we argue that both duals are sharp (i.e., give rise to a stochastically minimal strong stationary time for the \(P\)-chain), the absorption time \(T^{*}\) of \(P^{*}\) has the same distribution as the absorption time \(\widehat{T}\) of \(\widehat{P}\), and the latter distribution is manifestly the convolution of geometric distributions.
+
+**Remark 1.4****.**: (a) Although our proof of Theorem 1.2 is stochastic, it leaves open [or, rather, left open—see part (c) of this remark] the question of whether the absorption time itself can be represented as an independent sum of explicit geometric random variables; the proof establishes only equality in distribution. The difficulty with our approach is that there can be many different stochastically minimal strong stationary times for a given chain.
+
+(b) However, for either of the two steps of our argument we can give sample-path constructions relating the two chains (either \(P^{*}\) and \(P\), or \(P\) and \(\widehat{P}\)). This has already been carried out in detail for the first step in [4]. For the second step, what this means is that we can show how to watch the \(P\)-chain \(X\) run and contemporaneously construct from it a chain \(\widehat{X}\) with kernel \(\widehat{P}\) in such a way that the absorption time \(\widehat{T}\) of \(\widehat{P}\) is a fastest strong stationary time for \(X\).
+
+(c) Subsequent to the work leading to the present paper, Diaconis and Miclo [5] gave another stochastic proof of Theorem 1.1. Their proof, which provides an “intertwining” between the kernels \(P^{*}\) and \(\widehat{P}\) (in our notation), yields a construction of exponentials summing to the absorption time, but the construction is, by their own estimation, “quite involved”. In a forthcoming paper [9], we will exhibit a much simpler such construction, with extensions to skip-free processes.
+
+Section 2 is devoted to a brief review of strong stationary duality and a proof that any discrete-time birth-and-death kernel with positive eigenvalues satisfies (1.1). In Section 3, we construct \(P\) from \(P^{*}\). In Section 4 we construct \(\widehat{P}\) from \(P\) and (in Section 4.1) describe the sample-path construction discussed in Remark 1.4(b). In Section 5 for completeness we provide continuous-time analogs of our discrete-time auxiliary results, which we find interesting in their own right and which combine to give a direct stochastic proof of Theorem 1.1. Section 6 shows how to extend Theorem 1.1 from the hitting time of state \(d\) to the occupation-time vector for the states \(\{0,\dots,d-1\}\) and connects the present paper with work of Kent [13] and the celebrated Ray–Knight theorem [17, 14].
+
+## 2 A quick review of strong stationary duality
+
+The main purpose of this background section is to review the theory of strong stationary duality only to the extent necessary to understand the proof of Theorem 1.2. For a more general and more detailed treatment, consult [4], especially Sections 2–4. To a reasonable extent, the notation of this paper matches that of [4]. Strong stationary duality has been used to bound mixing times of Markov chains and also to build perfect simulation algorithms [8].
+
+### Strong stationary duality in general
+
+Let \(X\) be an ergodic (irreducible and aperiodic) Markov chain on a finite state space; call its stationary distribution \(\pi\). A _strong stationary time_ is a randomized stopping time \(T\) for \(X\) such that \(X_{T}\) has the distribution \(\pi\) and it independent of \(T\). Aldous and Diaconis [1, Proposition 3.2] prove that for any such \(X\) there exists a fastest (i.e., stochastically minimal) strong stationary time, although it is well known that such a fastest time is not (generally) unique. (Such a fastest time is called a _time to stationarity_ in [4], but this terminology has not been widely adopted and so will not be used here.)
+
+A systematic approach to building strong stationary times is provided by the framework of strong stationary duality. The following specialization of the treatment in Section 2 of [4] (see especially Theorem 2.17 and Remark 2.39 there) is sufficient for our purposes.
+
+**Theorem 2.1****.**: _Let \(\pi_{0}\) and \(\pi^{*}_{0}\) be probability mass functions on \(\{0,1,\dots d\}\), regarded as row vectors, and let \(P\), \(P^{*}\), and \(\Lambda\) be transition matrices on \(S\). Assume that \(P\) is ergodic with stationary distribution \(\pi\), that state \(d\) is absorbing for \(P^{*}\), and that the row \(\Lambda(d,\cdot)\) equals \(\pi\). If \((\pi^{*}_{0},P^{*})\) is a strong stationary dual of \((\pi_{0},P)\) with respect to the link \(\Lambda\) in the sense that_
+
+\[\pi_{0}=\pi^{*}_{0}\Lambda\quad\mbox{and}\quad\Lambda P=P^{*}\Lambda,\] (2.1)
+
+_then there exists a bivariate Markov chain \((X^{*},X)\) such that_
+
+1. (a)\(X\) _is marginally Markov with initial distribution_ \(\pi_{0}\) _and transition matrix_ \(P\)_;_
+2. (b)\(X^{*}\) _is marginally Markov with initial distribution_ \(\pi^{*}_{0}\) _and transition matrix_ \(P^{*}\)_;_
+3. (c)_the absorption time_ \(T^{*}\) _of_ \(X^{*}\) _is a strong stationary time for_ \(X\)_._
+
+_Moreover, if \(\Lambda(i,d)=0\) for \(i=0,\dots,d-1\), then the dual is sharp in the sense that \(T^{*}\) is a fastest strong stationary time for \(X\)._
+
+**Remark 2.2****.**: In both our applications of Theorem 2.1 (Sections 3 and 4),
+
+1. (i)the initial distributions \(\pi_{0}\) and \(\pi^{*}_{0}\) are both taken to be unit mass \(\delta_{0}\) at \(0\), and \(\Lambda(0,\cdot)=\delta_{0}\), too, so only the second equation in (2.1) needs to be checked; and
+2. (ii)the link \(\Lambda\) is lower triangular, from which we observe that the corresponding dual is sharp and (if also the diagonal elements of \(\Lambda\) are all positive) that, given \(P\), there is at most one stochastic matrix \(P^{*}\) satisfying (2.1), namely, \(P^{*}=\Lambda P\Lambda^{-1}\).
+
+### Classical (set-valued) strong stationary duals
+
+Let \(P\) be ergodic with stationary distribution \(\pi\), and let \(H\) denote the corresponding cumulative distribution function (cdf):
+
+\[H_{j}=\sum_{i\leq j}\pi_{i}.\]
+
+Let \(\Lambda\) be the link of truncated stationary distributions:
+
+\[\Lambda(x^{*},x)={\bf 1}(x\leq x^{*})\pi_{x}/H_{x^{*}}.\] (2.2)
+
+If \(P\) is a monotone birth-and-death chain (more generally, if \(P\) is arbitrary and the time reversal of \(P\) is monotone—see [4, Theorem 4.6]), then a dual \(P^{*}\) exists [and is sharp and unique by Remark 2.2(ii)]:
+
+**Theorem 2.3****.**: _Let \(P\) be a monotone ergodic birth-and-death chain on \(\{0,\dots,d\}\) with stationary cdf \(H\). Then \(P\) has a sharp strong stationary dual \(P^{*}\) with respect to the link of truncated stationary distributions. The chain \(P^{*}\) is also birth-and-death, with death, hold, and birth probabilities (respectively)_
+
+\[q^{*}_{i}=\frac{H_{i-1}}{H_{i}}p_{i}\qquad r^{*}_{i}=1-(p_{i}+q_{i+1}),\qquad p^{*}_{i}=\frac{H_{i+1}}{H_{i}}q_{i+1}.\] (2.3)
+
+See Sections 3–4 of [4] for an explanation as to why the dual in Theorem 2.3 is called “set-valued”; in this paper we shall refer to it as the “classical” SSD. The equations (2.3) reproduce [4, (4.18)].
+
+### Positivity of eigenvalues and stochastic monotonicity for birth-and-death chains
+
+When we prove Theorem 1.2 assuming that \(P^{*}\) has positive eigenvalues, we will utilize the strengthened monotonicity condition (1.1). Part (a) of the following lemma provides justification.
+
+**Lemma 2.4****.**: _Let \(P^{*}\) be the kernel of any birth-and-death chain on \(\{0,\dots,d\}\)._
+
+1. (a)_If_ \(P^{*}\) _has positive eigenvalues, then (_1.1_) holds._
+2. (b)_If_ \(P^{*}\) _has nonnegative eigenvalues, then_ \(P^{*}\) _is monotone._
+
+Proof.: (a) By perturbing \(P^{*}\) if necessary, we may assume that \(P^{*}\) is ergodic. Then \(P^{*}\) is diagonally similar to a positive definite matrix whose principal minor corresponding to rows and columns \(i-1\) and \(i\) is \(r^{*}_{i-1}r^{*}_{i}-p^{*}_{i-1}q^{*}_{i}\), so
+
+\[0<r^{*}_{i-1}r^{*}_{i}-p^{*}_{i-1}q^{*}_{i}\leq(1-p^{*}_{i-1})(1-q^{*}_{i})-p^{*}_{i-1}q^{*}_{i}=1-p^{*}_{i-1}-q^{*}_{i}.\]
+
+(b) This follows by perturbation from part (a). ∎
+
+**Remark 2.5****.**: Both converse statements are false. For any given \(d\geq 2\), the condition (1.1) does not imply nonnegativity of eigenvalues, not even for chains \(P^{*}\) satisfying the hypotheses of Theorem 1.2. An explicit counterexample for \(d=2\) is
+
+\[P^{*}=\left[\begin{array}[]{lll}0.50&0.50&0\\ 0.49&0.02&0.49\\ 0&0&1\end{array}\right],\]
+
+whose smallest eigenvalue is \((26-\sqrt{3026})/100<0\). For general \(d\geq 2\), perturb the direct sum of this counterexample with the identity matrix.
+
+## 3 An anti-dual \(P\) of the given \(P^{*}\)
+
+As discussed in Section 1, the main discrete-time theorem, Theorem 1.2, follows from the chief results, Theorems 3.1 and 4.2, of this section and the next.
+
+Under the strengthened monotonicity condition (1.1) (with no assumption here about nonnegativity of the eigenvalues), the anti-dual construction of Theorem 3.1 exhibits the given chain (call its kernel \(P^{*}\)) as the classical SSD of another birth-and-death chain.
+
+**Theorem 3.1****.**: _Consider a discrete-time birth-and-death chain \(P^{*}\) on \(\{0,\dots,d\}\) started at \(0\), and suppose that \(d\) is an absorbing state. Write \(q^{*}_{i}\), \(r^{*}_{i}\), and \(p^{*}_{i}\) for its death, hold, and birth probabilities, respectively. Suppose that \(p^{*}_{i}>0\) for \(0\leq i\leq d-1\), that \(q^{*}_{i}>0\) for \(1\leq i\leq d-1\), and that \(p^{*}_{i-1}+q^{*}_{i}<1\) for \(1\leq i\leq d\). Then \(P^{*}\) is the classical (and hence sharp) SSD of some monotone ergodic birth-and-death kernel \(P\) on \(\{0,\dots,d\}\)._
+
+Proof.: In light of Remark 2.2(i), we have dispensed with initial distributions. The claim is that \(P^{*}\) is related to some monotone ergodic \(P\) with stationary cdf \(H\) via (2.3). We will _begin_ our proof by defining a suitable function \(H\), and then we will construct \(P\).
+
+We inductively define a strictly increasing function \(H:\{0,\dots,d\}\to(0,1]\). Let \(H_{d}:=1\), and define \(H_{d-1}\in(0,1)\) in (for now) arbitrary fashion. Having defined \(H_{d},\dots,H_{i}\) (for some \(1\leq i\leq d-1\)), choose the value of \(H_{i-1}\in(0,H_{i})\) so that
+
+\[\left(\frac{H_{i}}{H_{i-1}}-1\right)q^{*}_{i}=\left(1-\frac{H_{i}}{H_{i+1}}\right)p^{*}_{i};\] (3.1)
+
+this is clearly possible since the right side of (3.1) is in \((0,1)\) and the left side, as a function of the variable \(H_{i-1}\), decreases from \(\infty\) at \(H_{i-1}=0+\) to \(0\) at \(H_{i-1}=H_{i}\). It is also clear that by choosing \(H_{d-1}\) sufficiently close to \(1\), we can make _all_ the ratios \(H_{i}/H_{i-1}\) (\(i=1,\dots d\)) as (uniformly) close to \(1\) as we wish.
+
+Next, define \(q_{0}:=0\),
+
+\[p_{0}:=\left(1-\frac{H_{0}}{H_{1}}\right)p^{*}_{0},\] (3.2)
+
+and, for \(1\leq i\leq d\),
+
+\[p_{i}:=\frac{H_{i}}{H_{i-1}}q^{*}_{i},\qquad q_{i}:=\frac{H_{i-1}}{H_{i}}p^{*}_{i-1}.\] (3.3)
+
+When the \(H\)-ratios are taken close enough to \(1\), then for \(0\leq i\leq d\) we have \(p_{i}+q_{i}<1\) and we define
+
+\[r_{i}:=1-p_{i}-q_{i}>0.\]
+
+The kernel \(P\) with death, hold, and birth probabilities \(q_{i}\), \(r_{i}\), and \(p_{i}\) is irreducible and aperiodic, and thus ergodic. To complete the proof, will also show
+
+1. (a)\(P\) is monotone (recall: equivalent to \(p_{i}+q_{i+1}\leq 1\) for \(0\leq i\leq d-1\)),
+2. (b)\(P\) has stationary cdf \(H\), and
+3. (c)\(P^{*}\) is the classical SSD of \(P\).
+
+For (a) we simply observe, using (3.3) and (3.1), that
+
+\[p_{i}+q_{i+1}=\frac{H_{i}}{H_{i-1}}q^{*}_{i}+\frac{H_{i}}{H_{i+1}}p^{*}_{i}=q^{*}_{i}+p^{*}_{i}\leq 1\] (3.4)
+
+for \(1\leq i\leq d-1\); and similarly that
+
+\[p_{0}+q_{1}=\left(1-\frac{H_{0}}{H_{1}}\right)p^{*}_{0}+\frac{H_{0}}{H_{1}}p^{*}_{0}=p^{*}_{0}\leq 1.\]
+
+For (b) we observe, again using (3.3) and (3.1), that the detailed balance condition
+
+\[(H_{i}-H_{i-1})p_{i}=(H_{i}-H_{i-1})\frac{H_{i}}{H_{i-1}}q^{*}_{i}=(H_{i+1}-H_{i})\frac{H_{i}}{H_{i+1}}p^{*}_{i}=(H_{i+1}-H_{i})q_{i+1}\]
+
+holds for \(1\leq i\leq d-1\); by (3.2) and (3.3), it also holds for \(i=0\):
+
+\[H_{0}p_{0}=(H_{1}-H_{0})\frac{H_{0}}{H_{1}}p^{*}_{0}=(H_{1}-H_{0})q_{1}.\]
+
+For (c), we simply verify that (2.3) holds: for \(0\leq i\leq d\) (with \(H_{-1}:=0\)), from (3.3) and (3.4),
+
+\[\frac{H_{i-1}}{H_{i}}p_{i}=q^{*}_{i},\qquad\frac{H_{i+1}}{H_{i}}q_{i+1}=p^{*}_{i},\qquad p_{i}+q_{i+1}=q^{*}_{i}+p^{*}_{i}=1-r^{*}_{i}.\qquad~{}\qed\] (3.5)
+
+**Remark 3.2****.**: Once the value of \(H_{d-1}\) is chosen, the definitions of \(H\) and \(P\) are forced; indeed, if the detailed balance condition and (3.5) are to hold, then we must have (3.1)–(3.3).
+
+## 4 A pure birth “spectral” dual of \(P\)
+
+In this section we construct a sharp pure birth dual \(\widehat{P}\) for any ergodic birth-and-death chain \(P\) on \(\{0,\dots d\}\) with nonnegative eigenvalues started in state \(0\). When this construction is applied in the proof of Theorem 1.2 to the chain \(P\) resulting from \(P^{*}\) by application of Theorem 3.1, assuming nonnegativity of the eigenvalues of \(P^{*}\) yields the required nonnegativity of the eigenvalues of \(P\) in Theorem 4.2; indeed, as noted in Remark 2.2(ii), the matrices \(P\) and \(P^{*}\) are similar.
+
+Our construction of the pure birth dual specializes a SSD construction of Matthews [15] for general reversible chains with nonnegative eigenvalues; that construction is closely related to the spectral decomposition of the transition matrix. For completeness and the reader’s convenience, and because for birth-and-death chains (a) we can give a more streamlined presentation with minimal reference to eigenvectors and (b) we wish to establish the new result that the resulting dual is sharp, we do not presume familiarity with [15].
+
+To set up our construction we need some notation. Let \(P\) be an ergodic birth-and-death chain on \(\{0,\dots,d\}\) with stationary probability mass function \(\pi\) (note that \(\pi\) is everywhere positive) and nonnegative eigenvalues, say \(0\leq\theta_{0}\leq\theta_{1}\leq\dots\leq\theta_{d-1}<\theta_{d}=1\). (It is well known [12][4, Theorem 4.20] that the eigenvalues are all distinct, but we will not need this fact.) Let \(I\) denote the identity matrix and define
+
+\[Q_{k}:=(1-\theta_{0})^{-1}\cdots(1-\theta_{k-1})^{-1}(P-\theta_{0}I)\cdots(P-\theta_{k-1}I),\quad k=0,\dots,d,\] (4.1)
+
+with the natural convention \(Q_{0}:=I\). Note that for \(k=0,\dots,d-1\) we have
+
+\[Q_{k}P=\theta_{k}Q_{k}+(1-\theta_{k})Q_{k+1}.\] (4.2)
+
+**Lemma 4.1****.**: _The matrices \(Q_{k}\) are all stochastic, and every row of \(Q_{d}\) equals \(\pi\)._
+
+Proof.: For the first assertion it is clear that the rows of \(Q_{k}\) all sum to \(1\), so the only question is whether \(Q_{k}\) is nonnegative. But \(P=D^{-1/2}SD^{1/2}\), where \(D={\rm diag}(\pi)\) and \(S\) is symmetric, so the nonnegativity of \(Q_{k}\) follows from that of
+
+\[S_{k}:=(S-\theta_{0}I)\cdots(S-\theta_{k-1}I),\]
+
+which in turn is an immediate consequence of (the rather nontrivial) Theorem 3.2 in [16] using only that \(S\) is nonnegative and symmetric.
+
+For the second assertion, write
+
+\[S=\sum_{r=0}^{d}\theta_{r}u_{r}u_{r}^{T},\]
+
+where the column vectors \(u_{0},\dots,u_{d}\) form an orthogonal matrix and \(u_{d}\) has \(i\)th entry \(\sqrt{\pi_{i}}\). Then, as noted at (2.6) of [16],
+
+\[S_{k}=\sum_{r=k}^{d}\left[\prod_{t=0}^{k-1}(\theta_{r}-\theta_{t})\right]u_{r}u_{r}^{T}.\]
+
+In particular, \(S_{d}=(1-\theta_{0})\cdots(1-\theta_{d-1})u_{d}u_{d}^{T}\), so every row of \(Q_{d}\) equals \(\pi\). ∎
+
+Now let \(\delta_{0}\) denote unit mass at \(0\) (regarded as a row vector), and define the probability mass functions
+
+\[\lambda_{k}:=\delta_{0}Q_{k},\quad k=0,\dots,d.\] (4.3)
+
+Let \(\widehat{\Lambda}\) [so named to distinguish it from the classic link \(\Lambda\) of (2.2)] be the lower-triangular square matrix with successive rows \(\lambda_{0},\dots,\lambda_{d}\), and define \(\widehat{P}\) to be the pure-birth chain transition matrix on \(\{0,\dots,d\}\) with holding probability \(\theta_{i}\) at state \(i\) for \(i=0,\dots,d\); that is,
+
+\[\hat{p}_{ij}:=\begin{cases}\theta_{i}&\mbox{if $j=i$}\\ 1-\theta_{i}&\mbox{if $j=i+1$}\\ 0&\mbox{otherwise}.\end{cases}\] (4.4)
+
+**Theorem 4.2****.**: _Let \(P\) be an ergodic birth-and-death chain on \(\{0,\dots,d\}\) with nonnegative eigenvalues. In the above notation, \(\widehat{P}\) is a sharp strong stationary dual of \(P\) with respect to the link \(\widehat{\Lambda}\)._
+
+Proof.: We have again dispensed with initial distributions by Remark 2.2(i). The desired equation \(\widehat{\Lambda}P=\widehat{P}\widehat{\Lambda}\) is equivalent to
+
+\[\lambda_{k}P=\theta_{k}\lambda_{k}+(1-\theta_{k})\lambda_{k+1},\quad k=0,\dots,d-1;\qquad\lambda_{d}P=\lambda_{d},\]
+
+which is true because \(\lambda_{d}=\pi\) and, for \(k=0,\dots,d-1\),
+
+\[\lambda_{k}P=\delta_{0}Q_{k}P=\theta_{k}\lambda_{k}+(1-\theta_{k})\lambda_{k+1}\]
+
+by (4.2). The SSD is sharp because \(\widehat{\Lambda}\) is lower triangular; recall Remark 2.2(ii). ∎
+
+**Remark 4.3****.**: Lemma 4.1 is interesting and, as we have now seen, gives rise to the construction of a new “spectral” SSD for a certain subclass of monotone birth-and-death chains, namely, chains with nonnegative eigenvalues [recall Lemma 2.4(b)]. But for the proof of Theorem 1.2 one could make do without the nonnegativity of the matrix \(\widehat{\Lambda}\) by taking the approach of Matthews [15] and considering the chain \(P\) started in a suitable mixture of \(\delta_{0}\) and the stationary distribution \(\pi\). We omit further details.
+
+### Sample-path construction of the spectral dual
+
+Let \(X\) be an ergodic birth-and-death chain on \(\{0,\dots,d\}\) with kernel \(P\) having nonnegative eigenvalues, assume \(X_{0}=0\), and let \(T\) be any fastest strong stationary time for \(X\). Independent of interest in Theorems 1.1 and 1.2, Theorem 4.2 gives the first stochastic interpretation of the individual geometrics in the representation of the distribution of \(T\) as a convolution of geometric distributions. In this subsection we carry this result one step further by showing how to construct, sample path by sample path, a particular fastest strong stationary time \(\widehat{T}\) which is the sum of explicitly identified independent geometric random variables.
+
+The idea is simple. Theorem 4.2 shows that \(\widehat{P}\) of (4.4) is an “algebraic” dual of \(P\) in the sense that the matrix-equation \(\widehat{\Lambda}P=\widehat{P}\widehat{\Lambda}\) holds. But whenever algebraic duality holds for any finite-state ergodic chain with respect to any link (\(\widehat{\Lambda}\) in our case), Section 2.4 of [4] shows explicitly how to construct, from \(X\) and independent randomness, a dual Markov chain (\(\widehat{X}\) in our case, with kernel \(\widehat{P}\)) such that the absorption time \(\widehat{T}\) of \(\widehat{X}\) is a strong stationary time for \(X\); since \(\widehat{\Lambda}\) is lower triangular, \(\widehat{T}\) will be stochastically optimal. So to describe our construction of \(\widehat{X}\) (and hence \(\widehat{T}\)) we need only specialize the construction of [4, Section 2.4] [see especially (2.36) there].
+
+The chain \(X\) starts with \(X_{0}=0\) and we set \(\widehat{X}_{0}=0\). Inductively, we will have \(\widehat{\Lambda}(\widehat{X}_{t},X_{t})>0\) (and so \(X_{t}\leq\widehat{X}_{t}\)) at all times \(t\). The value we construct for \(\widehat{X}_{t}\) depends only on the values of \(\widehat{X}_{t-1}\) and \(X_{t}\) and independent randomness. Indeed, given \(\widehat{X}_{t-1}=\hat{x}\) and \(X_{t}=y\), if \(y\leq\hat{x}\) then our construction sets \(\widehat{X}_{t}=\hat{x}+1\) with probability
+
+\[\frac{\widehat{P}(\hat{x},\hat{x}+1)\widehat{\Lambda}(\hat{x}+1,y)}{(\widehat{P}\widehat{\Lambda})(\hat{x},y)}=\frac{(1-\theta_{\hat{x}})\widehat{\Lambda}(\hat{x}+1,y)}{\theta_{\hat{x}}\widehat{\Lambda}(\hat{x},y)+(1-\theta_{\hat{x}})\widehat{\Lambda}(\hat{x}+1,y)}=\frac{(1-\theta_{\hat{x}})Q_{\hat{x}+1}(0,y)}{(Q_{\hat{x}}P)(0,y)}\] (4.5)
+
+and \(\widehat{X}_{t}=\hat{x}\) with the complementary probability; if \(y=\hat{x}+1\) (which is the only other possibility, since \(y=X_{t}\leq X_{t-1}+1\leq\hat{x}+1\) by induction), then we set \(\widehat{X}_{t}=\hat{x}+1\) with certainty.
+
+The independent geometric random variables, with sum \(\widehat{T}\), are the waiting times between successive births in the chain \(\widehat{X}\) we have built. Thus it is no longer true that the individual geometric distributions “have no known interpretation in terms of the underlying [ergodic] birth and death chain” [6, Section 4, Remark 1]; likewise, for continuous time consult Section 5.1 herein.
+
+Example 4.4**.**: Consider the well-studied Ehrenfest chain, with holding probability \(1/2\):
+
+\[q_{i}=\frac{i}{2d},\quad r_{i}=\frac{1}{2},\quad p_{i}=\frac{d-i}{2d},\qquad i=0,\dots,d.\]
+
+The eigenvalues are \(\theta_{i}\equiv i/d\). A straightforward proof by induction using (4.3) and (4.2) confirms that \(\lambda_{k}\) is the binomial distribution with parameters \(k\) and \(1/2\):
+
+\[\widehat{\Lambda}(\hat{x},x)\equiv{\hat{x}\choose x}2^{-\hat{x}}.\] (4.6)
+
+Thus the probability (4.5) reduces to
+
+\[\frac{(d-\hat{x})(\hat{x}+1)}{2\hat{x}(\hat{x}+1-y)+(d-\hat{x})(\hat{x}+1)}.\]
+
+The chain we have described lifts naturally to random walk on the set \({\bf Z}^{d}_{2}\) of binary \(d\)-tuples whereby one of the \(d\) coordinates is chosen uniformly at random and its entry is then replaced randomly by \(0\) or \(1\). It is interesting to note that the sharp pure-birth SSD chain constructed in this example does _not_ correspond to the well-known “coordinate-checking” sharp SSD (see Example 3.2 of [4]). Indeed, expressed in the birth-and-death chain domain, the coordinate-checking dual is a pure-birth chain, call it \(\widehat{X}^{\prime}\), such that the construction of \(\widehat{X}^{\prime}_{t}\) depends not only on \(\widehat{X}^{\prime}_{t-1}\) and \(X_{t}\) but also on \(X_{t-1}\). The construction rules are that if \(\widehat{X}^{\prime}_{t-1}=\hat{x}\), \(X_{t-1}=x\), and \(X_{t}=y\), then \(\widehat{X}^{\prime}_{t}\) is set to \(\hat{x}+1\) with probability
+
+\(0\) if \(y=x-1\), \(1-\frac{\hat{x}}{d}\) if \(y=x\), and \(\frac{d-\hat{x}}{d-x}\) if \(y=x+1\),
+
+and otherwise \(\widehat{X}^{\prime}_{t}\) holds at \(\hat{x}\). Both duals correspond to the same link (4.6) and the (marginal) transition kernels for \(\widehat{X}\) and \(\widehat{X}^{\prime}\) are the same, but the bivariate constructions of \((\widehat{X},X)\) and \((\widehat{X}^{\prime},X)\) are different.
+
+The freedom for such differences was noted in [4, Remark 2.23(c)] and exploited in the creation of an interruptible perfect simulation algorithm (see [8, Remark 9.8]). In fact, \(\widehat{X}^{\prime}\) (when lifted to \({\bf Z}^{d}_{2}\)) corresponds to the construction used in [8]. An advantage of the \(\widehat{X}\)-construction of the present paper is that it allows (both in our Ehrenfest example and in general) for holding probabilities that are arbitrary (subject to nonnegativity of eigenvalues); in the paragraph containing (4.5), all that changes when a weighted average of the transition kernel and the identity matrix is taken are the eigenvalues \(\theta_{0},\dots,\theta_{d-1}\). __
+
+## 5 Continuous-time analogs of other results
+
+As discussed in Section 1, the continuous-time Theorem 1.1 follows immediately from the discrete-time Theorem 1.2. Another way to prove Theorem 1.1 is to repeat the proof of Theorem 1.2 by establishing continuous-time analogs (namely, the next three results) of the auxiliary results (Theorem 3.1, Lemma 4.1, and Theorem 4.2) in the preceding two sections; we find these interesting in their own right. The continuous-time results are easy to prove utilizing the continuous-time SSD theory of [7], either by repeating the discrete-time proofs or by applying the discrete-time results to the appropriate kernel \(P^{*}(\varepsilon)=I+\varepsilon G^{*}\) or \(P(\varepsilon)=I+\varepsilon G\), with \(\varepsilon>0\) chosen sufficiently small to meet the hypotheses of those results; so we state the results without proof.
+
+In Section 5.1 we will present the analog of Section 4.1 for continuous time.
+
+Here, first, is the analog of Theorem 3.1.
+
+**Theorem 5.1****.**: _Consider a continuous-time birth-and-death chain with generator \(G^{*}\) on \(\{0,\dots,d\}\) started at \(0\), and suppose that \(d\) is an absorbing state. Write \(\mu^{*}_{i}\) and \(\lambda^{*}_{i}\) for its death and birth rates, respectively. Suppose that \(\lambda^{*}_{i}>0\) for \(0\leq i\leq d-1\) and that \(\mu^{*}_{i}>0\) for \(1\leq i\leq d-1\). Then \(G^{*}\) is the classical set-valued (and hence sharp) SSD of some ergodic birth-and-death generator \(G\) on \(\{0,\dots,d\}\)._
+
+To set up the second result we need a little notation. Let \(G\) be the generator of a continuous-time ergodic birth-and-death chain on \(\{0,\dots,d\}\) with stationary probability mass function \(\pi\) and eigenvalues \(\nu_{0}\geq\nu_{1}\geq\dots\geq\nu_{d-1}>\nu_{d}=0\) for \(-G\). (Again, we don’t need the fact [12] that the eigenvalues are distinct.) Define
+
+\[Q_{k}:=\nu_{0}^{-1}\cdots\nu_{k-1}^{-1}(G+\nu_{0}I)\cdots(G+\nu_{k-1}I),\quad k=0,\dots,d,\] (5.1)
+
+with the natural convention \(Q_{0}:=I\).
+
+**Lemma 5.2****.**: _The matrices \(Q_{k}\) are all stochastic, and every row of \(Q_{d}\) equals \(\pi\)._
+
+Now define \(\widehat{\Lambda}\) in terms of the \(Q_{k}\)’s as in the paragraph preceding Theorem 4.2, and let \(\widehat{G}\) be the pure-birth generator on \(\{0,\dots,d\}\) with birth rate \(\nu_{i}\) at state \(i\) for \(i=0,\dots,d\).
+
+**Theorem 5.3****.**: _Let \(G\) be the generator of an ergodic birth-and-death chain on \(\{0,\dots,d\}\). In the above notation, \(\widehat{G}\) is a sharp strong stationary dual of \(G\) with respect to the link \(\widehat{\Lambda}\):_
+
+\[\widehat{\Lambda}G=\widehat{G}\widehat{\Lambda}.\]
+
+### Sample-path construction of the continuous-time spectral dual
+
+Let \(X\) be an ergodic continuous-time birth-and-death chain on \(\{0,\dots,d\}\), adopt all the notation of Section 5 thus far, and assume \(X_{0}=0\). In this subsection by a routine application of Section 2.3 of [7] we give a simple sample-path construction of a “spectral dual” pure birth chain \(\widehat{X}\) with generator \(\widehat{G}\) as described just before Theorem 5.3; its absorption time \(\widehat{T}\) is then a fastest strong stationary time for \(X\) and the independent exponential random variables with sum \(\widehat{T}\) are simply the waiting times for the successive births for \(\widehat{X}\). We thus obtain a stochastic proof, with explicit identification of individual exponential random variables, of Theorem 5 in [7].
+
+The chain \(X\) starts with \(X(0)=0\) and we set \(\widehat{X}(0)=0\). Let \(n\geq 1\) and suppose that \(\widehat{X}\) has been constructed up through the epoch \(\tau_{n-1}\) of the \((n-1)\)st transition for the bivariate process \((\widehat{X},X)\); here \(\tau_{0}:=0\). We describe next, in terms of an exponential random variable \(\widehat{V}\), how to define \(\tau_{n}\) and \(\widehat{X}(\tau_{n})\); we will have \(\widehat{\Lambda}(\widehat{X}(\tau_{n}),X(\tau_{n}))>0\) and hence \(X(\tau_{n})\leq\widehat{X}(\tau_{n})\). Write \((\hat{x},x)\) for the value of \((\widehat{X},X)\) at time \(\tau_{n-1}\); by induction we have \(\widehat{\Lambda}(\hat{x},x)>0\).
+
+Let \(\widehat{V}_{n}\) be exponentially distributed with rate
+
+\[r=\nu_{\hat{x}}\widehat{\Lambda}(\hat{x}+1,x)/\widehat{\Lambda}(\hat{x},x),\] (5.2)
+
+independent of \(\widehat{V}_{1},\dots,\widehat{V}_{n-1}\) and the chain \(X\). Consider two (independent) exponential waiting times begun at epoch \(\tau_{n-1}\): a first for the next transition of the chain \(X\), and a second with rate \(r\). How we proceed breaks into two cases:
+
+1. (i)If the first waiting time is smaller than the second, then \(\tau_{n}\) is the epoch of this next transition for \(X\) and we set \(\widehat{X}(\tau_{n})=\hat{x}=\widehat{X}(\tau_{n-1})\) (with certainty) except in one circumstance: if \(X(\tau_{n})=\hat{x}+1\), then we set \(\widehat{X}(\tau_{n})=\hat{x}+1\), too.
+2. (ii)If the second waiting time is smaller, then \(\tau_{n}=\tau_{n-1}+\widehat{V}_{n}\) and we set \(\widehat{X}(\tau_{n})=\hat{x}+1\).
+
+Example 5.4**.**: Consider the continuous-time version of the Ehrenfest chain with death rates \(\mu_{i}\equiv i\) and birth rates \(\lambda_{i}\equiv d-i\), \(0\leq i\leq d\); the eigenvalues are \(\nu_{i}\equiv 2(d-i)\). Then \(\widehat{\Lambda}\) is again the link (4.6) of binomial distributions, and the rate (5.2) reduces to
+
+\[r=\frac{(d-\hat{x})(\hat{x}+1)}{\hat{x}+1-x}.\]
+
+## 6 Occupation times and connection with Ray–Knight Theorem
+
+Our final section utilizes work of Kent [13]; see the historical note at the end of Section 1 of [5] for closely related material. We show how to extend the continuous-time Theorem 1.1 from the hitting time of state \(d\) first to the occupation-time vector for the states \(\{0,\dots,d-1\}\) and then to the the local time of Brownian motion, thereby proving the Ray–Knight theorem [17, 14].
+
+### From hitting time to occupation times
+
+Consider a continuous-time irreducible birth-and-death chain with generator \(G^{*}\). It is then immediate from the Karlin–McGregor theorem (Theorem 1.1) that the hitting time \(T^{*}\) of state \(d\) has Laplace transform
+
+\[{\bf E}\,e^{-uT^{*}}=\frac{\det(-G_{0})}{\det(-G_{0}+uI)},\] (6.1)
+
+with \(G_{0}\) obtained from \(G^{*}\) by leaving off the last row and column.
+
+Equation (6.1) gives the distribution of the total time elapsed before the chain hits state \(d\). But how is that time apportioned to the states \(0,\dots,d-1\)? This question can be answered from (6.1) using a neat trick of Kent [13] [see the last sentence of his Remark (1) on page 164]. To find the multivariate distribution of the occupation-time vector \({\bf T}=(T_{0},T_{1},\dots,T_{d-1})\), where \(T_{i}\) denotes the occupation time of (i.e., amount of time spent in) state \(i\), it of course suffices to compute the value \({\bf E}\,e^{-\langle{\bf u},{\bf T}\rangle}\) of the Laplace transform for any vector \({\bf u}=(u_{0},\dots,u_{d-1})\) with strictly positive entries. But the distribution of the random variable \(\langle{\bf u},{\bf T}\rangle=\sum u_{i}T_{i}\) is that of the time to absorption for the time-changed generator \(G_{\bf u}\) (say) obtained by dividing the \(i\)th row of \(G^{*}\) by \(u_{i}\) for \(i=0,\dots,d-1\). Therefore, by (6.1) and the scaling property of determinants,
+
+\[{\bf E}\,e^{-\langle{\bf u},{\bf T}\rangle}=\frac{\det(-G_{\bf u})}{\det(-G_{\bf u}+I)}=\frac{\det(-G_{0})}{\det(-G_{0}+U)},\]
+
+where \(U:=\mbox{diag}(u_{0},\dots,u_{d-1})\).
+
+### From occupation times to the Ray–Knight Theorem
+
+Call the stationary distribution \(\pi\). Then the matrix \(S:=D(-G_{0})D^{-1}\) is (strictly) positive definite, where \(D:=\mbox{diag}(\sqrt{\pi})\). Let \(\Sigma:=\frac{1}{2}S^{-1}\). By direct calculation, \({\bf T}\) has the same law as \({\bf Y}+{\bf Z}\), where \({\bf Y}\) and \({\bf Z}\) are independent random vectors with the same law and \({\bf Y}\) is the coordinate-wise square of a Gaussian random vector \({\bf V}\sim{\rm N}(0,\Sigma)\).
+
+Kent [13] uses and extends this “double derivation” of \({\cal L}({\bf T})\) to prove the theorem of Ray [17] and Knight [14] expressing the local time of Brownian motion as the sum of two independent 2-dimensional Bessel processes (i.e., as the sum of two independent squared Brownian motions).
+
+**Acknowledgments.** We thank Persi Diaconis for helpful discussions, and Raymond Nung-Sing Sze and Chi-Kwong Li for pointing out the reference [16].
+
+## References
+
+* [1] Aldous, D. and Diaconis, P. Strong uniform times and finite random walks. _Adv. in Appl. Math._**8** (1987), 69–97.
+* [2] Brown, M. and Shao, Y. S. Identifying coefficients in the spectral representation for first passage time distributions. _Probab. Eng. Inform. Sci._**1** (1987), 69–74.
+* [3] Cox, J. T. and Rösler, U. A duality relation for entrance and exit laws for Markov monotone Markov processes. _Ann. Probab._**13** (1985), 558–565.
+* [4] Diaconis, P. and Fill, J. Strong stationary times via a new form of duality. _Ann. Probab._**18** (1990), 1483–1522.
+* [5] Diaconis, P. and Miclo, L. On times to quasi-stationarity for birth and death processes. _J. Theoret. Probab._ (2009), to appear.
+* [6] Diaconis, P. and Saloff-Coste, L. Separation cut-offs for birth and death chains. _Ann. Appl. Probab._**16** (2006), 2098–2122.
+* [7] Fill, J. A. Strong stationary duality for continuous-time Markov chains. Part I: Theory. _J. Theoret. Probab._**5** (1992), 45–70.
+* [8] Fill, J. A. An interruptible algorithm for perfect sampling via Markov chains. _Ann. Appl. Probab._**8** (1998), 131–162.
+* [9] Fill, J. A. On hitting times and fastest strong stationary times for skip-free and more general chains. _J. Theoret. Probab._ (2009), to appear.
+* [10] Karlin, S. and McGregor, J. Coincidence properties of birth and death processes. _Pacific J. Math._**9** (1959), 1109–1140.
+* [11] Keilson, J. Log-concavity and log-convexity in passage time densities for of diffusion and birth-death processes. _J. Appl. Probab._**8** (1971), 391–398.
+* [12] Keilson, J. _Markov Chain Models—Rarity and Exponentiality._ Springer, New York, 1979.
+* [13] Kent, J. T. The appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes. In _Probability, statistics and analysis_, volume 79 of _London Math. Soc. Lecture Note Ser._, pages 161–179, Cambridge Univ. Press, Cambridge, 1983.
+* [14] Knight, F. B. Random walks and a sojourn density process of Brownian motion. _Trans. Amer. Math. Soc._**109** (1963), 56–86.
+* [15] Matthews, P. Strong stationary times and eigenvalues. _J. Appl. Probab._**29** (1992), 228–233.
+* [16] Micchelli, C. A. and Willoughby, R. A. On functions which preserve the class of Stieltjes matrices. _Lin. Alg. Appl._**23** (1979), 141–156.
+* [17] Ray, D. Sojourn times of diffusion processes. _Illinois J. Math._**7** (1963), 615–630.
+

arxiv_ground_truth/0709.2178.md

@@ -0,0 +1,167 @@
+# Long Memory and Volatility Clustering: is the empirical evidence consistent across stock markets?
+
+S nia R. Bentes
+
+*Iscal, Av. Miguel Bombarda, 20, 1069-035 Lisboa Portugal, soniabentes@clix.pt; **ISCTE, Av. Forcas Armadas, 1649-025 Lisboa, Portugal
+
+###### Abstract
+
+Long memory and volatility clustering are two stylized facts frequently related to financial markets. Traditionally, these phenomena have been studied based on conditionally heteroscedastic models like ARCH, GARCH, IGARCH and FIGARCH, inter alia. One advantage of these models is their ability to capture nonlinear dynamics. Another interesting manner to study the volatility phenomena is by using measures based on the concept of entropy. In this paper we investigate the long memory and volatility clustering for the SP 500, NASDAQ 100 and Stoxx 50 indexes in order to compare the US and European Markets. Additionally, we compare the results from conditionally heteroscedastic models with those from the entropy measures. In the latter, we examine Shannon entropy, Renyi entropy and Tsallis entropy. The results corroborate the previous evidence of nonlinear dynamics in the time series considered.
+
+keywords: Long memory, volatility clustering, ARCH type models, nonlinear dynamics, entropy , Rui Menezes**, Diana A. Mendes**
+
+## Introduction
+
+The study of stock market volatility and the reasons that lie beyond price movements have always played a central role in financial theory, given rise to an intense debate in which long memory and volatility clustering have proven to be particularly significant. Since long memory reflects long run dependencies between stock market returns, and volatility clustering describes the tendency of large changes in asset prices to follow large changes and small changes to follow small changes, these concepts are interrelated and frequently studied in complementarity. Several models based on heteroscedastic conditionally variance have been proposed to capture their properties. This constitutes what we consider the traditional approach. It includes the autoregressive conditionally heteroscedastic model (ARCH) proposed by Engle [1], the Generalized ARCH (GARCH) due to Bollerslev [2] and Taylor [3], the Integrated GARCH (IGARCH) derived by Engle and Bollerslev [4] and the Fractionally Integrated GARCH (FIGARCH) introduced by Baillie _et al._[5]. These models account for nonlinear dynamics, which are shown by the seasonal or cyclical behavior of many stock market returns and constitute their main advantage. However, they are not fully satisfactory, especially when modeling volatility of intra-daily financial returns. For a comprehensive debate on this matter see Bordignon _et al._[6].
+
+In this paper we propose an alternative way to study stock market volatility based on econophysics models. The application of concepts of physics to explain economic phenomena is relatively recent and started when some regularities between economic/financial and physical data were found in a consistent way (see e.g [7] and [8]). In this sense, one concept of physics that can be helpful to measure the nonlinear volatility of stock markets is the concept of entropy. Regarding this, we discuss three different measures: Shannon entropy, Renyi entropy and Tsallis entropy, and compare the main results.
+
+The plan for the remainder of the paper is as follows: Section 2 puts together the ARCH/GARCH type models and the entropy models. Next, in Section 3 we describe the empirical findings. Finally, Section 4 presents the conclusions.
+
+## 1 Traditional Volatility Models versus Econophysics Models
+
+According to the traditional approach the presence of conditionally heteroscedastic variance in stock market returns gives support to the use of ARCH/GARCH models when studying stock market volatility. Although there are other econometric models to seek for long memory and volatility clustering the most common ones are the GARCH \((p,q)\) model and their derivations - IGARCH \((p,q)\) and FIGARCH \((p,d,q)\) - which are summarized below.
+
+Consider a time series \(y_{t}\) with the associated error
+
+\[e_{t}=y_{t}-E_{t-1}y_{t},\] (1)
+
+where \(E_{t-1}\) is the expectation operator conditioned on time \(t-1\). A GARCH model where
+
+\[e_{t}=z_{t}\sigma_{t},\qquad z_{t}\sim N\left(0,1\right)\] (2)
+
+was developed such as
+
+\[\sigma_{t}^{2}=\omega+\alpha\left(L\right)\varepsilon_{t}^{2}+\beta\left(L\right)\sigma_{t}^{2},\] (3)
+
+where \(\omega>0\), and \(\alpha\left(L\right)\) and \(\beta\left(L\right)\) are polynomials in the lag operator \(L\)\(\left(L^{i}x_{i}=x_{t-i}\right)\) of order \(q\) and \(p,\) respectively. Expression (3) can be rewritten as the infinite-order ARCH \((p)\) process,
+
+\[\Phi\left(L\right)e_{t}^{2}=\omega+\left[1-\beta\left(L\right)\right]\upsilon_{t},\] (4)
+
+where \(\upsilon_{t}\equiv e_{t}^{2}-\sigma_{t}^{2}\) and \(\Phi\left(L\right)=\left[1-\alpha\left(L\right)-\beta\left(L\right)\right]\). Even though this process is frequently used to describe volatility clustering, it shows some limitations when dealing with long memory since it assumes that shocks decay at a fast geometric rate allowing only for short term persistence. To overcome this drawback Engle and Bollerslev [4] developed the IGARCH specification given by
+
+\[\Phi\left(L\right)\left(1-L\right)e_{t}^{2}=\omega+\left[1-\beta\left(L\right)\right]\upsilon_{t}.\] (5)
+
+Motivated by the presence of apparent long memory in the autocorrelations of squared or absolute returns of several financial assets, Baillie _et al._[5] introduced the FIGARCH model defined as
+
+\[\Phi\left(L\right)\left(1-L\right)^{d}e_{t}^{2}=\omega+\left[1-\beta\left(L\right)\right]\upsilon_{t},\] (6)
+
+where \(0\leq d\leq 1\) is the fractional difference parameter. An interesting feature of this model is that it nests both the GARCH model for \(d=0\) and IGARCH for \(d=1\). Alternatively, for \(0<d<1\) the FIGARCH model implies a long memory behavior, _i.e._, a slow decay of the impact of a volatility shock. Also, we shall note that this type of processes is not covariance stationary but instead strictly stationary and ergodic for \(d\in\left[0,1\right].\)
+
+An alternative way to study stock market volatility is by applying concepts of physics which significant literature has already proven to be helpful in describing financial or economic problems. One measure that can be applied to describe the nonlinear dynamics of long memory and volatility clustering is the concept of entropy. This concept was originally introduced in 1865 by Clausius to explain the tendency of temperature, pressure, density and chemical gradients to flatten out and gradually disappear over time. Based on this Clausius developed the Second Law of Thermodynamics which postulates that the entropy of an isolated system tends to increase continuously until it reaches its equilibrium state. Although there are many different understandings of this concept the most commonly used in literature is as a measure of ignorance, disorder, uncertainty or even lack of information (see [9]). Later, in a subsequent investigation Shannon [10] provided a new insight on this matter showing that entropy wasn’t only restricted to thermodynamics but could be applied in any context where probabilities can be defined. In fact, thermodynamic entropy can be viewed as a special case of the Shannon entropy since it measures probabilities in the full state space. Based on the Hartley’s [11] formula, Shannon derived his entropy measure and established the foundations of information theory.
+
+For the probability distribution \(p_{i}\equiv p\left(X=i\right)\), \(\left(i=1,...,n\right)\) of a given random variable \(X,\) Shannon entropy \(S(X)\) for the discrete case, can be defined as
+
+\[S\left(X\right)=-\sum\limits_{i=1}^{n}p_{i}\ln p_{i},\] (7)
+
+with the conventions \(0\ln\left(0/z\right)=0\) for \(z\geq 0\) and \(z\ln\left(z/0\right)=\infty\).
+
+As a measure of uncertainty the properties of entropy are well established in literature (see [12]). For the non-trivial case where the probability of an event is less than one, the logarithm is negative and the entropy has a positive sign. If the system only generates one event, there is no uncertainty and the entropy is equal to zero. By the same token, as the number of likely events duplicates the entropy increases one unit. Similarly, it attains its maximum value when all likely events have the same probability of occurrence. On the other hand, the entropy of a continuous random variable may be negative. The scale of measurements sets an arbitrary zero corresponding to a uniform distribution over a unit volume. A distribution which is more confined than this has less entropy and will be negative.
+
+By replacing linear averaging in Shannon entropy with the Kolmogorov-Nagumo average or quasi-linear mean and further imposing the additivity constraints, Renyi [13] proposed the first formal generalization of the Shannon entropy. The need for a new information measure was due to the fact that there were a number of situations that couldn’t be explained by Shannon entropy. As Jizba and Arimitsu [14] pointed out Shannon’s information measure represents mere idealized information appearing only in situations when the storage capacity of a transmitting channel is finite.
+
+Using this formalism Renyi [13] developed his information measure, known as Renyi entropy or Renyi information measure of \(\alpha\) order, \(S_{\alpha}(X)\). For discrete variables it comes
+
+\[S_{\alpha}(X)=\frac{1}{1-\alpha}\ln\left(\sum\limits_{i=1}^{n}p_{k}^{\alpha}\right),\] (8)
+
+for \(\alpha>0\) and \(\alpha\neq 1\). In the limit \(\alpha\to 1\), Renyi entropy reduces to Shannon entropy and can be viewed as a special case of the latter. Additionally, evidence was found that Renyi’s entropies of order greater than 2 are related to search problems (see for example [15]). Even though this measure can be used in a variety of problems, empirical evidence has shown that it has a built-in predisposition to account for self-similar systems and, so, it naturally aspires to be an effective tool to describe equilibrium and non-equilibrium phase transitions (see [16] and [14]). Despite its relevance, Renyi entropy didn’t experience the same success of its predecessor’s which can be explained basically by two factors: ambiguous renormalization of Renyi’s entropy for non-discrete distributions and little insight into the meaning of Renyi’s \(\alpha\) index (see [16]). A new insight into this matter was brought by Csiszár [17], who has identified the \(\alpha\) index as the \(\beta\) cutoff rate for hypothesis testing problems.
+
+With the aim of studying physical systems that entail long-range interaction, long-term memories and multi-fractal structures, Tsallis [18] derived a new generalized form of entropy, known as Tsallis entropy. Although this measure was first introduced by Havrda and Charvát [19] in cybernetics and late improved by Daróczy [20], it was Tsallis [18] who really developed it in the context of physical statistics and, therefore, it is also known as Havrda-Charvát-Daróczy-Tsallis entropy.
+
+For any nonnegative real number \(q\) and considering the probability distribution \(p_{i}\equiv p\)\((X=i)\), \(i=1,...,n\) of a given random variable \(X,\) Tsallis entropy denoted by \(S_{q}\left(X\right)\), is defined as
+
+\[S_{q}\left(X\right)=\frac{1-\sum\limits_{i=1}^{n}p_{i}^{q}}{q-1}.\] (9)
+
+As \(q\to 1,\)\(S_{q}\) recovers \(S_{q}\left(X\right)\) because the \(q\)-logarithm uniformly converges to a natural logarithm as \(q\to 1\). This index may be thought as a biasing parameter since \(q<1\) privileges rare events and \(q>1\) privileges common events (see [21]). A concrete consequence of this is that while Shannon entropy yields exponential equilibrium distributions, Tsallis entropy yields power-law distributions. As Tatsuaki and Takeshi [22] have already pointed out the index \(q\) plays a similar role as the light velocity \(c\) in special relativity or Planck’s constant \(\hbar\) in quantum mechanics in the sense of a one-parameter extension of classical mechanics, but unlike \(c\) or \(\hbar\), \(q\) does not seem to be a universal constant. Further, we shall also mention that for applications of finite variance \(q\) must lie within the range \(1\leq q<5/3\).
+
+## 2 Empirical evidence
+
+This section examines the results obtained from both perspectives. In order to compare the volatility of the US and European stock market returns we have collected data from SP 500, NASDAQ 100 and Stoxx 50 indexes and constituted a sample spanning over the period June 2002-January 2007. The values were gathered on a daily basis without considering the re-investment of dividends. Based on them, we computed the stock market returns given by the log-ratio of the index values at time \(t\) and time \(t-1\) and performed the estimates.
+
+Within the traditional approach we have considered the GARCH \(\left(1,1\right)\), IGARCH \(\left(1,1\right)\) and FIGARCH \(\left(1,d,1\right)\) specifications, whose main results are listed in Table 1. The conclusions are similar to all the three indexes considered. Specifically, for the GARCH \(\left(1,1\right)\) it was found evidence of heteroscedastic conditional variance. Also, the fact that \(\alpha+\beta\simeq 1\) could denounce the presence of nonlinear persistence in the log-returns of the stock market indexes led us to estimate the IGARCH \((1,1)\) model. However, evidence has shown that many coefficients were not statistically significant. Then, the next step was to adjust the FIGARCH specification \((1,d,1)\) with the restriction \(d\neq 1\) whose main results corroborate the long memory hypothesis.
+
+\begin{table}
+\begin{tabular}{c l c c c}
+\hline \hline
+Coef. & \multicolumn{1}{c}{Indexes} & GARCH & IGARCH & FIGARCH \\
+\hline
+ & Stoxx 50 & \(1.12E-06\)* & \(0.014544\) & \(6.573658\)** \\
+\(\omega\) & SP 500 & \(3.19E-07\)** & \(0.003906\)* & \(0.009144\) \\
+ & NASDAQ 100 & \(5.74e-07\)* & \(0.004543\) & \(8.749987\)** \\
+\hline
+ & Stoxx 50 & \(0.076581\)** & \(0.104788\)** & \(-0.119727\)** \\
+\(\alpha\) & SP 500 & \(0.051592\)** & \(0.057835\)** & \(-0.118990\) \\
+ & NASDAQ 100 & \(0.040982\)** & \(0.044868\)** & \(-0.037189\)** \\
+\hline
+ & Stoxx 50 & \(0.913627\)** & \(0.895212\) & \(0.643566\)** \\
+\(\beta\) & SP 500 & \(0.946445\)** & \(0.942165\) & \(0.514637\)* \\
+ & NASDAQ 100 & \(0.957592\)** & \(0.955132\) & \(0.700161\)** \\
+\hline
+ & Stoxx 50 & - & - & \(0.695285\)** \\
+\(d\) & SP 500 & - & - & \(0.579649\)** \\
+ & NASDAQ 100 & - & - & \(0.655988\)** \\
+\hline
+ & Stoxx 50 & \(10.15165\)** & \(25.044983\) & \(21.850493\)* \\
+Student & SP 500 & \(6.152842\)** & \(5.374068\)** & \(1219.021742\)** \\
+ & NASDAQ 100 & \(11.42494\)** & \(11.030880\)** & \(335.539275\)** \\
+\hline
+ & Stoxx 50 & \(3812.074\) & \(1290.958\) & \(1296.348\) \\
+Log-L & SP 500 & \(13891.33\) & \(13039.1\) & \(1482.22\) \\
+ & NASDAQ 100 & \(10775.58\) & \(10024.569\) & \(1248.274\) \\ \hline \hline
+\end{tabular}
+\end{table}
+Table 1: GARCH, IGARCH and FIGARCH models for Stoxx 50, SP 500 and NASDAQ 100 indexes; ** denotes significance at the 1% level, * denotes significance at the 5% level
+
+In the domain of the econophysics approach we have computed the Shannon, Renyi and Tsallis entropies which are depicted in Table 2.
+
+\begin{table}
+\begin{tabular}{c c c c c}
+\hline \hline
+Entropies & Index (\(\alpha\)/\(q\)) & Stoxx & SP 500 & NASDAQ 100 \\
+\hline
+Shannon & - & \(3.3624\) & \(3.3784\) & \(3.2981\) \\
+\hline
+ & \(1.4\) & \(10.2076\) & \(10.217\) & \(10.2085\) \\
+Renyi & \(1.45\) & \(10.2065\) & \(10.2163\) & \(10.2074\) \\
+ & \(1.5\) & \(10.2054\) & \(10.2155\) & \(10.2064\) \\
+\hline
+ & \(1.4\) & \(1.8354\) & \(1.8395\) & \(1.8102\) \\
+Tsallis & \(1.45\) & \(1.7204\) & \(1.7238\) & \(1.6981\) \\
+ & \(1.5\) & \(1.6161\) & \(1.619\) & \(1.5965\) \\ \hline \hline
+\end{tabular}
+\end{table}
+Table 2: Shannon, Renyi and Tsallis entropies
+
+All entropies were estimated with histograms based on equidistant cells. For the calculation of Tsallis entropy we set values at \(1.4\), \(1.45\) and \(1.5\) for the index \(q\), which is consistent with the finding that when considering financial data their values lie within the range \(q\simeq 1.4-1.5\) (see [21]). The same assumption was made for the Renyi’s index. Since all entropies are positive we shall conclude that the data show nonlinearities. This phenomenon is particularly evident for the SP 500 index, which always attained the highest levels regardless of the method applied in its calculation. As for the others the results are not conclusive since they vary according to the entropy method adopted.
+
+## 3 Conclusions
+
+In this paper we have investigated the properties of the realized volatility for the SP 500, NASDAQ 100 and Stoxx 50 indexes. Our main goal was to compare two different perspectives: the so-called traditional approach in which we have considered the GARCH \(\left(1,1\right)\), IGARCH \(\left(1,1\right)\) and FIGARCH \(\left(1,d,1\right)\) specifications and the econophysics approach based on the concept of entropy. For our purpose three variants of this notion were chosen: the Shannon, Renyi and Tsallis measures. The results from both perspectives have shown nonlinear dynamics in the volatility of SP 500, NASDAQ 100 and Stoxx 50 indexes and must be understood in complementarity.
+
+We consider that the concept of entropy can be of great help when analyzing stock market returns since it can capture the uncertainty and disorder of the time series without imposing any constraints on the theoretical probability distribution. By contrast, the ARCH/GARCH type models assume that all variables are independent and identically distributed (_i.i.d_). However, in order to capture global serial dependence one should use a specific measure such as, for example, mutual information. By analyzing the entropy values for different equally spaced sub-periods we could have a clearer idea about the extent of volatility clustering and long-memory effects, an issue that will be pursued in further work.
+
+## References
+
+* [1]R. Engle, Econometrica 50 \(\left({\small 1982}\right)\) 987-1008.
+* [2]T. Bollerslev, Journal of Econometrics 31 \(\left({\small 1986}\right)\) 307-327.
+* [3]S.J. Taylor, John Wiley and Sons, Chichester \(\left({\small 1986}\right)\).
+* [4]R. Engle, T.P. Bollerslev, Econometric Reviews 5 \(\left({\small 1986}\right)\) 1-50.
+* [5]R.T. Baillie, Bollerslev, T., Mikkelsen, H. Journal of Econometrics 74 \(\left({\small 1996}\right)\) 3-30.
+* [6]S. Bordignon, M. Caporin, F. Lisi, Computational Statistics and Data Analysis 51 \(\left({\small 2007}\right)\)5900-5912.
+* [7]H.E. Stanley, L.A.N Amaral, D. Canning, P. Gopikrishnan, Y. Lee and Y. Liu, Physica A 269 \(\left({\small 1999}\right)\) 156-159.
+* [8]V. Plerou, P. Gopikrishnan, B. Rosenow. L.A.N Amaral, H.E. Stanley, Physica A 279 \(\left(2000\right)\) 443-456.
+* [9]A. Golan, Journal of Econometrics 107 \(\left({\small 2002}\right){\small\ }\)1-15.
+* [10]C.E. Shannon, The Bell System Technical Journal 27 \(\left({\small 1948}\right)\) 379-423, 623-656.
+* [11]R.V.L. Hartley, Bell System Technical Journal 7 3 \(\left({\small 1928}\right)\) 535-563.
+* [12]C.E. Shannon, W. Weaver, The University of Illinois Press, Urbana \(\left({\small 1964}\right)\).
+* [13]A. Renyi, Proceedings of the Fourth Berkeley Symposium on Mathematics, statistics and Probability 1 \(\left({\small 1961}\right)\) 547-561.
+* [14]T. Jizba, T. Arimitsu, Physica A 340 \(\left({\small 2004b}\right)\) 110-116.
+* [15]C.E. Pfister., W.G. Sullivan, IEEE Trans. Inform. Theory 50 (11) \(\left({\small 2004}\right)\) 2794-2800.
+* [16]T. Jizba, T. Arimitsu, Annals of Physics 312 \(\left({\small 2004a}\right)\) 17-59.
+* [17]Csiszár, I., IEEE Transaction of Information Theory 41 \(\left({\small 1995}\right){\small\ }\)26-34.
+* [18]C. Tsallis, Journal of Statistical Physics 1/2 \(\left({\small 1988}\right)\) 479-487.
+* [19]J. Havdra, F. Chárvat, Kybernetica 3 \(\left({\small 1967}\right)\) 50-57.
+* [20]DZ. Daróczy, Information and Control 16 \(\left({\small 1970}\right)\) 36-51
+* [21]C. Tsallis, C. Anteneodo, L. Borland, R. Osorio, Physica A 324 \(\left({\small 2003}\right)\) 89-100.
+* [22]W. Tatsuaki, S. Takeshi, Physica A 301 \(\left({\small 2001}\right)\) 284-290.
+

arxiv_ground_truth/0709.2857.md

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+# Chern numbers and diffeomorphism types of projective varieties
+
+D. Kotschick
+
+Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany
+dieter@member.ams.org
+
+(Date: December 20, 2007; MSC 2000: primary 57R20, secondary 14J30, 14J35, 32Q55)
+
+###### Abstract.
+
+In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.
+
+_Herrn Prof. Dr. F. Hirzebruch zum 80. Geburtstag gewidmet._
+
+## 1. Introduction
+
+More than fifty years ago, Hirzebruch raised the question to what extent the Chern and Hodge numbers of projective algebraic manifolds are topologically invariant, see Problem 31 in [2]. He noted that Chern numbers of almost complex manifolds are not topologically invariant simply because there are too many almost complex structures, even on a fixed manifold. Then, in 1959, Borel and Hirzebruch gave an example of a \(10\)-dimensional closed oriented manifold with two projective algebraic structures for which \(c_{1}^{5}\) are different, see [1] Section 24.11. Until 1987, when the commentary in [4] was written, nothing further was discovered concerning this question. In particular, Hirzebruch wrote then that he did not know whether \(c_{1}^{3}\) and \(c_{1}^{4}\) are topological invariants of complex three- and four-folds respectively.
+
+In this paper we prove that in complex dimension \(3\) the only linear combinations of the Chern numbers \(c_{1}^{3}\), \(c_{1}c_{2}\) and \(c_{3}\) that are invariant under orientation-preserving diffeomorphims of simply connected projective algebraic manifolds are the multiples of the Euler characteristic \(c_{3}\). In dimension \(4\) the only linear combinations of Chern numbers that are invariant are the linear combinations of the Euler characteristic \(c_{4}\) and of the Pontryagin numbers \(p_{1}^{2}\) and \(p_{2}\). We also prove some partial results in arbitrary dimensions.
+
+These results stem from the fact that in complex dimension \(2\) the Chern number \(c_{1}^{2}\) is not invariant under orientation-reversing homeomorphisms; cf. [5]. By suitable stabilisation of the counterexamples, we find enough examples at least in dimensions \(3\) and \(4\) to detect the independent variation of all Chern numbers which are not combinations of the Euler and Pontryagin numbers. Our results suggest a weaker form of Hirzebruch’s problem, asking whether the topology determines the Chern numbers up to finite ambiguity. We have no counterexample to an affirmative answer to this weaker question in the projective algebraic case, although it is known to be false for non-Kähler complex manifolds; cf. [7].
+
+## 2. Preliminary results
+
+### Complex surfaces
+
+For complex surfaces there are two Chern numbers, \(c_{2}\) and \(c_{1}^{2}\), which turn out to be diffeomorphism invariants even without assuming that the diffeomorphism is orientation-preserving with respect to the orientations given by the complex structures:
+
+**Theorem 1****.**: _If two compact complex surfaces are diffeomorphic, then their Chern numbers coincide._
+
+Proof.: In this case \(c_{2}\) is the topological Euler characteristic \(e\). By Wu’s formula we have
+
+(1) \[c_{1}^{2}(X)=2e(X)+p_{1}(X)\ .\]
+
+The first Pontryagin number is \(3\) times the signature, and so the right-hand side of (1) is invariant not just under orientation-preserving diffeomorphisms, but even under orientation-preserving homotopy equivalences¹.
+
+Footnote 1: All this was known in 1954, and Hirzebruch [2] remarked that the Chern numbers of an algebraic surface are topological invariants (of the underlying oriented manifold).
+
+Now suppose that two compact complex surfaces are orientation-reversing diffeomorphic, with respect to the orientations defined by their complex structures. Then, using Seiberg–Witten theory, I proved in 1995, see Theorem 2 of [6], that the signatures of these surfaces vanish. Thus, their Chern numbers agree by (1). ∎
+
+The statement about orientation-reversing diffeomorphisms concerns projective algebraic surfaces only, because, by the classification of complex surfaces, a complex surface with positive signature is always projective.
+
+As we saw in the proof, \(c_{1}^{2}\) is invariant under orientation-preserving homotopy equivalences, and under orientation-reversing diffeomorphisms. But it is not invariant under orientation-reversing homeomorphisms. Already in 1991, I had proved the following:
+
+**Theorem 2******([5])**.**: _There are infinitely many pairs of simply connected projective algebraic surfaces \(X_{i}\) and \(Y_{i}\) of non-zero signature which are orientation-reversing homeomorphic._
+
+The proof is based on geography results for surfaces of general type due to Persson and Chen. The surfaces \(X_{i}\) and \(Y_{i}\) are projective algebraic because they are of general type. They can be chosen to contain embedded holomorphic spheres, in which case they can not be orientation-reversing diffeomorphic, although they are orientation-reversing homeomorphic. This was the motivation for the results of [6] quoted in the proof of Theorem 1.
+
+By Wu’s formula (1) the homeomorphic surfaces \(X_{i}\) and \(Y_{i}\) have different \(c_{1}^{2}\). Indeed, the homeomorphism in question preserves the Euler number and reverses the sign of the signature, so that (1) gives:
+
+(2) \[c_{1}^{2}(Y_{i})=4e(X_{i})-c_{1}^{2}(X_{i})\ .\]
+
+Wu’s formula (1) shows in particular that the unoriented homeomorphism type almost determines the Chern numbers of a compact complex surface: there are only two possible values for \(c_{1}^{2}\) (and only one for \(c_{2}\), of course).
+
+We shall use the examples from Theorem 2 as building blocks for our high-dimensional examples.
+
+### Inductive formulae for Chern classes
+
+We require the following easy calculation.
+
+**Lemma 1****.**: _Let \(A\) be a compact complex \(n\)-fold, and \(B=A\times\mathbb{C}P^{1}\). Then the Chern numbers of \(B\) are_
+
+(3) \[c_{r_{1}}\ldots c_{r_{k}}(B)=2\sum_{j=1}^{k}c_{r_{1}}\ldots c_{r_{j}-1}\ldots c_{r_{k}}(A)\ .\]
+
+Proof.: The Whitney sum formula \(c(TB)=c(TA)c(T\mathbb{C}P^{1})\) for the total Chern classes implies that, with respect to the Künneth decomposition of the cohomology of \(B\), the Chern classes of \(B\) are
+
+\[\begin{split}c_{1}(B)&=c_{1}(A)+c_{1}(\mathbb{C}P^{1})\\ c_{2}(B)&=c_{2}(A)+c_{1}(A)c_{1}(\mathbb{C}P^{1})\\ &\ldots\\ c_{n}(B)&=c_{n}(A)+c_{n-1}(A)c_{1}(\mathbb{C}P^{1})\\ c_{n+1}(B)&=c_{n}(A)c_{1}(\mathbb{C}P^{1})\ .\end{split}\]
+
+The claim follows using that the first Chern number of \(\mathbb{C}P^{1}\) equals \(2\). ∎
+
+We also need the following generalization of Lemma 1 to non-trivial \(\mathbb{C}P^{1}\)-bundles:
+
+**Lemma 2****.**: _Let \(B\) be a compact complex surface and \(E\longrightarrow B\) a holomorphic vector bundle of rank two. Then the projectivisation \(X=\mathbb{P}(E)\) has_
+
+(4) \[\begin{split}c_{3}(X)&=2c_{2}(B)\ ,\\ c_{1}c_{2}(X)&=2(c_{1}^{2}(B)+c_{2}(B))\ ,\\ c_{1}^{3}(X)&=6c_{1}^{2}(B)+2p_{1}(\mathbb{P}(E))\ .\end{split}\]
+
+Here \(p_{1}(\mathbb{P}(E))=c_{1}^{2}(E)-4c_{2}(E)\) is the first Pontryagin number for the group \(SO(3)=PU(2)\), which is the structure group of the sphere bundle \(X\longrightarrow B\). Notice that in the case that \(p_{1}(\mathbb{P}(E))=0\), the formulae reduce to those obtained for the trivial bundle.
+
+Proof.: The formulae for \(c_{3}\) and for \(c_{1}c_{2}\) are immediate from the multiplicativity of the topological Euler characteristic and of the Todd genus, recalling that the Todd genera in dimension \(2\) and \(3\) are \(\frac{1}{12}(c_{1}^{2}+c_{2})\), respectively \(\frac{1}{24}c_{1}c_{2}\). To compute \(c_{1}^{3}\) note that by the Leray-Hirsch theorem the cohomology ring of \(X\) is generated as a \(H^{*}(B)\)-module by a class \(y\in H^{2}(X)\) restricting as a generator to every fiber and satisfying the relation
+
+\[y^{2}+c_{1}(E)y+c_{2}(E)=0\ .\]
+
+Moreover, \(c_{1}(X)=c_{1}(B)+c_{1}(E)+2y\) because the vertical tangent bundle has first Chern class \(c_{1}(E)+2y\). The third power is computed straightforwardly using the relation and the fact that \(y\) evaluates to \(1\) on the fiber. ∎
+
+## 3. Complex three-folds
+
+A variant of Hirzebruch’s problem for three-folds was taken up by LeBrun in 1998, see [7], who proved that there are closed \(6\)-manifolds which admit complex structures with different \(c_{1}c_{2}\) and \(c_{1}^{3}\). He even proved that a fixed manifold can have complex structures realising infinitely many different values for \(c_{1}c_{2}\). However, for all the examples discussed in [7] only one of the complex structures is projective algebraic, or at least Kähler, and all the others are non-Kähler. Therefore, these examples say nothing about the topological invariance of Chern numbers for projective algebraic three-folds.
+
+Nevertheless, both \(c_{1}c_{2}\) and \(c_{1}^{3}\) are not diffeomorphism invariants of projective three-folds:
+
+**Proposition 1****.**: _There are infinitely many pairs of projective algebraic three-folds \(Z_{i}\) and \(T_{i}\) with the following properties:_
+
+1. (i)_Each_ \(Z_{i}\) _and_ \(T_{i}\) _admits an orientation-reversing diffeomorphism._
+2. (ii)_For each_ \(i\) _the manifolds underlying_ \(Z_{i}\) _and_ \(T_{i}\) _are diffeomorphic._
+3. (iii)_For each_ \(i\) _one has_ \(c_{1}c_{2}(Z_{i})\neq c_{1}c_{2}(T_{i})\) _and_ \(c_{1}^{3}(Z_{i})\neq c_{1}^{3}(T_{i})\)_._
+
+Proof.: Let \(X_{i}\) and \(Y_{i}\) be the algebraic surfaces from Theorem 2, constructed in [5], and take \(Z_{i}=X_{i}\times\mathbb{C}P^{1}\) and \(T_{i}=Y_{i}\times\mathbb{C}P^{1}\). Then the identity on the first factor times complex conjugation on the second factor gives an orientation-reversing selfdiffeomorphism of \(Z_{i}\) and of \(T_{i}\).
+
+Denote by \(\bar{Y}_{i}\) the smooth manifold underlying \(Y_{i}\), but endowed with the orientation opposite to the one induced by the complex structure. Then \(X_{i}\) and \(\bar{Y}_{i}\) are orientation-preserving homeomorphic simply connected smooth four-manifolds, and are therefore h-cobordant. If \(W\) is an h-cobordism between them, then \(W\times S^{2}\) is an h-cobordism between \(Z_{i}\) and \(\bar{T}_{i}=\bar{Y}_{i}\times\mathbb{C}P^{1}\). By Smale’s h-cobordism theorem, \(Z_{i}\) and \(\bar{T}_{i}\) are orientation-preserving diffeomorphic. As \(Z_{i}\) and \(T_{i}\) admit orientation-reversing diffeomorphisms, we conclude that they are both orientation-preserving and orientation-reversing diffeomorphic.
+
+For the Chern numbers (3) gives
+
+(5) \[\begin{split}c_{1}c_{2}(Z_{i})&=2(c_{1}^{2}+c_{2})(X_{i})\ ,\\ c_{1}^{3}(Z_{i})&=6c_{1}^{2}(X_{i})\ ,\end{split}\]
+
+and similarly for \(T_{i}\) and \(Y_{i}\). As \(X_{i}\) and \(Y_{i}\) have the same \(c_{2}\) but different \(c_{1}^{2}\), we conclude that \(Z_{i}\) and \(T_{i}\) have different \(c_{1}c_{2}\) and different \(c_{1}^{3}\). ∎
+
+Thus \(c_{1}c_{2}\) and \(c_{1}^{3}\) are not topological invariants of projective three-folds, but it is not yet clear that they vary independently. This is the content of the following:
+
+**Theorem 3****.**: _The only linear combinations of the Chern numbers \(c_{1}^{3}\), \(c_{1}c_{2}\) and \(c_{3}\) that are invariant under orientation-preserving diffeomorphisms of simply connected projective algebraic three-folds are the multiples of the Euler characteristic \(c_{3}\)._
+
+Proof.: First of all, let us dispose of the orientation question. If two complex three-folds are orientation-reversing diffeomorphic with respect to the orientations given by their complex structures, then they become orientation-preserving diffeomorphic after we replace one of the complex structures by its complex conjugate. As the conjugate complex structure has the same Chern numbers as the original one, we do not have to distinguish between orientation-preserving and orientation-reversing diffeomorphisms.
+
+All the examples constructed in the proof of Proposition 1 have the property that \(3c_{1}c_{2}-c_{1}^{3}\) agrees on \(Z_{i}\) and \(T_{i}\), as follows by combining (2) with (5) and the topological invariance of \(c_{2}\) for surfaces. But, by Proposition 1, linear combinations of \(3c_{1}c_{2}-c_{1}^{3}\) and of \(c_{3}\) are the only candidates left for combinations of Chern numbers that can be topological invariants of projective three-folds. In order to show that \(3c_{1}c_{2}-c_{1}^{3}\) is not an oriented diffeomorphism invariant we shall use certain ruled manifolds which are non-trivial \(\mathbb{C}P^{1}\)-bundles, rather than the products used above.
+
+Consider again a pair \(X_{i}\) and \(Y_{i}\) of simply connected algebraic surfaces as in Theorem 2. For simplicity we just denote them by \(X\) and \(Y\), with orientations implicitly given by the complex structures. The oriented manifolds \(X\) and \(\bar{Y}\) are orientation-preserving h-cobordant. Let \(M\) be the projectivisation of the holomorphic tangent bundle \(TY\) of \(Y\). Temporarily ignoring the complex structure of \(M\), we think of it as a smooth oriented two-sphere bundle over \(Y\), or over \(\bar{Y}\). If \(W\) is any h-cobordism between \(X\) and \(\bar{Y}\), then the two-sphere bundle \(M\longrightarrow\bar{Y}\) extends to a uniquely defined oriented two-sphere bundle \(V\longrightarrow W\). Let \(N\) be the restriction of this bundle to \(X\subset W\). If we give \(N\) the orientation induced from that of \(X\) and \(M\) the orientation induced from that of \(\bar{Y}\), then, by construction, \(V\) is an h-cobordism between \(N\) and \(M\). By Smale’s h-cobordism theorem, \(M\) and \(N\) are diffeomorphic.
+
+Because the bundle \(p\colon M\longrightarrow Y\) was defined as the projectivisation of the holomorphic tangent bundle of \(Y\), its characteristic classes are \(w_{2}(p)=w_{2}(Y)\) and \(p_{1}(p)=c_{1}^{2}(Y)-4c_{2}(Y)\). Considered as a bundle over \(\bar{Y}\), \(p\) has the same Stiefel–Whitney class, but the first Pontryagin number changes sign. It follows that \(q\colon N\longrightarrow X\) has
+
+\[p_{1}(q)=-c_{1}^{2}(Y)+4c_{2}(Y)=c_{1}^{2}(X)\ ,\]
+
+where the last equality is from (2). Moreover, \(w_{2}(q)=w_{2}(X)\), although \(X\) and \(\bar{Y}\) are not diffeomorphic. This follows for example from the cohomological characterisation of \(w_{2}(X)\) as the unique element of \(H^{2}(X;\mathbb{Z}_{2})\) which for all \(x\) satisfies
+
+\[w_{2}(X)\cdot x\equiv x^{2}\pmod{2}\ .\]
+
+The bundle \(q\) is determined by \(w_{2}(q)=w_{2}(X)\) and \(p_{1}(q)=c_{1}^{2}(X)\), and so we can think of it as the projectivisation of the holomorphic rank two bundle \(\mathcal{O}(K)\oplus\mathcal{O}\longrightarrow X\). Therefore the total space \(N\) inherits a complex-algebraic structure from that of \(X\). Its Chern numbers are given by (4):
+
+(6) \[\begin{split}c_{3}(N)&=2c_{2}(X)\ ,\\ c_{1}c_{2}(N)&=2(c_{1}^{2}(X)+c_{2}(X))\ ,\\ c_{1}^{3}(N)&=8c_{1}^{2}(X)\ .\end{split}\]
+
+This \(N\) is diffeomorphic to \(M\), which has a complex-algebraic structure as the projectivisation of the holomorphic tangent bundle of \(Y\). (Recall from the beginning of the proof that we do not have to keep track of the orientations induced by complex structures, because we can always replace a structure by its complex conjugate.) The Chern numbers of \(M\) are also given by (4):
+
+(7) \[\begin{split}c_{3}(M)&=2c_{2}(Y)=2c_{2}(X)\ ,\\ c_{1}c_{2}(M)&=2(c_{1}^{2}(Y)+c_{2}(Y))=2(-c_{1}^{2}(X)+5c_{2}(X))\ ,\\ c_{1}^{3}(M)&=8c_{1}^{2}(Y)-8c_{2}(Y)=8(-c_{1}^{2}(X)+3c_{2}(X))\ ,\end{split}\]
+
+using (2) to replace the Chern numbers of \(Y\) by combinations of those of \(X\). Unlike for the examples in Proposition 1, the combination \(3c_{1}c_{2}-c_{1}^{3}\) is not the same for \(M\) and \(N\). This finally shows that \(c_{1}c_{2}\) and \(c_{1}^{3}\) vary independently (within a fixed diffeomorphism type). ∎
+
+Although the Chern numbers of a projective three-fold are not determined by the underlying differentiable manifold, this may still be the case up to finite ambiguity. By the Hirzebruch–Riemann–Roch theorem one has
+
+(8) \[\frac{1}{24}c_{1}c_{2}=1-h^{1,0}+h^{2,0}-h^{3,0}\ ,\]
+
+so that in the Kähler case \(c_{1}c_{2}\) is bounded from above and from below by linear combinations of Betti numbers. In particular, for Kähler structures on a fixed \(6\)-manifold \(c_{1}c_{2}\) can take at most finitely many values. We are left with the following:
+
+**Problem 1****.**: _Does \(c_{1}^{3}\) take on only finitely many values on the projective algebraic structures with the same underlying \(6\)-manifold?_
+
+The issue here is that there is no Riemann–Roch type formula expressing \(c_{1}^{3}\) as a combination of Hodge numbers and the other Chern numbers.
+
+For three-folds with ample canonical bundle one has \(c_{1}^{3}<0\), and Yau’s celebrated work [11] gives \(c_{1}^{3}\geq\frac{8}{3}c_{1}c_{2}\). As \(c_{1}c_{2}\) is bounded below by a linear combination of Betti numbers, we have a positive answer to Problem 1 for this restricted class of projective three-folds. Even in the non-Kähler category, there are no examples where infinitely many values are known to arise for \(c_{1}^{3}\).
+
+## 4. Higher dimensions
+
+It is now very easy to show that, except for the Euler number, no Chern number is diffeomorphism-invariant:
+
+**Theorem 4****.**: _For projective algebraic \(n\)-folds with \(n\geq 3\) the only Chern number \(c_{I}\) which is diffeomorphism-invariant is the Euler number \(c_{n}\)._
+
+Note that by Theorem 1 this is false for \(n=2\), because in that case \(c_{1}^{2}\) is also diffeomorphism-invariant. On the other hand, by Theorem 2 it is not homeomorphism-invariant, so that Theorem 4 is true for \(n=2\) if we replace diffeomorphism-invariance by homeomorphism-invariance. As in the case of Proposition 1, the examples we exhibit in the proof of Theorem 4 admit orientation-reversing diffeomorphisms, so that one cannot restore diffeomorphism-invariance of \(c_{I}\neq c_{n}\) by restricting to orientation-preserving diffeomorphisms only.
+
+Proof.: For \(n=3\) this was already proved in Proposition 1. For \(n>3\) we take the examples \(T_{i}\) and \(Z_{i}\) from Proposition 1 and multiply them by \(n-3\) copies of \(\mathbb{C}P^{1}\). Call these products \(T_{i}^{\prime}\) and \(Z_{i}^{\prime}\). Using formula (3) and induction, we see that, on the one hand, \(c_{n}\) is a universal multiple of the \(c_{2}\) of the surfaces we started with. On the other hand, \(c_{1}^{n}(T_{i}^{\prime})\) and \(c_{1}^{n}(Z_{i}^{\prime})\) are universal multiples of \(c_{1}^{2}(X_{i})\) and of \(c_{1}^{2}(Y_{i})\) respectively, and so are different. All other Chern numbers \(c_{I}\) are universal linear combinations of \(c_{2}(X_{i})\) and \(c_{1}^{2}(X_{i})\), respectively \(c_{2}(Y_{i})\) and \(c_{1}^{2}(Y_{i})\), with the coefficients of both \(c_{2}\) and \(c_{1}^{2}\) strictly positive. As \(X_{i}\) and \(Y_{i}\) have the same \(c_{2}\) but different \(c_{1}^{2}\), the result follows. ∎
+
+Although the individual Chern numbers are not diffeomorphism-invariant, certain linear combinations are invariant once we restrict to orientation-preserving diffeomorphisms. Of course, as remarked by Hirzebruch [4], the Pontryagin numbers \(p_{J}\) have this invariance property², but this only helps when the complex dimension is even.
+
+Footnote 2: Note that, unlike the Euler number, the Pontryagin numbers change sign under a change of orientation.
+
+**Problem 2****.**: _Prove that, in arbitrary dimensions, the only combinations of Chern numbers that are invariant under orientation-preserving diffeomorphisms of smooth complex projective varieties are linear combinations of Euler and Pontryagin numbers._
+
+For complex dimension \(3\) this is Theorem 3 above. Theorem 5 below deals with the case of complex dimension \(4\).
+
+It would be interesting to know whether each of the Chern numbers \(c_{I}\neq c_{n}\) takes on only finitely many values on a fixed smooth manifold. In the Kähler case this is known to be true for \(c_{1}c_{n-1}\) by a result of Libgober and Wood, who showed that this Chern number is always a linear combination of Hodge numbers, see Theorem 3 in [8]. In the non-Kähler case \(c_{1}c_{n-1}\) can take on infinitely many values on a fixed manifold. This follows as in the proof of Theorem 4 by taking products of LeBrun’s examples [7] mentioned in the previous section with \(\mathbb{C}P^{1}\), and using formula (3). Because \(c_{1}c_{2}\) takes on infinitely many values on a fixed \(6\)-manifold, the same conclusion holds for \(c_{1}c_{n-1}\) in real dimension \(2n\geq 8\).
+
+Returning to the Kähler case, the Riemann–Roch theorem expresses the \(\chi_{p}\)-genus³
+
+Footnote 3: Our notation is consistent with [10], changing the traditional superscript in \(\chi^{p}\) from [3] to a subscript.
+
+\[\chi_{p}=\sum_{q=0}^{n}(-1)^{q}h^{p,q}\]
+
+as a linear combination of Chern numbers, and it follows that the combinations of Chern numbers which appear in this way can take on only finitely many values on a fixed manifold, as they are bounded above and below by linear combinations of Betti numbers.
+
+_Remark 1__._: If the complex dimension \(n\) is odd, then the Todd genus expressing the Euler characteristic \(\chi_{0}=(-1)^{n}\chi_{n}\) of the structure sheaf as a combination of Chern numbers does not involve \(c_{1}^{n}\). This follows from the _Bemerkungen_ in Section 1.7 of [3]. On the one hand, in any dimension, the coefficient of \(c_{n}\) in the Todd genus agrees with the coefficient of \(c_{1}^{n}\). On the other hand, the Todd genus is divisible by \(c_{1}\) if \(n\) is odd.
+
+Generalising this Remark, and what we saw for \(n=3\) in the previous section, we now prove:
+
+**Proposition 2****.**: _If \(M\) is Kähler of odd complex dimension \(n>1\), then all \(\chi_{p}\) are linear combinations of Chern numbers which do not involve \(c_{1}^{n}\)._
+
+This shows that for odd \(n\) there is no general way to extract the value of \(c_{1}^{n}\) from the Hodge numbers. In particular, one can not obtain a finiteness result for \(c_{1}^{n}\) in this way.
+
+Proof.: The Kähler symmetries imply \(\chi_{p}=(-1)^{n}\chi_{n-p}\), so that it is enough to prove the claim for \(p>\frac{n}{2}\). We shall do this by descending induction starting at \(p=n\).
+
+Salamon [10] proved that for \(2\leq k\leq n\) the number
+
+(9) \[\sum_{p=k}^{n}(-1)^{p}{p\choose k}\chi_{p}\]
+
+is a linear combination of Chern numbers each of which involves a \(c_{i}\) with \(i>n-2[\frac{k}{2}]\), see [10] Corollary 3.3. Using this for \(n\) odd and \(k=n\), we obtain once more the claim for \(\chi_{n}\) treated already in the Remark above.
+
+Suppose now that the claim has been proved for \(\chi_{n}\), \(\chi_{n-1}\), …, \(\chi_{j}\) with \(j>\frac{n}{2}+1\). Then we consider (9) with \(k=j-1\). (Note that this still satisfies \(k\geq 2\).) As \(\chi_{p}\) with \(p\geq j\) does not involve \(c_{1}^{n}\) by the induction hypothesis, Salamon’s result implies that \(\chi_{j-1}\) does not involve \(c_{1}^{n}\) either. ∎
+
+## 5. Four-folds
+
+In the case of four-folds, in addition to the Euler number \(c_{4}\), the following are invariants of the underlying oriented smooth manifold:
+
+(10) \[\begin{split}p_{1}^{2}&=(c_{1}^{2}-2c_{2})^{2}=c_{1}^{4}-4c_{1}^{2}c_{2}+4c_{2}^{2}\\ p_{2}&=c_{2}^{2}-2c_{1}c_{3}+2c_{4}\ .\end{split}\]
+
+The vector space of Chern numbers of four-folds is \(5\)-dimensional, containing the \(3\)-dimensional subspace spanned by \(c_{4}\), \(p_{1}^{2}\) and \(p_{2}\). It turns out that all combinations of Chern numbers that are invariant under orientation-preserving diffeomorphisms are contained in this subspace:
+
+**Theorem 5****.**: _The only linear combinations of Chern numbers that are invariant under orientation-preserving diffeomorphisms of simply connected projective algebraic four-folds are linear combinations of the Euler characteristic and of the Pontryagin numbers._
+
+Proof.: This is a rather formal consequence of our results for complex three-folds. Consider the vector space of Chern number triples \((c_{3},c_{2}c_{1},c_{1}^{3})\). Whenever we have a smooth six-manifold with two different complex structures, the difference of the two Chern vectors must be in the kernel of any linear functional corresponding to a topologically invariant combination of Chern numbers. In the proof of Theorem 3 we produced two kinds of examples for which these difference vectors were linearly independent. Therefore the space of topologically invariant combinations of Chern numbers is at most one-dimensional, and as it contains \(c_{3}\) it is precisely one-dimensional.
+
+Consider now the four-folds obtained by multiplying the three-dimensional examples by \(\mathbb{C}P^{1}\). If the difference of Chern vectors in a three-dimensional example is \((0,a,b)\), then by (3) the difference of Chern vectors \((c_{4},c_{1}c_{3},c_{2}^{2},c_{1}^{2}c_{2},c_{1}^{4})\) for the product with \(\mathbb{C}P^{1}\) is \((0,2a,4a,4a+2b,8b)\). Two examples in dimension three with linearly independent difference vectors lead to examples in dimension four which also have linearly independent difference vectors. Thus, in the five-dimensional space spanned by the Chern numbers of complex projective four-folds, the subspace invariant under orientation-preserving diffeomorphisms has codimension at least two. As it contains the linearly independent elements \(c_{4}\), \(p_{1}^{2}\) and \(p_{2}\), it is exactly three-dimensional. ∎
+
+Concerning the weaker question which Chern numbers of projective or Kähler four-folds are determined by the topology up to finite ambiguity, this is so for \(c_{1}c_{3}\) on general grounds, see the discussion above and [8, 10]. The formula for \(p_{2}\) then shows that \(c_{2}^{2}\) is also determined up to finite ambiguity. Using either the formula for \(p_{1}^{2}\) or the Riemann–Roch formula for the structure sheaf, we conclude that \(c_{1}^{4}-4c_{1}^{2}c_{2}\) is also determined up to finite ambiguity, but it is not clear whether this is true for \(c_{1}^{4}\) and \(c_{1}^{2}c_{2}\) individually. Note that a negative answer to Problem 1, giving infinitely many values for \(c_{1}^{3}\) on a fixed \(6\)-manifold, would show that \(c_{1}^{4}\) also takes on infinitely many values on a fixed \(8\)-manifold by taking products with \(\mathbb{C}P^{1}\).
+
+For four-folds with ample canonical bundle one has \(c_{1}^{4}>0\), and Yau’s work [11] gives \(c_{1}^{4}\leq\frac{5}{2}c_{1}^{2}c_{2}\). Therefore
+
+\[0<c_{1}^{4}\leq\frac{5}{3}(4c_{1}^{2}c_{2}-c_{1}^{4})\ .\]
+
+As the right-hand side takes on only finitely many values, the same is true for \(c_{1}^{4}\), and then for \(c_{1}^{2}c_{2}\) as well.
+
+_Remark 2__._: Pasquotto [9] recently raised the question of the topological invariance of Chern numbers of symplectic manifolds, particularly in (real) dimensions \(6\) and \(8\). Our results for Kähler manifolds of course show that Chern numbers of symplectic manifolds are not topological invariants. In the Kähler case we have used Hodge theory to argue that the variation of Chern numbers is quite restricted, often to finitely many possibilities. It would be interesting to know whether any finiteness results hold in the symplectic non-Kähler category.
+
+## References
+
+* [1] A. Borel and F. Hirzebruch, _Characteristic classes and homogeneous spaces, II_, Amer. J. Math. **81** (1959), 315–382.
+* [2] F. Hirzebruch, _Some problems on differentiable and complex manifolds_, Ann. Math. **60** (1954), 213–236.
+* [3] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, 2. ergänzte Auflage, Springer Verlag 1962.
+* [4] F. Hirzebruch, Gesammelte Abhandlungen, Band I, Springer-Verlag 1987.
+* [5] D. Kotschick, _Orientation–reversing homeomorphisms in surface geography_, Math. Annalen **292** (1992), 375–381.
+* [6] D. Kotschick, _Orientations and geometrisations of compact complex surfaces_, Bull. London Math. Soc. **29** (1997), 145–149.
+* [7] C. LeBrun, _Topology versus Chern numbers for complex \(3\)-folds_, Pacific J. Math. **191** (1999), 123–131.
+* [8] A. S. Libgober and J. W. Wood, _Uniqueness of the complex structure on Kähler manifolds of certain homotopy types_, J. Differential Geometry **32** (1990), 139–154.
+* [9] F. Pasquotto, _Symplectic geography in dimension \(8\)_, Manuscripta math. **116** (2005), 341–355.
+* [10] S. M. Salamon, _On the cohomology of Kähler and hyper-Kähler manifolds_, Topology **35** (1996), 137–155.
+* [11] S.-T. Yau, _Calabi’s conjecture and some new results in algebraic geometry_, Proc. Natl. Acad. Sci. USA **74** (1977), 1798–1799.
+

arxiv_ground_truth/0709.4466.md

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+# Serially Concatenated IRA Codes
+
+Taikun Cheng, Krishnamoorthy Sivakumar, and Benjamin J. Belzer
+
+The authors are with the School of Electrical Engineering and Computer Science, Washington State University, P.O. Box 642752, Pullman, WA 99164-2752, USA tcheng,belzer,siva@eecs.wsu.edu
+
+###### Abstract
+
+We address the error floor problem of low-density parity check (LDPC) codes on the binary-input additive white Gaussian noise (AWGN) channel, by constructing a serially concatenated code consisting of two systematic irregular repeat-accumulate (IRA) component codes connected by an interleaver. The interleaver is designed to prevent stopping-set error events in one of the IRA codes from propagating into stopping set events of the other code. Simulations with two 128-bit rate 0.707 IRA component codes show that the proposed architecture achieves a much lower error floor at higher SNRs, compared to a 16384-bit rate 1/2 IRA code, but incurs an SNR penalty of about 2 dB at low to medium SNRs. Experiments indicate that the SNR penalty can be reduced at larger blocklengths.
+
+## I Introduction
+
+LDPC codes, introduced by Gallager in the early 1960s [1], have received great interest since researchers in the late 1990s and early 2000s ([2, 3, 4]) showed that they can perform within less than 0.1dB of the Shannon limit for a number of important communication channels, including the binary erasure channel and the binary-input AWGN channel. However, for the above-cited codes, near-capacity performance typically holds only above bit error rates (BERs) of \(10^{-5}\) or \(10^{-6}\); at lower BERs, the nearly vertical (and highly negative) slope of the BER vs. SNR curve levels off into an “error floor” with a smaller magnitude slope.
+
+As there are several important applications that require BERs of \(10^{-12}\) or lower (e.g., mass storage, broadband satellite communications), a number of recent publications have proposed LDPCs specially designed to reduce the error floor. IRA codes, introduced in [5] by Jin, Khandekar, and McEliece, feature a section \(H_{2}\) of the parity check matrix \(H\) that contains only weight-two columns (except for one weight-1 column), and consists of “1”s down the main diagonal and the diagonal just below it. A lemma proved in [6] shows that if the \(H_{2}\) section contains all the weight two columns of \(H\), then it helps lower the error floor because \(H_{2}\) contains the maximum number of degree-two variable nodes without a cycle among them. Extended IRA (e-IRA) codes, introduced in [6], are a generalization of systematic IRA codes wherein the remaining section (“\(H_{1}\)”) of the \(H\) matrix assumes a more general form; design rules for lowering the error floor of e-IRA codes by optimizing the degree distributions of \(H_{1}\) are given in [6]. IRA codes and e-IRA codes have the low decoding complexity characteristic of LDPC codes, and the low encoding complexity characteristic of turbo codes [5, 6, 7].
+
+LDPC error floors are caused by connected sets of cycles called “stopping sets” [8]. Codes with larger stopping sets generally have lower error floors. The design technique in [9] attempts to maximize stopping set size by maximizing the average number of connections leading outside small cycles, referred to as the ACE distance \(d_{ACE}\); simulations showed that LDPC codes with larger \(d_{ACE}\) had lower error floors. More recently, the authors of [10] proposed a method of directly estimating the variable and check nodes in the smallest stopping sets, along with a code design algorithm to directly maximize the size of these sets. The design algorithm in [10] resulted in codes with significantly lower error floors than those designed according to [9].
+
+The contribution of the present paper is a method of designing serially concatenated IRA codes that achieve lower error floors than single IRA codes of equivalent rate and block size. Two systematic component codes, with block length and rate equal to the square roots of those of a comparable full-length IRA code, are connected in series, with an interleaver between them. This architecture is similar to that of turbo product codes [11], except that, rather than employing the row-column interleaver of product codes, we design the interleaver to avoid the convergence problems that lead to error floors. We use the method of [10] to estimate the stopping sets of the component codes. Then the stopping set data is used to design the interleaver so that, as much as possible, stopping set error events of one of the component codes are not mapped into stopping set variable nodes of the other code. Since each component code has the ability to successfully decode the other code’s non-convergent blocks, convergence problems are greatly reduced, resulting in a lowered error floor at high SNR. Because of the IRA component codes, the concatenated system has relatively low encoding complexity compared to a general irregular LDPC code. The decoding complexity is about twice that of the comparable full-length IRA code, due to the need for outer iterations between the component codes.
+
+This paper is organized as follows. Section II summarizes the encoder and decoder architectures. Section III presents the interleaver design. Section IV presents simulation results, and section V concludes the paper.
+
+## II Concatenated IRA Encoder and Decoder
+
+Figure 1: Block diagram of the concatenated encoder with systematic IRA component codes connected by interleaver (denoted by \(\pi\)).
+
+A block diagram of the concatenated encoder is shown in Fig. 1. It consists of two systematic IRA component codes connected by an interleaver (denoted by \(\pi\)). In the following discussion, we visualize the concatenated system as a product code, with the two encoders operating on rows and columns. The source data is arranged in a two-dimensional block of size \(K\times K\). The rows of the source block are first encoded with the outer \([N,K]\) systematic IRA code, yielding a \(K\times N\) coded block in which the first \(K\) elements of each row are systematic bits. Then the \(K\times N\) coded block is passed through the interleaver. The purpose of the interleaver is to minimize the intersection between the stopping set error events of the row and column component codes. After the interleaver, each \(K\)-bit column is encoded with the inner \([N,K]\) systematic IRA code, producing an \(N\times N\) codeword block. The overall code rate is \(R=K^{2}/N^{2}\). The identical variable-node degree distributions of the two component codes are chosen to optimize their performances in the waterfall region according to the design algorithm given in [5], subject to the constraint that all weight-2 columns appear in the \(H_{2}\) section of the parity check matrix; the constraint helps lower the error floors of the component codes. All example codes designed in this paper used a fixed check node degree of 10. The variable-to-check node connections in the component codes are optimized using the ACE algorithm of [9], in order to further lower the error floors. In our examples, the variable-to-check node connections in the component codes are different, so that the codes have different stopping sets; however, the interleaver design described in section III also works if the component codes are identical.
+
+The decoder for the concatenated system employs iterative message passing between the decoders for the two component codes. The decoder block diagram is shown in Fig. 2. It consists of column and row decoders connected by the interleaver and de-interleaver. The received channel data is decoded column by column by a standard \([N,K]\) IRA decoder employing the sum-product algorithm (SPA, [12]) on the code’s Tanner graph; the column decoder uses the extrinsic information from the row decoder as _a priori_ information. The column decoder outputs a \(K\times N\) block of extrinsic information LLRs. The column decoder’s output extrinsic information is then passed through the interleaver and used as prior information by the row decoder. The row decoder makes use of the de-interleaved channel information and the prior information to decode the data row by row, and outputs a \(K\times N\) block of LLRs to be used for final decoding decisions, along with a \(K\times N\) block of extrinsic LLRs for the column decoder to use during the next iteration.
+
+Figure 2: Block diagram of the concatenated decoder.
+
+## III Interleaver Design
+
+The reasons to encode/decode using the structure described above rather than using a single \([N^{2},K^{2}]\) IRA code are as follows. The performance of an LDPC code at high SNR (i.e., in the error floor region) is not determined by the code’s minimum distance, but rather by sets of interconnected short cycles (called stopping sets) that prevent the decoder from converging to a valid codeword. If we can design the interleaver to prevent the mapping of stopping set error events from one of the component codes into stopping set nodes of the other code, then the concatenated structure will help improve the performance at high SNR.
+
+The definition of a stopping set used in this paper is as follows. A variable-node set is called a stopping set if all its neighbors are connected to this set at least twice [9]. In LDPC codes at high SNR, error events occur on the smallest stopping sets with higher probability than on larger stopping sets or non-stopping sets. To simplify, if a variable node is a part of a stopping set, we call it a sensitive node.
+
+Here is an example of how an error event from one IRA component code could propagate into the other one. Suppose variable nodes \((6,9,25)\) are sensitive nodes of the column component code and that errors occur on these positions. Since each column uses the same component code, errors will occur on these positions on most columns, i.e., at the end of column decoding, most positions of rows \((6,9,25)\) are errors. If we do nothing but directly input these rows to the row decoder, the outputs will have a large number of errors (perhaps even larger then the number of input errors) due of the bad prior information. If we pass the output extrinsic information from the column decoder through an interleaver before it is fed to the row decoder, the errors will not be concentrated on rows \((6,9,25)\) and hence can be corrected more easily. Therefore, we postulate two interleaver design rules for the concatenated system:
+
+1. 1.Spread concentrated errors all over the data block.
+2. 2.Avoid mapping the sensitive nodes of the row (column) component code into the sensitive nodes of the column (row) component code.
+
+The sensitive positions of a component code can be determined by employing the stopping set detection algorithm of [10]. For a given starting variable node, the algorithm in [10] finds a stopping set containing that node, but does not guarantee that the detected set is minimal; thus, some relatively less-sensitive nodes may be included in the set. To find the most sensitive nodes, we repeatedly run the detection algorithm by starting from every variable node in the code, and count the accumulated times each node appears in a stopping set; the higher the count, the more sensitive the node. Fig. 3 shows the results of running the algorithm of [10] over the \([181,128]\) row component IRA code by starting from each variable node. In Fig. 3, the maximum sensitivity count is \(181\). It is clear from the figure that some nodes are highly sensitive, and that most of the parity bits (bits 129-181) have high sensitivity counts.
+
+Figure 3: Sensitivity measurement via stopping set detection. The sensitivity counts on the vertical axis are accumulated by running the algorithm of [10] on every possible starting variable node, and then counting the number of times any given node appears in the detected stopping sets.
+
+Based on the above design rules, we design the interleaver by starting with a random interleaver and imposing additional constraints. First, a relatively good \(K\times N\) random interleaver is found by simulation. Then the stopping sets of the row and column component codes are detected using the method of [10]. For given sensitive nodes \(i\) and \(j\) of the row/column component codes \(i\in\{I_{0},I_{1},\cdots,I_{n}\}\) and \(j\in\{J_{0},J_{1},\cdots,J_{m}\}\), where \(\{I_{0},I_{1},\cdots,I_{n}\}\) and \(\{J_{0},J_{1},\cdots,J_{m}\}\) are the sensitive nodes of the row and column component codes respectively, we modify the random interleaver so that no element in the \(j\)th row before passing through the interleaver is located in the \(i\)th column after passing through the interleaver. If the random interleaver maps any element in row \(j\) to column \(i\) (the “bad mapping” condition), then that element is re-mapped to a random position in the output block, and the element formerly at that random position is mapped into the position of the element in row \(j\); this re-mapping continues until either no bad mappings are found or all the possible positions in the interleaver have been checked, in which case no interleaver solution is possible. Since the stopping set detection algorithm yields a large set, we select only the most sensitive nodes (i.e., the nodes with highest sensitivity counts in a histogram like that of Fig. 3) to design the interleaver at the beginning. Then we increase the number of selected sensitive nodes step by step until we cannot find a solution for the interleaver.
+
+## IV Simulation Results
+
+The Monte Carlo simulation results for the proposed concatenated IRA code structure on the binary-input AWGN channel are shown in Fig. 4. In the figure, the right-most curve (marked by ‘+’ symbols) is the performance of a single IRA component code with source block length \(K=128\) bits and code rate \(0.707\). The second rightmost curve (marked ‘x’) is the proposed concatenated code with block size \(K^{2}=16384\), rate \(0.5\), and a random interleaver; the random interleaver was found by (non-exhaustive) search over a large number of randomly generated interleavers. The solid line with circle markers is the same code structure as the second curve, but uses an interleaver based on the design rules proposed in section III; this designed interleaver used the random interleaver of the ‘x’ curve as the design’s starting point. The dashed line with star markers is the BER of a rate 1/2, block length \(K^{2}=65536\) concatenated IRA code with optimized interleaver. For comparison, we also simulated single long block length IRA codes with rate \(0.5\); the solid line is with source block length \(16384\) and the dashed line is with source block length \(65536\).
+
+All the single IRA code simulations were run until either a valid codeword was decoded, or 100 iterations were performed. For both the 16384-bit concatenated curves the decoder was run for a total of 10 outer iterations between the component codes, and the component codes were each iterated 10 times per outer iteration. Component decoding (on a given row or column) was terminated before 10 iterations if a valid codeword was decoded. The concatenated iteration schedule was determined experimentally, and therefore may not be optimal. (Further optimization of the iteration schedule using, e.g., EXIT charts [13], will be the focus of future work.) The complexity of the 16384-bit concatenated decoder is thus approximately twice that of the 16384-bit single IRA code, although at higher SNR the complexity of the concatenated system is relatively higher because termination events for the concatenated code eliminate only single rows or columns from the iteration, not the entire codeword. The 65536-bit concatenated decoder was run for a total of 10 outer iterations with 20 inner iterations per outer iteration, so its decoding complexity is about four times that of the single 65536-bit IRA code.
+
+Figure 4: Simulation results. All codes are rate 1/2, except for the \(K=128\) IRA code, which is rate 0.707.
+
+From the figure it is clear that, although the concatenated 16384-bit IRA code has an SNR penalty in the waterfall region (about 2.1 dB SNR at BER \(10^{-5}\)) compared to the single 16384-bit IRA code of equivalent rate, it has a much lower error floor. There is a crossover point between these two codes’ BER curves at a BER of about \(10^{-7}\), and the BER of the concatenated IRA code decreases much faster than that of the single IRA code at high SNR. By comparing the 16384-bit concatenated codes’ performance with different interleavers, we see that the proposed interleaver design can achieve significant gains (about \(0.7\) dB at \(10^{-5}\) and \(0.3\) dB at \(10^{-7}\)) over the random interleaver used as the design starting point, which means the idea of separating the component codes’ stopping sets works.
+
+The \(K=128\) example component codes are quite short. We conjecture that when the block length is increased the penalty in the waterfall region will decrease, since the component IRA codes will asymptotically approach capacity as the block length increases. This conjecture is partly supported by the smaller SNR penalty (about 1.7 dB at BER \(10^{-5}\)) of the 65536-bit rate-1/2 concatenated code compared to the equivalent-rate 65536-bit IRA code, although part of the improvement over the 16384-bit codes may be due to the increased decoder iterations allocated to the 65536-bit concatenated system.
+
+## V Conclusions
+
+This paper has demonstrated that serial concatenation of two IRA codes connected by an appropriately designed interleaver can greatly lower the level and slope of the BER curve in the high SNR region, compared to a single IRA code of equivalent length and rate. We believe that the proposed approach will also work with more general LDPCs as component codes, including, e.g., e-IRA codes or codes optimized with the error-floor lowering algorithm of [10]. Future work will focus on reducing the SNR penalty of the concatenated codes in the waterfall region through more rigorous optimization of the iteration schedule, and through use of longer block length component codes.
+
+## Acknowledgment
+
+The authors would like to thank the National Science Foundation for providing support for the work presented in this paper, under grant CCF-0635390.
+
+## References
+
+* [1] R. G. Gallager, _Low-Density Parity-Check Codes._ Cambridge, MA: MIT Press, 1963.
+* [2] D. J. C. MacKay, “Good error correcting codes based on very sparse matrices,” _IEEE Trans. Inform. Theory_, vol. 45, pp. 399-431, Mar. 1999.
+* [3] T. J. Richardson and R. L. Urbanke, “The Capacity of low-density parity-check codes under message-passing decoding,” _IEEE Trans. Inform. Theory_, vol. 47, pp. 599-618, Feb. 2001.
+* [4] T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” _IEEE Trans. Inform. Theory_, vol. 47, pp. 619-637, Feb. 2001.
+* [5] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat-accumulate codes,” in _Proc. 2nd. Int. Symp. Turbo Codes and Related Topics_, Brest, France, Sept. 2000, pp. 1-8.
+* [6] M. Yang, W. E. Ryan, and Y. Li, “Design of efficiently encodable moderate-length high-rate irregular LDPC codes,” _IEEE Trans. Commun._, vol. 52, pp. 564-571, Apr. 2004.
+* [7] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” _IEEE Trans. Commun._, vol. 44, pp. 1261-1271, Oct. 1996.
+* [8] C. Di, D. Proietti, E. Telatar, T. Richardson, and R. Urbanke, “Finite length analysis of low-density parity-check codes on the binary erasure channel,” _IEEE Trans. Inform. Theory_, vol. 48, pp. 1570-1579, June 2002.
+* [9] T. Tian, C. R. Jones, J. D. Villasenor, and R. D. Wesel, “Selective avoidance of cycles in irregular LDPC code construction,” _IEEE Trans. Commun._, vol. 52, pp. 1242-1247, Aug. 2004.
+* [10] S. H. Lee, K. S. Kim, J. K. Kwon, Y. H. Kim, and J. Y. Ahn, “Design of an LDPC code with low error floor,” in _Proc. IEEE Int. Symp. Info. Theory (ISIT 2005)_, Adelaide, Australia, Sept. 2005, pp. 990-994.
+* [11] R. M. Pyndiah, “Near-optimum decoding of product codes: block turbo codes,” _IEEE Trans. Commun._, vol. 46, pp. 1003-1010, Aug. 1998.
+* [12] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” _IEEE Trans. Inform. Theory_, vol. 47, pp. 498-519, Feb. 2001.
+* [13] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” _IEEE Trans. Commun._, vol. 49, pp. 1727-1737, Oct. 2001.
+

arxiv_ground_truth/0710.2265.md

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+# Controlling spatiotemporal chaos and spiral turbulence in excitable media: A review
+
+Sitabhra Sinha
+
+sitabhra@imsc.res.in
+
+S. Sridhar
+
+The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai - 600 113 India
+
+###### Abstract
+
+Excitable media are a generic class of models used to simulate a wide variety of natural systems including cardiac tissue. Propagation of excitation waves in this medium results in the formation of characteristic patterns such as rotating spiral waves. Instabilities in these structures may lead to spatiotemporal chaos through spiral turbulence, which has been linked to clinically diagnosed conditions such as cardiac fibrillation. Usual methods for controlling such phenomena involve very large amplitude perturbations and have several drawbacks. There have been several recent attempts to develop low-amplitude control procedures for spatiotemporal chaos in excitable media which are reviewed in this paper. The control schemes have been broadly classified by us into three types: (i) global, (ii) non-global spatially-extended and (iii) local, depending on the way the control signal is applied, and we discuss the merits and drawbacks for each.
+
+pacs: 87.19.Hh, 05.45.Gg, 05.45.Jn, 87.18.Hf
+
+## I Introduction
+
+Excitable media denotes a class of systems that share a set of features which make their dynamical behavior qualitatively similar. These features include (i) the existence of two characteristic dynamical states, comprising a stable _resting state_ and a metastable _excited state_, (ii) a _threshold_ value associated with one of the dynamical variables characterising the system, on exceeding which, the system switches from the resting state to the excited state, and (iii) a _recovery period_ following an excitation, during which the response of the system to a supra-threshold stimulus is diminished, if not completely absent [1]. Natural systems which exhibit such features include, in biology, cells such as neurons, cardiac myocytes and pancreatic beta cells, all of which are vital to the function of a complex living organism. Other examples of dynamical phenomena associated with excitable media include cAMP waves observed during aggregation of slime mold, calcium waves observed in Xenopus oocytes, muscle contractions during childbirth in uterine tissue, chemical waves observed in the Belusov-Zhabotinsky reaction and concentration patterns in CO-oxidation reaction on Pt(110) surface. Excitation in such systems is observed as the characteristic _action potential_, where a variable associated with the system (e.g., membrane potential, in the case of biological cells) increases very fast from its resting value to the peak value corresponding to the excited state, followed by a slower process during which it gradually returns to the resting state.
+
+The simplest model system capable of exhibiting all these features is the generic Fitzhugh-Nagumo set of coupled differential equations:
+
+\[de/dt=e(1-e)(e-b)-g,~{}~{}dg/dt=\epsilon(ke-g),\] (1)
+
+which, having only two variables, is obviously incapable of exhibiting chaos. However, when several such sets are coupled together diffusively to simulate a spatially extended media (e.g., a piece of biological tissue made up of a large number of cells), the resulting high-dimensional dynamical system can display chaotic behavior. The genesis of this _spatiotemporal chaos_ lies in the distinct property of interacting waves in excitable media, which mutually annihilate on colliding. This is a result of the fact that an excitation wavefront is followed by a region whose cells are all in the recovery period, and which, therefore, cannot be stimulated by another excitation wavefront, as for example when two waves cross each other [2]. Interaction between such waves result in the creation of spatial patterns, referred to variously as _reentrant excitations_ (in 1D), _vortices_ or _spiral waves_ (in 2D) and _scroll waves_ (in 3D), which form when an excitation wavefront is broken as the wave propagates across partially recovered tissue or encounters an inexcitable obstacle [3]. The free ends of the wavefront gradually curl around to form spiral waves. Once formed, such waves become self-sustained sources of high-frequency excitation in the medium, and usually can only be terminated through external intervention. The existence of nonlinear properties of wave propagation in several excitable media can lead to complex non-chaotic spatiotemporal rhythms. Thus, spiral waves are associated with periodic as well as quasiperiodic patterns of temporal activity.
+
+However, in this paper, we shall not be discussing the many schemes proposed to terminate single spiral waves, but instead, focus on the control of spatiotemporally chaotic patterns seen in excitable media (in 2 or 3 dimensions), that occur when under certain conditions, spiral or scroll waves become unstable and break up. Various mechanisms of such breakup have been identified [4], including meandering of the spiral focus. If the meandering is sufficiently high, the spiral wave can collide with itself and break up spontaneously, resulting in the creation of multiple smaller spirals (Fig. 1). The process continues until the spatial extent of the system is spanned by several coexisting spiral waves that activate different regions without any degree of coherence. This state of _spiral turbulence_ marks the onset of spatiotemporal chaos, as indicated by the Lyapunov spectrum and Kaplan-Yorke dimension [5].
+
+Figure 1: Onset of spatiotemporal chaos in the 2-dimensional Panfilov model of linear dimension \(L=256\). The initial condition is a broken plane wave that is allowed to curl around into a spiral wave (left). Meandering of the spiral focus causes wavebreaks to occur (centre) that eventually result in spiral turbulence, with multiple independent sources of high-frequency excitation (right).
+
+Controlling spatiotemporal chaos in excitable media has certain special features. Unlike other chaotic systems, here the response to a control signal is not proportional to the signal strength because of the existence of a threshold. As a result, an excitable system shows discontinuous response to control. For instance, regions that have not yet recovered from a previous excitation or where the control signal is below the threshold, will not be affected by the control stimulus at all. Also, the focus of control in excitable media is to eliminate all activity rather than to stabilize unstable periodic behavior. This is because the problem of chaos termination has great practical importance in the clinical context, as the spatiotemporally chaotic state has been associated with the cardiac problem of ventricular fibrillation (VF). VF involves incoherent activation of the heart that results in the cessation of pumping of blood, and is fatal within minutes in the absence of external intervention. At present, the only effective treatment is electrical defibrillation, which involves applying very strong electrical shocks across the heart muscles, either externally using a defibrillator or internally through implanted devices. The principle of operation for such devices is to overwhelm the natural cardiac dynamics, so as to drive all the different regions of the heart to rest simultaneously, at which time the cardiac pacemaker can take over once again. Although the exact mechanism by which this is achieved is still not completely understood, the danger of using such large amplitude control (involving \(\sim kV\) externally and \(\sim 100V\) internally) is that, not only is it excruciatingly painful to the patient, but by causing damage to portions of cardiac tissue which subsequently result in scars, it can potentially increase the likelihood of future arrhythmias. (i.e., abnormalities in the heart’s natural rhythm). Therefore, devising a low-power control method for spatiotemporal chaos in excitable media promises a safer treatment for people at risk from potentially fatal cardiac arrhythmias.
+
+In this paper, we have discussed most of the recent control methods that have been proposed for terminating spatiotemporal chaos in excitable media [6]. These methods are also often applicable to the related class of systems known as oscillatory media, described by complex Landau-Ginzburg equation [8], which also exhibit spiral waves and spatiotemporal chaos through spiral breakup. We have broadly classified all control schemes into three types, depending on the nature of application of the control signal. If every region of the media is subjected to the signal (which, in general, can differ from region to region) it is termed as _global control_; on the other hand, if the control signal is applied only at a small, localised region from which its effects spread throughout the media, this is called _local control_. Between these two extremes lie control schemes where perturbations are applied simultaneously to a number of spatially distant regions. We have termed these methods as _non-global, spatially extended control_. While global control may be the easiest to understand, involving as it does the principle of synchronizing the activity of all regions, it is also the most difficult to implement in any practical situation. On the other hand, local control will be the easiest to implement (requiring a single control point) but hardest to achieve.
+
+In the next section we describe a few of the more commonly used models for studying control of spatiotemporal chaos in excitable media. Section 3 discusses proposed methods of global control, while Section 4 discusses other spatially extended schemes. The next section deals with local control methods, and we conclude with a brief section containing general discussions about chaos control and its implications.
+
+## II Models of Spatiotemporal Chaos in Excitable Media
+
+Figure 2: (top) Dynamics in the phase-space of the Fitzhugh-Nagumo model, with the resulting time evolution of the action potential shown in the inset. The resting state corresponds to \(e=0,g=0\). (Bottom) The result of applying a positive (“+”) or negative (“\(-\)”) additive perturbation of the same duration to the \(e\) variable: “+” control decreases the threshold and makes excitation more likely, while “\(-\)” control decreases the duration of the action potential and allows the system to recover faster. For the duration of the control signal, the \(e\)-nullcline shifts upward (downward) for positive (negative) perturbation as indicated by the dashed (dash-dotted) curve.
+
+The generic Fitzhugh-Nagumo model for excitable media (Eq. 1) exhibits a structure that is common to most models used in the papers discussed here. Typically, the dynamics is described by a fast variable, \(e({\bf x},t)\), and a slow variable, \(g({\bf x},t)\), the ratio of timescales being given by \(\epsilon\). The resulting phase space behavior is shown in Fig. 2 (left). For biological cells, the fast variable is often associated with the transmembrane potential, while the slow (recovery) variable represents an effective membrane conductance that replaces the complexity of several different types of ion channels. For the spatially extended system, the fast variable of neighboring cells are coupled diffusively. There are several models belonging to this general class of excitable media which display breakup of spiral waves (in 2D) and scroll waves (in 3D), including the one proposed by Panfilov [9; 10]
+
+\[{\partial e}/{\partial t}={\nabla}^{2}e-f(e)-g,~{}~{}{\partial g}/{\partial t}={\epsilon}(e,g)(ke-g).\] (2)
+
+Here, \(f(e)\) is the function specifying the initiation of the action potential and is piecewise linear: \(f(e)=C_{1}e\), for \(e<e_{1}\), \(f(e)=-C_{2}e+a\), for \(e_{1}\leq e\leq e_{2}\), and \(f(e)=C_{3}(e-1)\), for \(e>e_{2}\). The physically appropriate parameters given in Ref. [10] are \(e_{1}=0.0026\), \(e_{2}=0.837\), \(C_{1}=20\), \(C_{2}=3\), \(C_{3}=15\), \(a=0.06\) and \(k=3\). The function \(\epsilon(e,g)\) determines the time scale for the dynamics of the recovery variable: \(\epsilon(e,g)=\epsilon_{1}\) for \(e<e_{2}\), \(\epsilon(e,g)=\epsilon_{2}\) for \(e>e_{2}\), and \(\epsilon(e,g)=\epsilon_{3}\) for \(e<e_{1}\) and \(g<g_{1}\) with \(g_{1}=1.8\/\), \(\epsilon_{1}=1/75\/\), \(\epsilon_{2}=1.0\/\), and \(\epsilon_{3}=0.3\)
+
+Simpler variants that also display spiral wave breakup in 2D include (i) the Barkley model [11]:
+
+\[{\partial e}/{\partial t}={\nabla}^{2}e+{\epsilon}^{-1}e(1-e)(e-\frac{g+b}{a}),~{}~{}{\partial g}/{\partial t}=e-g,\] (3)
+
+the appropriate parameter values being given in Ref. [12], and (ii) the Bär-Eiswirth model [13], which differs from (3) only in having \({\partial g}/{\partial t}=f(e)-g\), the functional form of \(f(e)\) and parameter values being as in Ref. [14]. The Aliev-Panfilov model [15] is a modified form of the Panfilov model, that takes into account nonlinear effects such as the dependence of the action potential duration on the distance of the wavefront to the preceding waveback. It has been used for control in Refs. [16; 17].
+
+All the preceding models tend to disregard several complex features of actual biological cells, e.g., the different types of ion channels that allow passage of electrically charged ions across the cellular membrane. There exists a class of models inspired by the Hodgkin-Huxley formulation describing action potential generation in the squid giant axon, which explicitly takes such details into account. While the simple models described above do reproduce generic features of several excitable media seen in nature, the more realistic models describe many properties of specific systems, e.g., cardiac tissue. The general form of such models are described by a partial differential equation for the transmembrane potential \(V\),
+
+\[\frac{\partial V}{\partial t}+\frac{I_{ion}}{C}=D\nabla^{2}V,\] (4)
+
+where \(C\) is the membrane capacitance density and \(D\) is the diffusion constant, which, if the medium is isotropic, is a scalar. \(I_{ion}\) is the instantaneous total ionic-current density, and different realistic models essentially differ in its formulation. For example, in the Luo-Rudy I model [19] of guinea pig ventricular cells, \(I_{ion}\) is assumed to be composed of six different ionic current densities, which are themselves determined by several time-dependent ion-channel gating variables whose time-evolution is governed by ordinary differential equations of the form:
+
+\[\frac{d\xi}{dt}=\frac{\xi_{\infty}-\xi}{\tau_{\xi}}.\] (5)
+
+Here, \(\xi_{\infty}=\alpha_{\xi}/(\alpha_{\xi}+\beta_{\xi})\) is the steady state value of \(\xi\) and \(\tau_{\xi}=\frac{1}{\alpha_{\xi}+\beta_{\xi}}\) is its time constant. The voltage-dependent rate constants, \(\alpha_{\xi}\) and \(\beta_{\xi}\), are complicated functions of \(V\) obtained by fitting experimental data.
+
+## III Global Control
+
+The first attempt at controlling chaotic activity in excitable media dates back almost to the beginning of the field of chaos control itself, when proportional perturbation feedback (PPF) control was used to stabilize cardiac arrhythmia in a piece of tissue from rabbit heart [20]. This method applied small electrical stimuli, at intervals calculated using a feedback protocol, to stabilize an unstable periodic rhythm. Unlike in the original proposal for controlling chaos [21], where the location of the stable manifold of the desired unstable periodic orbit (UPO) was moved using small perturbations, in the PPF method it is the state of the system that is moved onto the stable manifold. However, it has been later pointed out that PPF does not necessarily require the existence of UPOs (and, by extension, deterministic chaos) and can be used even in systems with stochastic dynamics [22]. Later, PPF method was used to control atrial fibrillation in human heart [23]. However, the effectiveness of such control in suppressing spatiotemporal chaos, when applied only at a local region, has been questioned, especially as other experimental attempts in feedback control have not been able to terminate fibrillation by applying control stimuli at a single spatial location [7].
+
+More successful, at least in numerical simulations, have been schemes where control stimuli is applied throughout the system. Such global control schemes either apply small perturbations to the dynamical variables (\(e\) or \(g\)) or one of the parameters (usually the excitation threshold). The general scheme involves introducing an external control signal \(A\) into the model equations, e.g., in the Panfilov model [Eq. (2)]:
+
+\[{\partial e}/{\partial t}={\nabla}^{2}e-f(e)-g+A,\] (6)
+
+for a control duration \(\tau\). If \(A\) is a small, positive perturbation, added to the fast variable, the result is an effective reduction of the threshold (Fig. 2, bottom), thereby making simultaneous excitation of different regions more likely. In general, \(A\) can be periodic, consisting of a sequence of pulses. Fig. 3 shows the results of applying a pulse of fixed amplitude but varying durations. While in general, increasing the amplitude, or the duration, increases the likelihood of suppressing spatiotemporal chaos, it is not a simple, monotonic relationship. Depending on the initial state at which the control signal is applied, even a high amplitude (or long duration) control signal may not be able to uniformly excite all regions simultaneously. As a result, when the control signal is withdrawn, the inhomogeneous activation results in a few regions becoming active again and restarting the spatiotemporal chaotic behavior.
+
+Figure 3: Global control of the 2-dimensional Panfilov model with \(L=256\) starting from a spatiotemporally chaotic state (top left). Pseudo-gray-scale plots of excitability \(e\) show the result of applying a pulse of amplitude \(A=0.833\) between \(t=\) 11 ms and 27.5 ms (top centre) that eventually leads to elimination of all activity (top right). Applying the pulse between \(t=\) 11 ms and 33 ms (bottom left) results in some regions becoming active again after the control pulse ends (bottom centre) eventually reinitiating spiral waves (bottom right).
+
+Most global control schemes are variations or modifications of the above scheme. Osipov and Collins [24] have shown that a low-amplitude signal used to change the value of the slow variable at the front and back of an excitation wave can result in different wavefront and waveback velocities which destabilizes the traveling wave, eventually terminating all activity, and, hence, spatiotemporal chaos. Gray [25] has investigated the termination of spiral wave breakup by using both short and long-duration pulses applied on the fast variable, in 2D and 3D systems. This study concluded that while short duration pulses affected only the fast variable, long duration pulses affected both fast and slow variables and that the latter is more efficient, i.e., uses less power, in terminating spatiotemporal chaos. The external control signal can also be periodic [\(A=Fsin(\omega t)\)], in which case the critical amplitude \(F_{c}\) required for terminating activity has been found to be a function of the signal frequency \(\omega\)[16].
+
+Figure 4: Spatiotemporal chaos (top row) and its control (bottom row) in the 2-dimensional Luo-Rudy I model with \(L=90\) mm. Pseudo-gray-scale plots of the transmembrane potential \(V\) show the evolution of spiral turbulence at times \(T\) = 30 ms, 90 ms, 150 ms and 210 ms. Control is achieved by applying an external current density \(I=150\mu A/cm^{2}\) for \(\tau\) = 2.5 msec over a square mesh with each block of linear dimension \(L/K=1.35\) cm. Within 210 msec of applying control, most of the simulation domain has reached a transmembrane potential close to the resting state value; moreover, the entire domain is much below the excitation threshold. The corresponding uncontrolled case shows spatiotemporal chaos across the entire domain.
+
+Other schemes have proposed applying perturbations to the parameter controlling the excitation threshold, \(b\). Applying a control pulse on this parameter (\(b=b_{f}\), during duration of control pulse;\(b=b_{0}\), otherwise) has been shown to cause splitting of an excitation wave into a pair of forward and backward moving waves [14]. Splitting of a spiral wave causes the two newly created spirals to annihilate each other on collision. For a spatiotemporally chaotic state, a sequence of such pulses may cause termination of all excitation, there being an optimal time interval between pulses that results in fastest control. Another control scheme that also applies perturbation to the threshold parameter is the uniform periodic forcing method suggested by Alonso _et al_[12; 26] for controlling scroll wave turbulence in three-dimensional excitable media. Such turbulence results from negative tension between scroll wave filaments, i.e., the line joining the phase singularities about which the scroll wave rotates. In this control method, the threshold is varied in periodic manner [\(b=b_{0}+b_{f}cos(\omega t)\)] and the result depends on the relation between the control frequency \(\omega\) and the spiral rotation frequency. If the former is higher than the latter, sufficiently strong forcing is seen to eliminate turbulence; otherwise, turbulence suppression is not achieved. The mechanism underlying termination has been suggested to be the effective increase of filament tension due to rapid forcing, such that, the originally negative tension between scroll wave filaments is changed to positive tension. This results in expanding scroll wave filaments to instead shrink and collapse, eliminating spatiotemporal chaotic activity. In a variant method, the threshold parameter has been perturbed by spatially uncorrelated Gaussian noise, rather than a periodic signal, which also results in suppression of scroll wave turbulence [27].
+
+As already mentioned, global control, although easy to understand, is difficult to achieve in experimental systems. A few cases in which such control could be implemented include the case of eliminating spiral wave patterns in populations of the Dictyostelium amoebae by spraying a fine mist of cAMP onto the agar surface over which the amoebae cells grow [28]. Another experimental system where global control has been implemented is the photosensitive Belusov-Zhabotinsky reaction, where a light pulse shining over the entire system is used as a control signal [29]. Indeed, conventional defibrillation can be thought of as a kind of global control, where a large amplitude control signal is used to synchronize the phase of activity at all points by either exciting a previously unexcited region (advancing the phase) or slowing the recovery of an already excited region (delaying the phase) [30].
+
+## IV Non-Global Spatially Extended Control
+
+The control methods discussed so far apply control signal to all points in the system. As the chaotic activity is spatially extended, one may naively expect that any control scheme also has to be global. However, we will now discuss some schemes that, while being spatially extended, do not require the application of control stimuli at all points of the system.
+
+### Applying control over a mesh
+
+The control method of Sinha _et al_[31] involving suprathreshold stimulation along a grid of points is based on the observation that spatiotemporal chaos in excitable media is a long-lived transient that lasts long enough to establish a non-equilibrium statistical steady state displaying spiral turbulence. The lifetime of this transient, \({\tau}_{L}\), increases rapidly with linear size of the system, \(L\), e.g., increasing from 850 ms to 3200 ms as \(L\) increases from 100 to 128 in the two-dimensional Panfilov model. This accords with the well-known observation that small mammals do not get life-threatening VF spontaneously whereas large mammals do [32] and has been experimentally verified by trying to initiate VF in swine ventricular tissue while gradually reducing its mass [33]. A related observation is that non-conducting boundaries tend to absorb spiral excitations, which results in spiral waves not lasting for appreciable periods in small systems.
+
+The essential idea of the control scheme is that a domain can be divided into electrically disconnected regions by creating boundaries composed of recovering cells between them. These boundaries can be created by triggering excitation across a thin strip. For two-dimensional media, the simulation domain (of size \(L\times L\)) is divided into \(K^{2}\) smaller blocks by a network of lines with the block size (\(L/K\times L/K\)) small enough so that spiral waves cannot form. For control in a 3D system, the mesh is used only on one of the faces of the simulation box. Control is achieved by applying a suprathreshold stimulation via the mesh for a duration \(\tau\). A network of excited and subsequently recovering cells then divides the simulation domain into square blocks whose length in each direction is fixed at a constant value \(L/K\) for the duration of control. The network effectively simulates non-conducting boundary conditions (for the block bounded by the mesh) for the duration of its recovery period, in so far as it absorbs spirals formed inside this block. Note that \(\tau\) need not be large at all because the individual blocks into which the mesh divides the system (of linear size \(L/K\)) are so small that they do not sustain long spatiotemporally chaotic transients. Nor does \(K\), which is related to the mesh density, have to be very large since the transient lifetime, \(\tau_{L}\), decreases rapidly with decreasing \(L\). The method has been applied to multiple excitable models, including the Panfilov and Luo-Rudy models (Fig. 4).
+
+An alternative method [17] for controlling spiral turbulence that also uses a grid of control points has been demonstrated for the Aliev-Panfilov model. Two layers of excitable media are considered, where the first layer represents the two-dimensional excitable media exhibiting spatiotemporal chaos that is to be controlled, and the second layer is a grid structure also made up of excitable media. The two layers are coupled using the fast variable but with asymmetric coupling constants, with excitation pulses travelling \(\sqrt{D}\) times faster in the second layer compared to the first. As the second layer consists only of grid lines, it is incapable of exhibiting chaotic behavior in the uncoupled state. If the coupling from the second layer to the first layer is sufficiently stronger than the other way round, the stable dynamics of the second layer (manifested as a single rotating spiral) overcomes the spiral chaos in the first layer, and drives it to an ordered state characterized by mutually synchronized spiral waves.
+
+### Applying control over an array of points
+
+An alternative method of spatially extended control is to apply perturbations at a series of points arranged in a regular array. Rappel _et al_[34] had proposed using such an arrangement for applying a time-delayed feedback control scheme. However, this scheme does not control spatiotemporal chaos and is outside the scope of this review.
+
+Figure 5: Control of the 2-dimensional Panfilov model (\(L=256\)) using an array of control points with spacing \(d=6\) and strength of control stimulus \(A=2.5\). Stimulation is started at the top left corner (\(T=0\) ms) and lasts at each control point, as the wave reaches that point, for 17.9 ms. By 200 ms, the spatiotemporal chaos has disappeared.
+
+More recently, the authors [35] have used an array of control points to terminate spatiotemporal chaos in the Panfilov model. Fig. 5 shows the result of applying a spatially non-uniform control scheme, which simulates an excitation wave traveling over the system, with the same wavefront velocity as in the actual medium. The control points are placed distance \(d\) apart along a regular array. At certain times, the control points at one corner of the system is stimulated, followed by the successive stimulation of the neighboring control points, such that a wave of stimulation is seen to move radially away from the site of original stimulation. This process is repeated after suitable intervals. Note that, simulating a traveling wave using the array is found to be more effective at controlling spatiotemporal chaos than the simultaneous activation of all control points. Using a traveling wave allows the control signal to engage all high-frequency sources of excitation in the spiral turbulence regime, ultimately resulting in complete elimination of chaos. If, however, the control had only been applied locally the resulting wave could only have interacted with neighboring spiral waves and the effects of such control would not have been felt throughout the system. The efficacy of the control scheme depends upon the spacing between the control points, as well as the number of simulated traveling waves. Traveling waves have previously been used in Ref. [36] to control spatiotemporal chaos, although in the global control context with a spatiotemporally periodic signal being applied continuously for a certain duration, over the entire system.
+
+## V Local Control of Spatiotemporal Chaos
+
+Figure 6: (left) Pacing response diagram for 2D Panfilov model (\(L=26\)) showing relative performance of different waveforms. The dash-dotted line represents a sine wave and the solid curve represents a wave of biphasic rectangular pulses, such that they have the same total energy. Successful control occurs if the effective frequency lies above the broken line representing the effective frequency of chaos (\(f_{c}\)), as seen for a larger system (\(L=500\)) at times \(T=1000\) (center) and \(T=3800\) (right) time units, where the control signal is applied only at the center of the simulation domain. The excitation wavefronts are shown in white, black marks the recovered regions ready to be excited, while the shaded regions indicate different stages of recovery.
+
+We now turn to the possibility of controlling spatiotemporal chaos by applying control at only a small localized region of the spatially extended system. Virtually all the proposed local control methods use _overdrive pacing_, generating a series of waves with frequency higher than any of the existing excitations in the spiral turbulent state. As low-frequency activity is progressively invaded by faster excitation, the waves generated by the control stimulation gradually sweep the chaotic activity to the system boundary where they are absorbed. Although we cannot speak of a single frequency source in the case of chaos, the relevant timescale is that of the spiral waves and is related to the recovery period of the medium. Control is manifested as a gradually growing region in which the waves generated by the control signal dominate, until the region expands to encompass the entire system. The time required to achieve termination depends on the frequency difference between the control stimulation and that of the chaotic activity, with control being achieved faster when this difference is greater.
+
+Stamp _et al_[37] has looked at the possibility of using low-amplitude, high-frequency pacing using a series of pulses to terminate spiral turbulence. However, using a series of pulses (having various waveform shapes) has met with only limited success in suppressing spatiotemporal chaos. By contrast, a periodic stimulation protocol [38] has successfully controlled chaos in the 2D Panfilov model, as well as other models [39]. The key mechanism underlying such control is the periodic alternation between positive and negative stimulation. A more general control scheme proposed in Ref. [40] uses _biphasic pacing_, i.e., applying a series of positive and negative pulses, that shortens the recovery period around the region of control stimulation, and thus allows the generation of very high-frequency waves than would have been possible using positive stimulation alone. A simple argument shows why a negative rectangular pulse decreases the recovery period for an excitable system. The stimulation vertically displaces the \(e\)-nullcline and therefore, the maximum value of \(g\) that can be attained is reduced. Consequently, the system will recover faster from the recovery period (Fig. 2, bottom).
+
+To understand how negative stimulation affects the response behavior of the spatially extended system, we can use _pacing response diagrams_ (Fig. 6, left) indicating the relation between the control stimulation frequency \(f\) and the effective frequency \(f_{eff}\) , measured by applying a series of pulses at one site and then recording the number of pulses that reach another site located at a distance without being blocked by a region in the recovery period. Depending on the relative value of \(f^{-1}\) and the recovery period, we observe instances of \(n:m\) response, i.e., \(m\) responses evoked by \(n\) stimuli. If, for any range of \(f\), the corresponding \(f_{eff}\) is significantly higher than the effective frequency of spatiotemporal chaos, then termination of spiral turbulence is possible. As shown in Ref. [40], there are indeed ranges of stimulation frequencies that give rise to effective frequencies that dominate chaotic activity. As a result, the periodic waves emerging from the stimulation region gradually impose control over the regions exhibiting chaos (Fig. 6). Note that, there is a tradeoff involved here. If \(f_{eff}\) is only slightly higher than the chaos frequency, control takes too long. On the other hand, if it is too high the waves suffer conduction block at inhomogeneities produced by chaotic activity which reduces the effective frequency, and therefore, control fails.
+
+Recently, another local control scheme has been proposed [41] that periodically perturbs the model parameter governing the threshold. In fact, it is the local control analog of the global control scheme proposed by Alonso _et al_[12] discussed in section III. As in the other methods discussed here, the local stimulation generates high-frequency waves that propagate into the medium and suppress spiral or scroll waves. Unlike the global control scheme, \(b_{f}>>b_{0}\), so that the threshold can be negative for a part of the time. This means that the regions in resting state can become spontaneously excited, which allow very high-frequency waves to be generated.
+
+## VI Discussion
+
+Most of the methods proposed for controlling spatiotemporal chaos in excitable media involve applying perturbations either globally or over a spatially extended system of control points covering a significant proportion of the entire system. However, in most practical situations this may not be a feasible option, either for issues of implementation, or because of the high power for the control signal that such methods would need. Moreover, if one is using such methods in the clinical context, e.g., terminating fibrillation, a local control scheme has the advantage that it can be readily implemented with existing hardware of the Implantable Cardioverter Defibrillator (ICD). This is a device implanted into patients at high risk from fibrillation that monitors the heart rhythm and applies electrical treatment when necessary through electrodes placed on the heart wall. A low-energy control method involving ICDs should therefore aim towards achieving control of spatiotemporal chaos by applying small perturbations from a few local sources.
+
+However, the problem with most local control schemes proposed so far is that they use very high-frequency waves to overdrive chaos. Such waves are themselves unstable and may break during propagation, resulting in reinitiation of spiral waves after the original chaotic activity has been terminated. The problem is compounded by the existence of inhomogeneities in real excitable media. Recently, Shajahan _et al_[42] have found complicated dependence of spatiotemporal chaos on the presence of non-conducting regions and other types of inhomogeneities in an excitable system. Such inhomogeneities make the proposed local control schemes more vulnerable, as it is known that high-frequency pacing interacting with, e.g., non-conducting obstacles, results in wave breaks and subsequent genesis of spatiotemporal chaos [43].
+
+The search is still on for a control algorithm for terminating spatiotemporal chaos in excitable media, that can be implemented using low power, or, that need be applied in only a small, local region of the system, and which will yet be robust, capable of terminating spiral turbulence without the control stimulation itself breaking up subsequently. The payoffs for coming up with such a method are enormous, as the potential benefits include an efficient device for cardiac defibrillation.
+
+**Acknowledgements:** We would like to thank colleagues and collaborators with whom some of the work described above has been carried out, especially, Rahul Pandit, Ashwin Pande, Avishek Sen, T. K. Shajahan, David J. Christini, Ken M. Stein, Johannes Breuer and Antina Ghose. We thank IFCPAR and IMSc Complex Systems Project (X Plan) for support.
+
+## References
+
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+

arxiv_ground_truth/0710.2740.md

@@ -0,0 +1,136 @@
+# Reliability of Module Based Software System
+
+ Rudrani Banerjee and Angshuman Sarkar
+**_Department of Statistics, Visva-Bharati University, India_**
+
+ Email:sangshu_2000@yahoo.com
+
+###### Abstract
+
+This paper consider the problem of determining the reliability of a software system which can be decomposed in a number of modules. We have derived the expression of the reliability of a system using the Markovian model for the transfer of control between modules in order. We have given the expression of reliability by considering both benign and catastrophic failure. The expression of reliability presented in this work is applicable for some control software which are designed to detect its own internal errors.
+
+## 1 Introduction
+
+Now a days large scale software systems are used in every walk of life. The price of software are much higher than the cost of hardware when we consider a huge computer intensive system. Moreover the penalty cost incurred by a false outcome of a system is enormous. To address such a challenge posed by this technological trend, during the last three decades extensive research has focused on the area of software reliability. The consideration of software reliability is increasing because of the growing emphasis on software that is reusable (as opposed to software that is written for a terminal mission), where it is essential to demonstrate that the system will perform reliably for a variety of end-user applications.
+A software system is defined here as a ” collection of programs and system files such that the system files are accessed and altered only by the programs in the collection ”. Each element in this collection will be called a module - for instance, a module might be a program, a subprogram, or a file. The performance ( and hence the reliability ) of the system clearly depends on that of each individual module and the relationship between these modules and the system; in this regard a software system is quite similar to any other system. However, the actual relationship between system and module reliabilities is quite unique and depends on the specific definition of software reliability as well as on the structure of the overall system. In this paper we focus on software systems that can be decomposed into a finite number of modules.
+In testing a software one may test the system as a whole, but in practice, different organizational entities are assigned responsibility of developing different modules. So it will be more beneficial in the context of both cost and time test the individual modules instead of testing them together. In order to do this, some mathematical models, often referred to as Software Reliability Growth Models (SRGM) are used to enable the software reliability practitioners to estimate the expected future reliability of a software under development and accordingly allocate time, money, human resources to a project. Often these reliability growth models express software system reliability in terms of the individual module reliabilities which is favorable under both time and cost constraints.
+Cheung (2), first expressed the system reliability in terms of the component reliabilities. Poore et al. (1) suggest allocating the targeted system reliability goal among the components and then testing the individual components to verify whether the component reliabilities meet the allocated goals at a specified level of confidence, where as Easterling, Mazumdar, Spencer and Diegert (6), has discussed this method may lead to estimates of overly conservative sample size requirements for component testing. Yang et. al. has implemented the idea of using testability to estimate software reliability. They have also provided the basic steps involve estimating testability, evaluating how well software was written, and assessing the relationship between testing and usage by assuming the modules are independently functioning. They have also compared their results with those obtained by using two reliability growth models. Rajgopal et. al. has used a Markovian model for the transfer of control between modules in order to develop the system reliability expression in terms of the module reliabilities in case of a dependent setup. They have also discussed a procedure for determining the minimum number of tests required of each module such that the probability of certifying a system whose reliability falls below a specified value \(R_{0}\) is less than a specified small fraction \(\beta\). Bondavalli et. al. has considered the concept of benign failure and catastrophic failure for determining the software reliability for a iterative program.
+In this paper we have expressed the system reliability in terms of testability of a particular module following Yang et. al. for dependent modules and have introduced the concepts of benign and catastrophic failure following Bondavalli et. al. in case of a system where it can be decomposed in a finite number of dependently functional modules. The section 2 discuss the notations and preliminaries, section 3 gives the expression of the probability of correct output for a specific input. Recent research [26] has shown a strong correlation between reliability and coverage criteria (Lott et al. (2005), Khun et. al. (2002), Yilmaz et. al. (2004) etc.), although it is very difficult to quantify this relation. Dalal et al. [6] and many more has examined this relationship between unit-test statement coverage and system-test faults later attributed to those units.
+Present work has been organized in 4 sections the section 2 gives the notation and preliminaries of software reliability in terms testability of a module. In the 3rd sections we have derived the probability of correct output of a particular system corresponding to a particular input considering both the case presence and absence of benign failure. In Section 4 we present a brief discussions about the procedure mentioned here.
+
+## 2 Notations and Preliminaries
+
+There is no rigorous definition of ’Quality’. But it can be weakly defined as the fitness of purpose of any product to its users. Similarly software quality is defined as the conformance to explicitly stated functions and performance requirements, explicitly documented development standards and implicit characteristics that are expected of all professionally crafted software(Cai Kai-Yuan Cai (3)). Alternatively, the quality of a software may be characterized by some quality factors of a software - reliability, efficiency, correctness, usability, testability etc.
+Reliability of a software system may be viewed as the expected value of probability of failure-free operation of a program for a randomly chosen set of input variables. The term failure in the context of software reliability implies a result other than what was expected from the software for a set of inputs. Following Voas et. al. (1995) we define the testability of a particular system as the probability of failure of the system for a particular input when it is assumed that there is at least one fault in the system. Suppose we have a software system which can be decomposed in \(N\) modules. Thus the testability of a particular module, say \(i\)th \((\forall i=1(1)N)\) module, is given by
+
+\[p_{i} =\] Prob[ that the \(i\)th module will give incorrect output \(\mid\) there is at least one fault, (1)
+probability distribution of input]
+
+The expression for the probability that the \(i\)th module will contain error if the module has tested \(n_{i}\) times successfully, is given by the following (Yang et. al. (1998))
+
+\[\alpha_{i}(t)=\frac{\alpha_{i}(0)(1-p_{i})^{n_{i}}}{\alpha_{i}(0)(1-p_{i})^{n_{i}}+1-\alpha_{i}(0)}\] (2)
+
+where \(\alpha_{i}(0)\) is the probability of failure of the system before testing. Let \(\pi_{t}(x)\) is the probability of a system giving correct output corresponding to a particular set of input \(x\). The expression of \(\pi_{t}(x)\) by assuming the independent setup is given by (Yang et. al. (1998))
+
+\[\pi_{t}(x)=\prod_{i\in S}(1-q_{i}\alpha_{i}(t))\] (3)
+
+where \(q_{i}\) is the revealibility of the \(i\) th module and \(S(x)\) is the set of those modules which will be executed by the input \(x\). The reliability of a software system is given by
+
+\[R_{t}=\int_{x\in X}\pi_{t}(x)\phi(x)dx\] (4)
+
+where \(X\) is the set of all possible inputs and \(\phi(x)\) is the probability distribution of \(x\).
+
+## 3 Detailed Expression of \(\pi_{t}(x)\) for Dependent Setup
+
+A software system is necessarily an iterative. In each iteration a particular module accepts a value and produce an output. The outcomes of an individual iteration may be: i) success, i.e., the delivery of a correct result, ii) a benign failure of the program, i.e., an output that is not correct but does not, by itself, cause the entire mission of the controlled system to fail, or iii) a catastrophic failure, i.e., an output that causes the immediate failure of the entire mission. The characterization of failures in benign and catastrophic is discussed with example by Bondavalli. et. al. (). In this section we derive the expression of \(\pi_{t}(x)\) first of all only considering the catastrophic failure and then in the subsequent subsection considering the benign and catastrophic failure simultaneously.
+
+### Expression of \(\pi_{t}(x)\): No Benign Failure in the System
+
+Consider the above software system with \(N\) modules. Let \(p_{ij}\) be the probability that the control from the \(i\)th module will be transferred to the \(j\)th module with correct execution \((\forall i=1(1)N,\forall j=1(1)N)\). Let \(S\) be a state of successful completion of the system. As \(S\) is achievable from any one of the module so we define \(p_{iS}\)\((\forall i=1(1)N)\) as the probability of successful completion of the mission from the \(i\)th module. Here we must have \(p_{iS}+\sum_{j=1}^{n}p_{ij}=1\).
+As we have a faulty system, that is, we have a system where there is at least one fault or if the faults can be classified into categories then there are at most one fault of each category. So we introduce another state \(F\), i.e., unsuccessful completion of the mission. As any module may be faulty so the state \(F\) also can be achieved from any of the module. We define \(p_{iF}\) as the probability of unsuccessful completion of the module \(i\)\((\forall i=1(1)N)\). The transition probability matrix takes the following form for the above setup.
+
+\[Q=\left(\begin{array}[]{cccccc}p_{11}(1-\alpha^{x}_{1}(t))&p_{12}(1-\alpha^{x}_{1}(t))&...&p_{1N}(1-\alpha^{x}_{1}(t))&p_{1S}(1-\alpha^{x}_{1}(t))&\alpha^{x}_{1}(t)\\ p_{21}(1-\alpha^{x}_{2}(t))&p_{22}(1-\alpha^{x}_{2}(t))&...&p_{2N}(1-\alpha^{x}_{2}(t))&p_{2S}(1-\alpha^{x}_{2}(t))&\alpha^{x}_{2}(t)\\ ...&...&...&...&...&...\\ p_{N1}(1-\alpha^{x}_{N}(t))&p_{N2}(1-\alpha^{x}_{N}(t))&...&p_{NN}(1-\alpha^{x}_{N}(t))&p_{NS}(1-\alpha^{x}_{N}(t))&\alpha^{x}_{N}(t)\\ 0&0&...&0&1&0\\ 0&0&...&0&0&1\\ \end{array}\right)\] (11)
+
+where \(\alpha^{x}_{i}(t)\) is the probability of faulty completion of the \(i\)th module for the input x. The expression of \(\alpha^{x}_{i}(t)\) is given by
+
+\[\alpha^{x}_{i}(t)=q_{i}\alpha_{i}(t)\] (12)
+
+If we assume that the first block is the control block then the probability of correct completion of the mission for the given input \(x\) is given by (Parzen (1962))
+
+\[\pi_{t}(x)=\sum_{i=1}^{N}(I_{N}-\hat{Q})^{-1}_{1i}p_{iS}(1-\alpha^{x}_{i}(t))\] (13)
+
+where \(\hat{Q}\) is the sub-matrix of \(Q\) deleting its last two columns and rows.
+
+### Expression of \(\pi_{t}(x)\): Benign Failure and Catastrophic Failure are in the System
+
+From the software viewpoint solely, and without referring to any specific application, we assume here that all detected failures (default safe values of the control outputs from the computer) do not prevent the mission to continue and are in this sense benign, whereas undetected failures are conservatively assumed to have a ”catastrophic” effect on the controlled system. Obviously, if knowledge of the consequences of software failures on the system was available for a specific system, the proper splitting of software failures into benign and catastrophic could be precisely made. We make the following assumption to model the system.
+Suppose \(SS\) is a state where the total system, that is all the \(N\) modules, runs without any fault of either kind. Let \(B_{i}\) be the state where the system is running in benign failure of \(i\)th level, that is after \(i\) iterations the system will enter in the state \(SS\). As the previous subsection \(S\) and \(F\) denotes the successful completion of the mission and completion of the mission with a failure respectively. The mission will fail if their is a catastrophic failure in the system. Let us also assume that if there is a benign failure of length greater than a threshold value, say \(n_{c}\), then the system will enter in a catastrophic failure region. Although this assumption will take the model a little away from reality, a model should be good enough to handle a benign failure of any arbitrary random length, but this assumption will make the calculation of reliability expression easier which will increase its practical application. At this point note that the state \(S\), that is the successful completion of the program, can be achieved only from the state \(SS\), where as the state \(F\) can be achieved from any of the state \(SS\) or \(B_{i}\)’s \((\forall i=1(1)N)\), but we assume here the control will be transferred from the state \(B_{i}\) to \(B_{i-1}\) only to reduce the number of parameters in the model.
+The transition probability matrix will be as follows
+
+\[Q=\left(\begin{array}[]{ccccccccc}Q_{00}&Q^{b}_{01}&Q^{b}_{02}&...&Q^{b}_{0(n_{c}-2)}&Q^{b}_{0(n_{c}-1)}&Q^{b}_{0n_{c}}&S^{0}&F^{0}\\ Q^{b}_{10}&O&O&...&O&O&O&\bar{0}&\bar{0}\\ O&Q^{b}_{21}&O&...&O&O&O&\bar{0}&\bar{0}\\ ...&...&...&...&...&...&...&...&...\\ O&O&O&...&Q^{b}_{(n_{c}-1)(n_{c}-2)}&O&O&\bar{0}&\bar{0}\\ O&O&O&...&O&Q^{b}_{n_{c}(n_{c}-1)}&O&\bar{0}&\bar{0}\\ \bar{0}^{\prime}&\bar{0}^{\prime}&\bar{0}^{\prime}&...&\bar{0}^{\prime}&\bar{0}^{\prime}&\bar{0}^{\prime}&1&0\\ \bar{0}^{\prime}&\bar{0}^{\prime}&\bar{0}^{\prime}&...&\bar{0}^{\prime}&\bar{0}^{\prime}&\bar{0}^{\prime}&0&1\\ \end{array}\right)\] (22)
+
+Here the matrix \(Q_{00}\) is a \(N\times N\) matrix which describes that the flow is running without entering in benign failure or catastrophic failure. The matrix \(Q^{b}_{0k}\) is also a \(N\times N\) matrix giving the transition probabilities of the flow of control from stable state to the \(k\)th level benign failure \((\forall k=1(1)n_{c})\). Similarly, the matrix \(Q^{b}_{kl}\) which is also \(N\times N\) denotes the transition probabilities of the control entering from the \(k\)th level benign failure to \(l\) th level \((\forall k=1(1)n_{c}\forall l=1(1)n_{c})\). From the \(k\)th level benign failure we can only achieve the \(k-1\)th level benign failure so \(Q^{b}_{kl}=O\) (\(\forall l\neq k-1\)). Where \(O\) is the null matrix of order \(N\times N\). \(S^{0}\) is a \(N\times 1\) vector of the transition probabilities of successful completion of the mission from the stable state. As the mission can terminate successfully only from the stable state so the rest of the entries in this column are all zero. \(\bar{0}\) denotes a null vector of length \(N\) and \(\bar{0}^{\prime}\) denotes transpose of \(\bar{0}\). Finally, \(F^{0}\) is a column vector of length \(N\) giving probabilities of reaching the state of catastrophic failure from the stable state.
+To give the structure of sub-matrices \(Q_{00}\), let us define \(p^{SS}_{ij}\) be the probability of the control to enter from the \(i\)th module to \(j\)th module in the state \(SS\). So the matrix \(Q_{00}\) is given by
+
+\[Q_{00}=\left(\begin{array}[]{cccc}p^{SS}_{11}&p^{SS}_{12}&...&p^{SS}_{1N}\\ p^{SS}_{21}&p^{SS}_{22}&...&p^{SS}_{2N}\\ ...&...&...&...\\ p^{SS}_{N1}&p^{SS}_{N2}&...&p^{SS}_{NN}\\ \end{array}\right)\] (27)
+
+Let us also define \(p^{SB}_{ij}\) be the probability that the control will be transferred from the module \(i\) to the module \(j\) from the state \(SS\) to any of benign failure. Let also \(p^{B}_{k}\) the probability that the control will enter in \(B_{k}\), thus the probability that the control will enter in the \(j\)th module from the \(i\)th module in the state \(B_{k}\) is given by \(p^{SB}_{ij}p^{B}_{k}\). So the matrix \(Q^{b}_{0k}\) will take the following form
+
+\[Q^{b}_{0k}=\left(\begin{array}[]{cccc}p^{SB}_{11}p^{B}_{k}&p^{SB}_{12}p^{B}_{k}&...&p^{SB}_{1N}p^{B}_{k}\\ p^{SB}_{21}p^{B}_{k}&p^{SB}_{22}p^{B}_{k}&...&p^{SB}_{2N}p^{B}_{k}\\ ...&...&...&...\\ p^{SB}_{N1}p^{B}_{k}&p^{SB}_{N2}p^{B}_{k}&...&p^{SB}_{NN}p^{B}_{k}\\ \end{array}\right)\] (32)
+
+If \(p_{iS}\) and \(p_{iF}\) is respectively the successful completion of the mission and achieving catastrophic failure from the \(i\)th module. Then we must have
+
+\[\sum_{j=1}^{N}p^{SS}_{ij}+\sum_{k=1}^{n_{c}}p^{B}_{k}\sum_{j=1}^{N}p^{SB}_{ij}+p_{iS}+p_{iF}=1\hskip 21.68121pt\forall i=1(1)N\] (33)
+
+The matrix \(Q^{b}_{kk-1}\) takes the following form
+
+\[Q^{b}_{kk-1}=\left(\begin{array}[]{cccc}p^{bb}_{11}&p^{bb}_{12}&...&p^{bb}_{1N}\\ p^{bb}_{21}&p^{bb}_{22}&...&p^{bb}_{2N}\\ ...&...&...&...\\ p^{bb}_{N1}&p^{bb}_{N2}&...&p^{bb}_{NN}\\ \end{array}\right)\] (38)
+
+Here we have
+
+\[\sum_{j=1}^{N}p^{bb}_{ij}=1\hskip 21.68121pt\forall i=1(1)N\] (39)
+
+Finally, the matrix \(Q^{b}_{10}\) is the matrix of transition probabilities, say \(p^{bS}_{ij}\), that the flow of control will be transferred from the \(i\)th to the \(j\)th module and from the \(B_{1}\) to \(SS\). Here also
+
+\[\sum_{j=1}^{N}p^{bS}_{ij}=1\hskip 21.68121pt\forall i=1(1)N\] (40)
+
+By assuming as before the first module as the control module the expression of \(\pi_{t}(x)\) is given
+
+\[\pi_{t}(x)=\sum_{i=1}^{N}(I_{Nn_{c}}-\hat{Q})^{-1}_{1i}p_{iS}\] (41)
+
+where \(\hat{Q}\) is once again the sub-matrix of \(Q\) deleting its last two columns and rows.
+
+## 4 Conclusions
+
+In this work we have given an expression of the reliability of a software system which can be divided in a finite number of modules. The transition probabilities we have considered can be easily estimated using maximum likelihood method of estimation.
+Consider the setup without benign failure, suppose \(i\)th block is tested \(n_{i}\) times, out of which \(x^{i}_{j}\) times the control is transferred to the \(j\)th state \((\forall i=1(1)N\&\forall j=1(1)N,S,F)\). The maximum likelihood estimates of \(p_{ij}\) is \(x^{i}_{j}/(\sum_{i=1}^{N}x^{i}_{j}+x^{i}_{S})\) and that of \(\alpha^{x}_{i}(t)\) is \(x^{i}_{F}/n_{i}\). Hence estimate of \(\pi_{t}(x)\) can be obtained and let it be denoted by \(\hat{\pi}_{t}(x)\). Finally the estimate of reliability of a system can be given by
+
+\[\hat{R_{t}}=\frac{1}{|W|}\sum_{x\in W}\hat{\pi}_{t}(x)\] (42)
+
+where \(W\) is the set of all inputs which are used for testing. This is an extension of some previous work and the model what we have considered are more realistic for some control software which are designed to detect its own internal errors and then issue a safe output and reset itself to a known state from which the program is likely to proceed correctly.
+
+## References
+
+* [1] A.Bertolino and L.Strigini (1996). On The Use of Testability Measures for Dependability Assessment. IEEE Trans on Software Engineering 22(2):97-108.
+* [2]C. Beightler and D.T. Phillips (1976). Applied Geometric Programming. John Wiley & Sons,Inc., New York.
+* [3] E.Cinlar (1975). Introduction to Stochastic Processes. Prentice-Hall, Inc., Englewood Clios, N.J.
+* [4]E.C.Soistman and K.B.Ragsdale (1984). Combined Hardware/Software Reliability Prediction Methodology. Rome Air Development Center Contract Report OR. 18 - 173, Vol.2.
+* [5] E.Parzen. (1962), Stochastic Processes, Holden-Day, San Francisco, Calif.
+* [6] J.H.Poore, H.D.Mills and D.Mutchler (1993). Planning and Certifying Software System Reliability. IEEE Software, 88-99.
+* [7] J.M.Voas and K.W.Miller (1995). Software Testability: The new verification. IEEE Software. 17-28.
+* [8]J.Rajgopal and D.L.Bricker (1995). An Algorithm for Solving The Polynomial GP Problem, Based on Generalized Programming. Department of Industrial Engineering, University of Pittsburgh, Technical Report No.TR95-10.
+* [9] J.Rajgopal and M.Mazumdar (1995). Designing Component Test Plans for Series System Reliability via Mathematical Programming. Technometrics 37,195-212.
+* [10] J.Rajgopal and M.Mazumdar (1997). Minimum Cost Component Test Plans for evaluating Reliability of a Highly Reliable Parallel System. Naval Research Logistics 44, 401-418.
+* [11]K.S.Al-Sultan, M.F.Hussain and J.S.Nizami (1996). A Genetic Algorithm for The Set Covering Problem. The Journal of the Operational Research Soc. 47, 5, 702-709.
+* [12]K.Siegrist(1998). Reliability of Systems with Markov Transfer of Control. IEEE Transaction on Software Engineering 14, 1049-1053.
+* [13]K.W.Miller, L.J.Morrel, R.E.Noonan, S.K.Park, D.M.Nicol, B.M.Murril and J.M.Voas (1992) Estimating the probability of failure when testing reveals no failure. IEEE Trans. on Software Engineering 18(1): 33-43.
+* [14]Mark C.K. Yang, W.Eric Wong, Alberto Pasquini (1998). Applying Testability to Reliability Estimation, Proc. of IEEE International Symposium on Software Reliability Engineering, Puderborn 90-99.
+* [15] M.Avriel,R.S.Dembo and U.Plassy (1975). Solution of Generalized Geometric Programs. International Journal for Numerical methods in Engineering 9, 149-168.
+* [16]R.C.Cheung (1980). A User-Oriented Reliability Model. IEEE Trans. Software Engineering, SE- 6(2): 118-125.
+* [17]R.C.Easterling, M.Mazumdar, F.W.Spencer and K.V.Diegert (1991). System Based Component Test Plan and Operating Characteristics: Binomial Data Technometrics 33,287- 298.
+* [18]S.Gal (1974). Optimal Test Design for Reliability Demonstration. Operational Research 22, 1236-1242.
+* [19]S.Ghosh, A.P.Mathur, J.R.Horgan, J.J.Li, W.E.Wong(1997). Software Fault Injection Testing on a Distributed System - A Case Study, Proc. of the 1st International Quality Week Europe, Brussels, Belgium.
+* [20] S.Wolfram(1996). The Mathematics Book (3rd edition). Cambridge University Press and Wolfram Media, Inc.Champaign, 3.
+* [21]W.Kuo (1992). Software Reliability. Maynards Industrial Engineering Handbook, 4th edition(W.K.Hodson, Editor-in-Chief), 11116-11122.
+* [22]W.Eric Wong, J.R.Horgan, S.London and Aditya P.Mathur (1998). Effect of test set minimization on Fault Detection Effectiveness. Software-Practice and Experience, 28(4): 347-369.
+

arxiv_ground_truth/0711.0528.md

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+# Web-based Interface in Public Cluster
+
+_Z. Akbar and L.T. Handoko_
+Group for Theoretical and Computational Physics, Research Center for Physics, Indonesian Institute of Sciences
+
+ _zaenal@teori.fisika.lipi.go.id_ _handoko@teori.fisika.lipi.go.id_ http://teori.fisika.lipi.go.id
+
+###### Abstract
+
+A web-based interface dedicated for cluster computer which is publicly accessible for free is introduced. The interface plays an important role to enable secure public access, while providing user-friendly computational environment for end-users and easy maintainance for administrators as well. The whole architecture which integrates both aspects of hardware and software is briefly explained. It is argued that the public cluster is globally a unique approach, and could be a new kind of e-learning system especially for parallel programming communities.
+
+## 1 Introduction
+
+LIPI Public Cluster (LPC) is a cluster-based computing facility maintained by Lembaga Ilmu Pengetahuan Indonesia - LIPI (the Indonesian Institute of Sciences) [1]. Although it is still a small scale cluster in the sense of number of nodes already installed, it has unique characteristics among existing clusters around the globe due to its openness. Here ”open” means everyone can access and use it anonymously for free to execute any types of parallel programmings [2].
+
+The development of LPC was initially motivated by real needs for high performance and advanced computing, especially in the research field of basic natural sciences. Even in Indonesia, the needs are growing along with the advances of scientific researches. In the last decades, clustering low specs (and low cost) machines becomes popular to realize an advanced computing system comparable to, or in most cases better than the conventional mainframe-based system with significant cost reduction [3].
+
+In general a cluster is designed to perform a single (huge) computational task at certain period. This makes the cluster system is usually exclusive and not at the level of appropriate cost for most potential users, neither young beginners nor small research groups, especially in the developing countries like Indonesia. It is clear that the cluster is in that sense still costly, although there are certain needs to perform such advanced computings. No need to say about educating young generations to be the future users familiar with parallel programmings. This background motivates us to further develop an open and free cluster environment for public [4].
+
+According to its nature LPC is, in contrast with any conventional clusters, designed to accommodate multiple users with their own parallel programmings executed independently at the same period. Therefore an issue on resource allocation is crucial, not only in the sense of allocating hardwares to the appropriate users but also to prevent any interferences among them. In LPC we have deployed a new algorithm to overcome this problem, namely the dependent [5] and independent multi-block approaches [6].
+
+Concerning its main objective as a training field to learn parallel programmings, the public cluster should be accessible and user-friendly for all users with various level of knowledges on parallel programming. It also should have enough flexibility regarding various ways of accessing the system in any platforms as well. This can be achieved by deploying web-based interfaces in all aspects. Presently we have resolved some main issues, such as security from anonymous users to prevent any kinds of interference among different tasks running simultaneously on multi blocks [5, 6], algorithm for resource allocation management [7] and the real-time monitoring and control over web for both administrators and end-users [8].
+
+In this paper we first present briefly the concept of LPC including the work flow from the initial registration to the execution of computational works. Thereafter we discuss the main part of this paper, that is the architecture of web-interface in LPC. Finally we conclude with some comments and discussion.
+
+## 2 The concept and work flow
+
+In order to overcome the issues mentioned in the preceeding section, it is clear that we should develope and utilize an integrated web-based interface in LPC. However, concerning real demands in Indonesia presently, we assume that our potential users would be beginners in parallel programmings who are going to use it moreless for educational or self-learning purposes. Although the cluster is also going to be used by some experts and larger research groups to perform more serious advanced computings, we do not expect any anonymous users with heavy computational works. The reason is mainly because of the limited resources, i.e. number of nodes and its specifications. Actually we rather prefer as many as people use our cluster as a training field to learn parallel programmings. In average we plan to provide only \(2\sim 4\) nodes in a relatively short period, namely less than 3 days, for each anonymous user.
+
+These characteristics completely differ with another existing clusters around the world. Because they are usually used by certain people or groups bringing similar types of computational works which are appropriate for the cluster after particular procedures like submitting proposals and any kinds of letter of agreements. In our case, incoming users are completely anonymous and then there is no way in any means to know the type of computational works being executed in the cluster. This fact consequently brings another problems as maintaining job executions owned by different users at same period simultaneously. Under these assumptions and conditions, the public cluster should fulfill very challenging requirements that might be irrelevant in any conventional clusters, that is :
+
+* •Security :
+This is the main issue we should encounter from the first. The users should have enough priviledges and freedom to optimize their opportunities to learn parallel programming, while their access must be limited at the maximum level for the sake of security of the whole system and another active users at the same period.
+* •Flexibility :
+It is impossible to provide the same level of flexibility for anonymous users as well-defined users with direct accesses through ssh, etc, but we should allow as much as possible the users to execute their codes on LPC. Also there should be a freedom on assigning the number of nodes for each user, since each user could require various number of nodes depending on the computational capacities they actually need.
+* •Stability :
+Simultaneous executions by different users with various programmes in different blocks of cluster without any interferences among them requires new innovations on cluster management. This problem includes some techniques to incorporate wide range of nodes with different specifications, that is ranging from Intel 486 to the latest Athlon based nodes.
+* •Efficiency :
+Since the cluster is dynamically divided into several blocks with various number of nodes inside according to the users requests, each node should be able to be completely turned on or off partially without any interruptions to another working nodes.
+
+All of these require modifications on some existing tools, and also new developments on alternative softwares and hardwares as well [4, 5, 6, 7, 8].
+
+Regarding to the above-mentioned concept, we have defined the work flow in LPC as follows :
+
+1. 1.A new user should complete an initial registration by providing personal data, the content of job will be performed and the number of nodes requested for the job.
+2. 2.The application is reviewed and verified by the administrator. After approval, the administrator will assign the nodes provided and its usage period.
+3. 3.After reconfirmations by user, proving their agreements with the provided nodes and usage period, the administrator will switch the nodes on that also activate all daemons automatically.
+4. 4.The users should adjust their parallel programmes to fit the provided nodes.
+5. 5.The user uploads all necessary programmes and libraries if any. At this stage the programmes can be executed immediately.
+6. 6.The administrator and automated system will monitor the usage of all running users.
+7. 7.Finished jobs and the results can be downloaded by owners.
+8. 8.Once the usage period is over, the nodes are turned off automatically.
+
+Again, these procedures can be done fully and remotely through web. We should mention that the types and number of nodes allocated for a newly assigned block is done automatically utilizing a decision making tool based on the extended genetic algorithm embedded in the web-interface [7]. Now we are ready to discuss the architecture of web-interface in LPC.
+
+## 3 The architecture of web-interface
+
+**Figure 1**: The communication architecture in LPC.
+
+As mentioned earlier, the web-interface takes over all aspects in LPC, from communication among the nodes till running related tools and softwares to perform computational works. It is also the only way connecting the internal system with the rest of the world.
+
+In term of communication, there are two kinds of communication in LPC : a) between users and gateway server through HTTP protocol, and b) between gateway server and nodes consisting blocks of cluster through ssh or scp. This architecture is intended to limit direct access to a block of cluster for security reason. On the other hand, this still allows the system to interact with another elements of cluster as IO server and master nodes of each block. As depicted in Fig. 1, we put a gateway server as a common entrance for users which serves all related web-based applications. The gateway server further sends and receives commands related to file or parallel programming operations using secure remote login ssh / scp.
+
+**Figure 2**: The architecture of three-tier web-based interface in LPC.
+
+The web application in LPC should accommodate the needs of any parallel programmings which vary and change along the time. Then we must make it as flexible as possible to guarantee its compatibilities in the future. Also, any future modifications in the web application must not alter another software components. This can be achieved by adopting three-tier software architecture [9]. This method is appropriate for distributed client / server applications which require high performances in terms of flexibility, maintainability, reusability and scalability, while at the same time it simplifies the complexities of users’ distributed processes.
+
+This architecture is shown in Fig. 2. As can be seen in the figure, each layer is completely separated which could then simplifies its developments and implementations. Presentation layer behaves as a front interface and interacts directly by receiving inputs from users. The HTML files in this layer could contain either static or dynamic (through application and logic layer) contents. For instance in the case of LPC they are generated dynamically by Python scripts.
+
+The application and logic layer handles all programming logics to serve user needs as uploading the jobs, executing codes and so forth. Further, the data access layer provides the access to data files from the database system, configuration files or generated by any commands in operating system.
+
+**Figure 3**: The flow of commands between nodes and gateway server in LPC.
+
+Next, the main task for web-interface in LPC is providing the users an appropriate environment for parallel programming. At time being, there are several parallel programming environments like MPI, PVM and so on [10]. Also some modified versions of them as MPICH (version 1 and 2) and LAM-MPI. The web-interface should allow users to choose the preferred one, or to change from one to another. In order to enable dynamic change of environment according to user preferences, we have deployed a mechanism based on the Modules package [11]. This package enables environment switching by changing its path and variables through a single command.
+
+Further issue is how to keep the whole performance of cluster during operation. This can be done by embedding ”real-time” control and monitoring system and displaying the results through web. We have developed a dedicated device for this purpose using microcontroller-based hardwares [8]. The control system manages the hardware aspect of each node, for instance turning them on or off. The microcontrollers are interfaced by Python and some external programmes in C++. The monitoring system takes the data of external physical observables such casing temperatures and humidities. The physical parameters for each node can be retrieved easily through BIOS. Both control and monitoring systems are integrated to realize an automated hardware control, i.e. shuting down a node if the temperature exceeds the predefined threshold, etc.
+
+Beside these hardware-based management systems, we have also implemented the software-based management system to control all nodes more efficiently. We have deployed a tool set called Cluster Command and Control (C3) [12] and Ganglia [13]. C3 bundles useful commands to enable automatic processes throughout multiple nodes, while Ganglia is used to monitor the current condition of cluster through web-interface. We have integrated these tools in our web-interface to create a more comprehensive web-based interface matches specific needs in LPC.
+
+Lastly, we should comment on the communication protocol among the nodes in LPC. All commands are sent to nodes through secure shell login ssh / scp passwordlessly using RSA / DSA keys. This handy solution can be used for any commands, either with or without standard outputs. The flow of commands between nodes and user through gateway server is diagrammatically given in Fig. 3
+
+## 4 Conclusion
+
+We have introduced the web-interface in LPC which integrates all aspects of its hardware and software. Although web-based interfaces in clusters are commonly known and actively developed by many groups, the web-interface for public clusters is quite unique and requires more careful developments. Moreover, in any existing conventional clusters, the web-interfaces are moreless complementary tools. While in public clusters like LPC, the web-interface is very crucial and the only solution to make it open for public.
+
+We would like to comment some points regarding the current status of LPC :
+
+* •In some specific cases, the work-flow in LPC can be automated to reduce administration works. For instance, it could be realised for a “public” cluster limited to a community with predefined users and types of computational works. However, in our case the present work flow is the simplest and most “automated” one regarding the security and its level of openness.
+* •Related to this, the main issue in daily operation is to prevent in advanced malicious users who try to submit huge never-ending computational tasks. Actually, this is themain reason we keep current work-flow which is quite powerfull to deal with this potential problem.
+* •At time being, the computational results should be downloaded by users. Although it is also possible to make a real-time monitoring for on-going works, we are not going to implement such services in the future. Because it is irrelevant in parallel programmings which could normally take few days computational time. Another reason is such services could exhaust the limited resources of the system, then it is not worthy compared to its importances.
+* •The usage time provided for an approved user is determined based on the their request since we do not confirm their code in detail. However, as our basic policy, we provide in most only several days usage time for each anonymous user in one new request.
+* •We have implemented our own resource allocation algorithm based on the extended genetic algorithm embedded in the web-interface [7]. This is different with the conventional cluster where the resource allocation should be performed during the computational period to achieve the best performance. In the public cluster like LPC, the conventional resource allocation is irrelevant, because it is done only at the first time of node assignment after the approval procedure. Once a block consisting of some resources has been allocated, it remains unchanged during the usage period. Of course, resource allocation within a block is still relevant, but it uses the existing tools chosen by users. Moreover, doing an appropriate load-balancing within a block used by user is an important aspect of learning parallel programming, so this task should be done by the users themself.
+* •The LPC so far enjoys numerous visits and many active users as can be seen in its web [1], ranging from high school to college students. This fact has definitely proven the usability of the system by (mostly) beginners in parallel programmings.
+* •The current system is in principle portable to other existing clusters. Especially the concept could be an interesting alternative for some old clusters that might still be usefull for public educational purposes.
+
+As future work, we are going to investigate more comprehensively the overall performances, especially in the case of simultaneous heavy computational jobs of multiple users. However, the current performances have been reported partially in our previous works [5, 6, 7]. Also following recent progress in high performance computing, we are in the step of connecting some blocks in LPC into grid computing.
+
+## Acknowledgement
+
+This work is financially supported by the Riset Kompetitif LIPI in fiscal year 2007 under Contract no. 11.04/SK/KPPI/II/2007 and the Indonesia Toray Science Foundation Research Grant 2007.
+
+## References
+
+* [1] LIPI Public Cluster, http://www.cluster.lipi.go.id.
+* [2] L.T. Handoko : ”Public Cluster : mesin paralel terbuka berbasis web”, Indonesian Copyright, No. B 268487 (2006).
+* [3] A. Goscinski, M. Hobbs and J. Silcock : ”A Cluster Operating System Supporting Parallel Computing”, Cluster Computing, Vol. 4, pp. 145–156 (2001).
+* [4] Z. Akbar, Slamet, B.I. Ajinagoro, G.J. Ohara, I. Firmansyah, B. Hermanto and L.T. Handoko : ”Open and Free Cluster for Public”, Proceeding of the International Conference on Rural Information and Communication Technology 2007, Bandung, Indonesia,2007.
+* [5] Z. Akbar, Slamet, B.I. Ajinagoro, G.J. Ohara, I. Firmansyah, B. Hermanto and L.T. Handoko : ”Public Cluster : parallel machine with multi-block approach”, Proceeding of the International Conference on Electrical Engineering and Informatics, Bandung, Indonesia, 2007.
+* [6] Z. Akbar and L.T. Handoko : ”Multi and Independent Block Approach in Public Cluster”, Proceeding of the 3rd Information and Communication Technology Seminar, Surabaya, Indonesia, 2007.
+* [7] Z. Akbar and L.T. Handoko : ”Resource Allocation in Public Cluster with Extended Optimization Algorithm”, Proceeding of the International Conference on Instrumentation, Communication, and Information Technology, Bandung, Indonesia, 2007.
+* [8] I. Firmansyah, B. Hermanto, Slamet, Hadiyanto and L.T. Handoko : ”Real-time control and monitoring system for LIPI Public Cluster”, Proceeding of the International Conference on Instrumentation, Communication, and Information Technology, Bandung, Indonesia, 2007.
+* [9] J. Foreman, J. Gross, R. Rosenstein, D. Fisher, K. Brune : ”C4 Software Technology Reference Guide - A Prototype”, Handbook (CMU/SEI-97-HB-001), Software Engineering Institute, Carnegie Mellon University, January 1997.
+* [10] For examples, see :
+Message Passing Interface Forum : ”MPI2: A Message Passing Interface standard”. International Journal of High Performance Computing Applications, 12(1–2):1–299, 1998.
+* [11] John L. Furlani ‘: ”Modules: Providing a Flexible User Environment”, Proceedings of the Fifth Large Installation Systems Administration Conference (LISA V), pp. 141-152, San Diego, CA, September 30 - October 3, 1991. http://modules.sourceforge.net.
+* [12] Ray Flanery, Al Geist, Brian Luethke and Stephen L. Scott : ”Cluster Command & Control (C3) Tool Suite”, 3rd Austrian-Hungarian Workship on Distributed and Parallel Systems (DAPSYS 2000) in conjunction with EuroPVM/MPI 2000, Balatonfured, Lake Balaton, Hungary, September 10-13, 2000. http://www.csm.ornl.gov/torc/C3/.
+* [13] Ganglia Monitoring System, http://ganglia.sourceforge.net.
+

arxiv_ground_truth/0711.3236.md

@@ -0,0 +1,377 @@
+**Confidence Intervals in Regression Utilizing Prior Information**
+
+**Paul Kabaila∗, Khageswor Giri**
+
+Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
+
+**Abstract**
+
+We consider a linear regression model with regression parameter \(\beta=(\beta_{1},\ldots,\beta_{p})\) and independent and identically \(N(0,\sigma^{2})\) distributed errors. Suppose that the parameter of interest is \(\theta=a^{T}\beta\) where \(a\) is a specified vector. Define the parameter \(\tau=c^{T}\beta-t\) where the vector \(c\) and the number \(t\) are specified and \(a\) and \(c\) are linearly independent. Also suppose that we have uncertain prior information that \(\tau=0\). We present a new frequentist \(1-\alpha\) confidence interval for \(\theta\) that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard \(1-\alpha\) confidence interval when the data strongly contradicts this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when \(\tau=0\). This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about \(\tau\) is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a \(2\times 2\) factorial experiment with 20 replicates. Suppose that the parameter of interest \(\theta\) is a specified _simple_ effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for \(\theta\) that utilizes this prior information.
+
+_Keywords:_ Frequentist confidence interval; Prior information; Linear regression.
+
+∗Corresponding author. Tel.: +61 3 9479 2594, fax: +61 3 9479 2466.
+
+E-mail address: P.Kabaila@latrobe.edu.au (Paul Kabaila).
+
+**1. Introduction**
+
+Consider the linear regression model \(Y=X\beta+\varepsilon\), where \(Y\) is a random \(n\)-vector of responses, \(X\) is a known \(n\times p\) matrix with linearly independent columns, \(\beta=(\beta_{1},\ldots,\beta_{p})\) is an unknown parameter vector and \(\varepsilon\sim N(0,\sigma^{2}I_{n})\) where \(\sigma^{2}\) is an unknown positive parameter. Suppose that the parameter of interest is \(\theta=a^{T}\beta\) where \(a\) is specified \(p\)-vector (\(a\neq 0\)). Define the parameter \(\tau=c^{T}\beta-t\) where the vector \(c\) and the number \(t\) are specified and \(a\) and \(c\) are linearly independent. Also suppose that previous experience with similar data sets and/or expert opinion and scientific background suggest that \(\tau=0\). In other words, suppose that we have uncertain prior information that \(\tau=0\). Of course, this includes the particular case that \(c=(0,\ldots,0,1)\) and \(t=0\), so that the uncertain prior information is that \(\beta_{p}=0\). Our aim is to find a frequentist \(1-\alpha\) confidence interval (i.e. a confidence interval whose coverage probability has infimum \(1-\alpha\)) for \(\theta\) that utilizes this prior information, based on an observation of \(Y\).
+
+An attempt to incorporate the uncertain prior information that \(\tau=0\) into the construction of a \(1-\alpha\) confidence interval for \(\theta\) is as follows. We carry out a preliminary test of the null hypothesis that \(\tau=0\) against the alternative hypothesis that \(\tau\neq 0\). If this null hypothesis is accepted then the confidence interval is constructed assuming that it was known _a priori_ that \(\tau=0\); otherwise the standard \(1-\alpha\) confidence interval for \(\theta\) is used. We call this the naive \(1-\alpha\) confidence interval for \(\theta\). This confidence interval is based on a false assumption and so we expect that its minimum coverage probability will not necessarily be \(1-\alpha\). This minimum coverage probability has been investigated by Giri and Kabaila (2008), Kabaila (1998, 2005a), Kabaila and Giri (2009a) and Kabaila and Leeb (2006). In many cases this minimum is far below \(1-\alpha\), showing that this confidence interval is completely inadequate. So, the naive \(1-\alpha\) confidence interval fails to utilize the prior information that \(\tau=0\).
+
+Whilst the naive \(1-\alpha\) confidence interval for \(\theta\) fails abysmally to utilize the prior information that \(\tau=0\), its form (as described in Section 2) will be used to provide some motivation for the new confidence interval described in Section 3. Similarly to Hodges and Lehmann (1952), Bickel (1983, 1984), Kabaila (1998), Kabaila (2005b), Farchione and Kabaila (2008), Kabaila and Tuck (2008) and Kabaila and Giri (2009b), our aim is to utilize the uncertain prior information in the frequentist inference of interest, whilst providing a safeguard in case this prior information happens to be incorrect. We assess a \(1-\alpha\) confidence interval for \(\theta\) using the ratio (expected length of this confidence interval)/(expected length of standard \(1-\alpha\) confidence interval). We call this ratio the scaled expected length of this confidence interval. In Section 3 we describe a new \(1-\alpha\) confidence interval for \(\theta\) that utilizes the prior information. This interval has endpoints that are continuous functions of the data and it has the following properties. It coincides with the standard \(1-\alpha\) confidence interval when the data strongly contradicts the prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when \(\tau=0\). This minimization leads to an interval that has the following desirable properties. This interval has scaled expected length that (a) is smaller than 1 when the prior information about \(\tau\) is correct and (b) has a maximum value that is not too much larger than 1. The idea of minimizing a weighted average expected length of a confidence interval, subject to a coverage probability inequality constraint, appears to have been first used by Pratt (1961).
+
+In Section 4 we consider the following scenario. Suppose that a \(2\times 2\) factorial experiment, with factors labeled A and B and with more than 1 replicate, has been conducted. Also suppose that our interest is solely in the _simple_ effect of changing factor A from low to high when factor B is low. Consider, for example, the case that factor A (B) being low or high corresponds to the absence or presence of treatment A (B), respectively. Our interest may be solely in the effect of treatment A compared to no treatment (cf. Hung et al (1995)). In other words, the parameter of interest \(\theta\) is the _simple_ effect (expected response when factor A is high and factor B is low) \(-\) (expected response when factor A is low and factor B is low). In this case, \(p=4\) and we identify \(\tau\) with the two-factor interaction. Suppose that previous experience with similar data sets and/or expert opinion and scientific background suggest that the two-factor interaction is zero. In a \(2\times 2\) factorial clinical trial comparing two drugs whose presumed effects are on completely different systems and/or diseases, it seems reasonable to suppose that we have uncertain prior information that the two-factor interaction is zero (Stampfer et al (1985), Steering Committee of the Physicians’ Health Study Research Group (1988)), Buring and Hennekens (1990) and Hung et al (1995)). For an example of the elicitation of uncertain prior information in a factorial experiment via expert opinion and scientific background in a chemical context see Dubé et al (1996).
+
+An attempt to utilize the uncertain prior information that the two-factor interaction is zero is to use a naive \(1-\alpha\) confidence interval for \(\theta\) constructed using the following preliminary test. The preliminary test is of the null hypothesis that the two-factor interaction is zero against the alternative hypothesis that the two-factor interaction is non-zero. This confidence interval has a minimum coverage probability that is far below \(1-\alpha\), showing that it is completely inadequate. As an illustration, consider the case that the number of replicates is 20, \(1-\alpha=0.95\) and the preliminary hypothesis test has level of significance 0.05. We find, using the methodology of Kabaila (1998, 2005a) or Giri and Kabaila (2008) or Kabaila and Giri (2009a), that the minimum coverage probability of this confidence interval is 0.7306. The poor coverage properties of the naive confidence interval are presaged by the poor properties of some other inferences carried out after this preliminary test, see Fabian (1991), Shaffer (1991) and Ng (1994) (cf. Neyman (1935), Bohrer and Sheft (1979) and Traxler (1976)).
+
+The properties of the new confidence interval, described in Section 3, are illustrated in Section 4 by a detailed analysis of the \(2\times 2\) factorial experiment example with 20 replicates and \(1-\alpha=0.95\). Define the parameter \(\gamma=\tau/\sqrt{\text{var}(\hat{\tau})}\), where \(\hat{\tau}\) denotes the least squares estimator of \(\tau\). As proved in Section 3, the coverage probability of the new confidence interval for \(\theta\) is an even function of \(\gamma\). The top panel of Figure 3 is a plot of the coverage probability of the new 0.95 confidence interval for \(\theta\) as a function of \(\gamma\). This plot shows that the new 0.95 confidence interval for \(\theta\) has coverage probability 0.95 throughout the parameter space. As proved in Section 3, the scaled expected length of the new confidence interval for \(\theta\) is an even function of \(\gamma\). The bottom panel of Figure 3 is a plot of the square of the scaled expected length of the new 0.95 confidence interval for \(\theta\) as a function of \(\gamma\). When the prior information is correct (i.e. \(\gamma=0\)), we gain since the square of the scaled expected length is substantially smaller than 1. The maximum value of the square of the scaled expected length is not too large. The new 0.95 confidence interval for \(\theta\) coincides with the standard \(1-\alpha\) confidence interval when the data strongly contradicts the prior information. This is reflected in Figure 3 by the fact that the square of the scaled expected length approaches 1 as \(\gamma\rightarrow\infty\).
+
+**2. The naive confidence interval**
+
+The naive \(1-\alpha\) confidence interval for \(\theta\) is constructed as follows. We carry out a preliminary test of the null hypothesis that \(\tau=0\) against the alternative hypothesis that \(\tau\neq 0\). If this null hypothesis is accepted then the confidence interval is constructed assuming that it was known _a priori_ that \(\tau=0\); otherwise the standard \(1-\alpha\) confidence interval for \(\theta\) is used. As noted in the introduction, this confidence interval will often have minimum coverage probability far below \(1-\alpha\), showing that it is completely inadequate. In this section we describe the naive confidence interval in a new form that will be used to provide some motivation for the new confidence interval described in the next section.
+
+Let \(\hat{\beta}\) denote the least squares estimator of \(\beta\). Let \(\hat{\Theta}\) denote \(a^{T}\hat{\beta}\) i.e. the least squares estimator of \(\theta\). Also, let \(\hat{\tau}\) denote \(c^{T}\hat{\beta}-t\) i.e. the least squares estimator of \(\tau\). Define the matrix \(V\) to be the covariance matrix of \((\hat{\Theta},\hat{\tau})\) divided by \(\sigma^{2}\). Let \(v_{ij}\) denote the \((i,j)\) th element of \(V\). The standard \(1-\alpha\) confidence interval for \(\theta\) is \(I=\big{[}\hat{\Theta}-t_{n-p,1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma},\quad\hat{\Theta}+t_{n-p,1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma}\big{]}\), where the quantile \(t_{m,a}\) is defined by \(P(T\leq t_{m,a})=a\) for \(T\sim t_{m}\) and \(\hat{\sigma}^{2}=(Y-X\hat{\beta})^{T}(Y-X\hat{\beta})/(n-p)\).
+
+The naive \(1-\alpha\) confidence interval for \(\theta\) is obtained as follows. The usual test statistic for testing the null hypothesis that \(\tau=0\) against the alternative hypothesis that \(\tau\neq 0\) is \(\hat{\tau}/(\hat{\sigma}\sqrt{v_{22}})\). Suppose that, for some given positive number \(q\), we fix \(\tau\) at 0 if \(|\hat{\tau}|/(\hat{\sigma}\sqrt{v_{22}})\leq q\); otherwise we allow \(\tau\) to vary freely. We use the notation \([a\pm b]\) for the interval \([a-b,a+b]\) (\(b>0\)). Also define \(\rho=v_{12}/\sqrt{v_{11}v_{22}}\). Note that \(\rho\) is the correlation between \(\hat{\Theta}\) and \(\hat{\tau}\) and so it satisfies \(-1\leq\rho\leq 1\). The naive \(1-\alpha\) confidence interval is as follows (Kabaila and Giri (2009a)). If \(|\hat{\tau}|/(\hat{\sigma}\sqrt{v_{22}})>q\) then this confidence interval is \(\big{[}\hat{\Theta}-t_{n-p,1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma},\quad\hat{\Theta}+t_{n-p,1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma}\big{]}\). If, on the other hand, \(|\hat{\tau}|/(\hat{\sigma}\sqrt{v_{22}})\leq q\) then this confidence interval is
+
+\[\left[\hat{\Theta}-\frac{v_{12}}{v_{22}}\hat{\tau}\,\pm\,t_{n-p+1,1-\frac{\alpha}{2}}\sqrt{\frac{(n-p)\hat{\sigma}^{2}+(\hat{\tau}^{2}/v_{22})}{n-p+1}}\sqrt{v_{11}-\frac{v_{12}^{2}}{v_{22}}}\right].\]
+
+This confidence interval can be expressed in the new form
+
+\[\bigg{[}\hat{\Theta}-\sqrt{v_{11}}\hat{\sigma}\,b\bigg{(}\frac{\hat{\tau}}{\hat{\sigma}\sqrt{v_{22}}}\bigg{)}\,\pm\,\sqrt{v_{11}}\hat{\sigma}\,s\bigg{(}\frac{|\hat{\tau}|}{\hat{\sigma}\sqrt{v_{22}}}\bigg{)}\bigg{]}\]
+
+where
+
+\[b(x)=\begin{cases}0&\ \ \ \ \ \text{for }\ \ \ |x|>q\\ \rho x&\ \ \ \ \ \text{for}\ \ \ \ |x|\leq q.\end{cases}\]
+\[s(x)=\begin{cases}t_{n-p,1-\frac{\alpha}{2}}&\text{for }\ \ \ x>q\\ t_{n-p+1,1-\frac{\alpha}{2}}\sqrt{1-\rho^{2}}\sqrt{\frac{n-p+x^{2}}{n-p+1}}&\text{for}\ \ \ \ 0<x\leq q.\end{cases}\]
+
+In Section 4 we will consider the example of a \(2\times 2\) factorial experiment with 20 replicates. Here \(p=4\). The parameter of interest \(\theta\) is the _simple_ effect (expected response when factor A is high and factor B is low) \(-\) (expected response when factor A is low and factor B is low). We identify \(\tau\) with the two-factor interaction, so that \(\rho=-1/\sqrt{2}=-0.7071068\). Suppose that we have uncertain prior information that the two-factor interaction is zero. Also suppose that we carry out a preliminary test of the null hypothesis that the two-factor interaction is zero against the alternative hypothesis that this interaction is non-zero. Let the level of significance of this test be 0.05, so that \(q=1.991673\). Figure 1 is a plot of the functions \(b\) and \(s\) for the resulting naive 0.95 confidence interval for \(\theta\). This confidence interval is completely inadequate, as its minimum coverage probability is 0.7306. It also has the unpleasant feature that its endpoints are discontinuous functions of the data.
+
+Figure 1: Plots of the functions \(b\) and \(s\) for the naive 0.95 confidence interval for the _simple_ effect \(\theta\) in the context of the \(2\times 2\) factorial experiment with 20 replicates. This confidence interval is based on a preliminary test of the null hypothesis that the two-factor interaction is zero against the alternative hypothesis that this interaction is non-zero, with level of significance 0.05.
+
+**3. New confidence interval utilizing prior information**
+
+In this section we describe a broad class of confidence intervals for \(\theta\). These confidence intervals are required to have endpoints that are smooth function of the data. They are also required to coincide with the standard \(1-\alpha\) confidence intervals when the data strongly contradict the prior information. We provide computationally convenient expressions for the coverage probability and the scaled expected length for confidence intervals from this class. These computationally convenient expressions were first described by Kabaila and Giri (2007a,b). We then describe a weight function for the difference ((scaled expected length of the confidence interval) \(-\) (scaled expected length of the standard \(1-\alpha\) confidence interval)). This weight function gives the largest weight to this difference when \(\tau=0\) i.e. when the prior information is correct. We find an interval that is optimal in the sense that it minimizes the weighted average of this difference subject to the constraint that it has minimum coverage probability \(1-\alpha\). Our choice of the weight function ensures that this interval utilizes the prior information.
+
+We introduce a confidence interval for \(\theta\) that is similar in form to the naive \(1-\alpha\) confidence interval, described in the previous section, but with a great “loosening up” of the forms that the functions \(b\) and \(s\) can take. Define the following confidence interval for \(\theta\)
+
+\[J(b,s)=\bigg{[}\hat{\Theta}-\sqrt{v_{11}}\hat{\sigma}\,b\bigg{(}\frac{\hat{\tau}}{\hat{\sigma}\sqrt{v_{22}}}\bigg{)}\,\pm\,\sqrt{v_{11}}\hat{\sigma}\,s\bigg{(}\frac{|\hat{\tau}|}{\hat{\sigma}\sqrt{v_{22}}}\bigg{)}\bigg{]}\] (1)
+
+where the functions \(b\) and \(s\) are required to satisfy the following restriction.
+
+Restriction 1
+\(b:\mathbb{R}\rightarrow\mathbb{R}\) is constrained to be an odd function and \(s:[0,\infty)\rightarrow[0,\infty)\).
+
+The motivation for restricting attention to this form of interval is provided by the new invariance arguments presented in Appendix A. We also require that the functions \(b\) and \(s\) satisfy the following restriction.
+
+Restriction 2
+\(b\) and \(s\) are continuous functions.
+
+This implies that the endpoints of the confidence interval \(J(b,s)\) are continuous functions of the data. Finally, we require the confidence interval \(J(b,s)\) to coincide with the standard \(1-\alpha\) confidence interval \(I\) when the data strongly contradict the prior information. The statistic \(|\hat{\tau}|/(\hat{\sigma}\sqrt{v_{22}})\) provides some indication of how far away \(\tau/(\sigma\sqrt{v_{22}})\) is from 0. We therefore require that the functions \(b\) and \(s\) satisfy the following restriction.
+
+Restriction 3
+\(b(x)=0\) for all \(|x|\geq d\) and \(s(x)=t_{n-p,1-\frac{\alpha}{2}}\) for all \(x\geq d\) where \(d\) is a (sufficiently large) specified positive number.
+
+Define \(\gamma=\tau/(\sigma\sqrt{v_{22}})\), \(G=(\hat{\Theta}-\theta)/(\sigma\sqrt{v_{11}})\) and \(H=\hat{\tau}/(\sigma\sqrt{v_{22}})\). Note that
+
+\[\left[\begin{matrix}G\\ H\end{matrix}\right]\sim N\left(\left[\begin{matrix}0\\ \gamma\end{matrix}\right],\left[\begin{matrix}1\quad\rho\\ \rho\quad 1\end{matrix}\right]\right).\] (2)
+
+where, as defined in Section 2, \(\rho=v_{12}/\sqrt{v_{11}v_{22}}\). Also define \(W=\hat{\sigma}/\sigma\). Note that \((G,H)\) and \(W\) are independent random vectors. Also, \(W\) has the same distribution as \(\sqrt{Q/(n-p)}\) where \(Q\sim\chi^{2}_{n-p}\). Let \(f_{W}\) denote the probability density function of \(W\).
+
+It is straightforward to show that the coverage probability \(P\big{(}\theta\in J(b,s)\big{)}\) is equal to \(P\big{(}\ell(H,W)\leq G\leq u(H,W)\big{)}\), where the functions \(\ell(\cdot,\cdot):\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}\) and \(u(\cdot,\cdot):\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}\) are defined by \(\ell(h,w)=b(h/w)\,w-s(h/w)\,w\) and \(u(h,w)=b(h/w)\,w+s(h/w)\,w\). For given \(b\), \(s\) and \(\rho\), the coverage probability of \(J(b,s)\) is a function of \(\gamma\). We denote this coverage probability by \(c(\gamma;b,s,\rho)\).
+
+Part of our evaluation of the confidence interval \(J(b,s)\) consists of comparing it with the standard \(1-\alpha\) confidence interval \(I\) using the criterion
+
+\[\frac{\text{expected length of $J(b,s)$}}{\text{expected length of $I$}}.\] (3)
+
+We call this the scaled expected length of \(J(b,s)\). This is equal to
+
+\[\frac{E\left(s\bigg{(}{\frac{|H|}{W}}\bigg{)}W\right)}{t_{n-p,1-\frac{\alpha}{2}}E(W)}.\]
+
+This is a function of \(\gamma\) for given \(s\). We denote this function by \(e(\gamma;s)\). Clearly, for given \(s\), \(e(\gamma;s)\) is an even function of \(\gamma\).
+
+Our aim is to find functions \(b\) and \(s\) that satisfy Restrictions 1–3 and such that (a) the minimum of \(c(\gamma;b,s,\rho)\) over \(\gamma\) is \(1-\alpha\) and (b)
+
+\[\int_{-\infty}^{\infty}(e(\gamma;s)-1)\,d\nu(\gamma)\] (4)
+
+is minimized, where the weight function \(\nu\) has been chosen to be
+
+\[\nu(x)=\lambda x+{\cal H}(x)\ \text{ for all }\ x\in\mathbb{R},\] (5)
+
+where \(\lambda\) is a specified nonnegative number and \({\cal H}\) is the unit step function defined by \({\cal H}(x)=0\) for \(x<0\) and \({\cal H}(x)=1\) for \(x\geq 0\). The larger the value of \(\lambda\), the smaller the relative weight given to minimizing \(e(\gamma;s)\) for \(\gamma=0\), as opposed to minimizing \(e(\gamma;s)\) for other values of \(\gamma\). Similarly to Farchione and Kabaila (2008), who consider a much simpler model, we expect the weight function (5) to lead to a \(1-\alpha\) confidence interval for \(\theta\) that has expected length that (a) is relatively small when \(\tau=0\) and (b) has maximum value that is not too large.
+
+The following theorem provides new computationally convenient expressions for the coverage probability and scaled expected length of \(J(b,s)\).
+
+**Theorem 1.**
+
+(a) Define the functions \(k^{{\dagger}}(h,w,\gamma,\rho)=\Psi\big{(}-t_{n-p,1-\frac{\alpha}{2}}w,t_{n-p,1-\frac{\alpha}{2}}w;\rho(h-\gamma),1-\rho^{2}\big{)}\) and \(k(h,w,\gamma,\rho)=\Psi\big{(}\ell(h,w),u(h,w);\rho(h-\gamma),1-\rho^{2}\big{)}\), where \(\Psi(x,y;\mu,v)=P(x\leq Z\leq y)\) for \(Z\sim N(\mu,v)\). The coverage probability of \(J(b,s)\) is denoted by \(c(\gamma;b,s,\rho)\) and is equal to
+
+\[(1-\alpha)+\int_{0}^{\infty}\int_{-d}^{d}\big{(}k(wx,w,\gamma,\rho)-k^{{\dagger}}(wx,w,\gamma,\rho)\big{)}\,\phi(wx-\gamma)\,dx\,w\,f_{W}(w)\,dw\] (6)
+
+where \(\phi\) denotes the \(N(0,1)\) probability density function. For given \(b\), \(s\) and \(\rho\), \(c(\gamma;b,s,\rho)\) is an even function of \(\gamma\).
+
+(b) The scaled expected length of \(J(b,s)\) is
+
+\[e(\gamma;s)=1+\frac{1}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int^{\infty}_{0}\int^{d}_{-d}\left(s(|x|)-t_{n-p,1-\frac{\alpha}{2}}\right)\phi(wx-\gamma)\,dx\,w^{2}\,f_{W}(w)\,dw.\] (7)
+
+Substituting (7) into (4), we obtain that (4) is equal to
+
+\[\frac{1}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int_{-\infty}^{\infty}\int^{\infty}_{0}\int^{d}_{-d}\left(s(|x|)-t_{n-p,1-\frac{\alpha}{2}}\right)\phi(wx-\gamma)\,dx\,w^{2}\,f_{W}(w)\,dw\,d\nu(\gamma)\]
+\[=\frac{1}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int^{\infty}_{0}\int^{d}_{-d}\left(s(|x|)-t_{n-p,1-\frac{\alpha}{2}}\right)\int_{-\infty}^{\infty}\phi(wx-\gamma)\,d\nu(\gamma)\,dx\,w^{2}\,f_{W}(w)\,dw\]
+\[=\frac{2}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int^{\infty}_{0}\int^{d}_{0}\left(s(x)-t_{n-p,1-\frac{\alpha}{2}}\right)(\lambda+\phi(wx))\,dx\,w^{2}\,f_{W}(w)\,dw\] (8)
+
+For computational feasibility, we specify the following parametric forms for the functions \(b\) and \(s\). We require \(b\) to be a continuous function and so it is necessary that \(b(0)=0\). Suppose that \(x_{1},\ldots,x_{q}\) satisfy \(0=x_{1}<x_{2}<\cdots<x_{q}=d\). Obviously, \(b(x_{1})=0\), \(b(x_{q})=0\) and \(s(x_{q})=t_{n-p,1-\frac{\alpha}{2}}\). The function \(b\) is fully specified by the vector \(\big{(}b(x_{2}),\ldots,b(x_{q-1})\big{)}\) as follows. Because \(b\) is assumed to be an odd function, we know that \(b(-x_{i})=-b(x_{i})\) for \(i=2,\ldots,q\). We specify the value of \(b(x)\) for any \(x\in[-d,d]\) by cubic spline interpolation for these given function values, subject to the constraint that \(b^{\prime}(-d)=0\) and \(b^{\prime}(d)=0\). We fully specify the function \(s\) by the vector \(\big{(}s(x_{1}),\ldots,s(x_{q-1})\big{)}\) as follows. The value of \(s(x)\) for any \(x\in[0,d]\) is specified by cubic spline interpolation for these given function values (without any endpoint conditions on the first derivative of \(s\)). We call \(x_{1},x_{2},\ldots x_{q}\) the knots.
+
+To conclude, the new \(1-\alpha\) confidence interval for \(\theta\) that utilizes the prior information that \(\tau=0\) is obtained as follows. For a judiciously-chosen set of values of \(d\), \(\lambda\) and knots \(x_{i}\), we carry out the following computational procedure.
+
+Computational Procedure
+Compute the functions \(b\) and \(s\), satisfying Restrictions 1–3 and taking the parametric forms described above, such that (a) the minimum over \(\gamma\geq 0\) of (6) is \(1-\alpha\) and (b) the criterion (S0.Ex6) is minimized. Plot \(e^{2}(\gamma;s)\), the square of the scaled expected length, as a function of \(\gamma\geq 0\).
+
+Based on these plots and the strength of our prior information that \(\tau=0\), we choose appropriate values of \(d\), \(\lambda\) and knots \(x_{i}\). The confidence interval corresponding to this choice is the new \(1-\alpha\) confidence interval for \(\theta\).
+
+Remark 3.1 Suppose that \(\lambda>0\) is fixed. Also suppose that we apply the Computational Procedure without any parametric restrictions of the form described above. The structure of the criterion (4) when \(\nu\) is given by (5) make it highly plausible that the resulting \(1-\alpha\) confidence interval for \(\theta\) will have a scaled expected length \(e(\gamma;s)\) that converges uniformly in \(\gamma\) to some limiting function as \(d\rightarrow\infty\). It is also highly plausible that this limiting function can be found to a very good approximation by applying this Computational Procedure for \(d\) sufficiently large and knots \(x_{i}\) sufficiently closely spaced.
+
+**4. Application to the analysis of data from a \(\mathbf{2\times 2}\) factorial 123experiment**
+
+In this section we consider a \(2\times 2\) factorial experiment with 20 replicates and parameter of interest \(\theta\) the _simple_ effect (expected response when factor A is high and factor B is low) \(-\) (expected response when factor A is low and factor B is low). We suppose that we have uncertain prior information that the two-factor interaction is zero. We use this example to illustrate the properties of the new \(1-\alpha\) confidence interval for \(\theta\) that utilizes this prior information, when \(1-\alpha=0.95\). All of the computations presented in this paper were performed with programs written in MATLAB, using the Optimization and Statistics toolboxes.
+
+Let \(x_{1}\) take the values \(-1\) and 1 when the factor A takes the values low and high respectively. Also let \(x_{2}\) take the values \(-1\) and 1 when the factor B takes the values low and high respectively. In other words, \(x_{1}\) and \(x_{2}\) are the coded values of the factors A and B respectively. The model for this experiment is
+
+\[Y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{12}x_{1}x_{2}+\varepsilon\] (9)
+
+where \(Y\) is the response, \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{12}\) are unknown parameters and the \(\varepsilon\) for different response measurements are independent and identically \(N(0,\sigma^{2})\) distributed. Thus \(\theta=2(\beta_{1}-\beta_{12})\). Let \(\hat{\beta}_{1}\) and \(\hat{\beta}_{12}\) denote the least squares estimators of \(\beta_{1}\) and \(\beta_{12}\) respectively. The least squares estimator of \(\theta\) is \(\hat{\Theta}=2(\hat{\beta}_{1}-\hat{\beta}_{12})\). Our uncertain prior information is that \(\beta_{12}=0\). Note that
+
+\[\left[\begin{matrix}\hat{\Theta}\\ \hat{\beta}_{12}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta\\ \beta_{12}\end{matrix}\right],\frac{\sigma^{2}}{80}\left[\begin{matrix}\phantom{1}8\quad-2\\ -2\quad\ \ 1\end{matrix}\right]\right).\]
+
+Hence \(\rho=-1/\sqrt{2}\).
+
+We followed the Computational Procedure, described at the end of the previous section, with \(d=6\), \(\lambda=0.2\) and evenly-spaced knots \(x_{i}\) at \(0,1,2,\ldots,6\). The resulting functions \(b\) and \(s\), which specify the new 0.95 confidence interval for \(\theta\), are plotted in Figure 2. The performance of this confidence interval is shown in Figure 3. This confidence interval has coverage probability 0.95 throughout the parameter space. When the prior information is correct (i.e. \(\gamma=0\)), we gain since \(e^{2}(0;s)=0.8683\). The maximum value of \(e^{2}(\gamma;s)\) is 1.1070. This confidence interval coincides with the standard \(1-\alpha\) confidence interval for \(\theta\) when the data strongly contradicts the prior information, so that \(e^{2}(\gamma;s)\) approaches 1 as \(\gamma\rightarrow\infty\). It is interesting to note the broad qualitative similarities between the functions plotted in Figures 1 and 2.
+
+These values of \(d=6\), \(\lambda=0.2\) and knots \(x_{i}\) were obtained after a search that we summarize as follows. Consider \(d=6\), evenly-spaced knots \(x_{i}\) at \(0,1,2,\ldots,6\) and \(\lambda=0.05\), 0.2 , 0.5 and 1. The Computational Procedure was applied for each of these values. As expected from the form of the weight function, for each of these values of \(\lambda\), \(e^{2}(\gamma;s)\) is minimized at \(\gamma=0\). For a given value of \(\lambda\), define the ‘expected gain’ to be \(\big{(}1-e^{2}(0;s)\big{)}\) and the ‘maximum potential loss’ to be \(\big{(}\max_{\gamma}e^{2}(\gamma;s)-1\big{)}\). As shown in Table 1, as \(\lambda\) increases (a) the expected gain decreases and (b) the ratio (expected gain)/(maximum potential loss) increases. By choosing \(\lambda=0.2\) we have both a reasonably large expected gain and a reasonably large value of the ratio (expected gain)/(maximum potential loss).
+
+\begin{table}
+\begin{tabular}{|c|c|c|c|c|}
+\hline
+\(\lambda\) & 0.05 & 0.2 & 0.5 & 1 \\
+\hline
+expected gain & 0.196 & 0.1317 & 0.0822 & 0.043 \\
+\hline
+maximum potential loss & 0.2610 & 0.1070 & 0.0503 & 0.0248 \\
+\hline
+(expected gain)/(maximum potential loss) & 0.7509 & 1.2308 & 1.6341 & 1.7338 \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Performance of the new 0.95 confidence interval for \(d=6\) and knots \(x_{i}\) at \(0,1,\ldots,6\) when we vary over \(\lambda\in\{0.05,0.2,0.5,1\}\).
+
+Now consider \(\lambda=0.2\) and evenly-spaced knots \(x_{i}\) at \(0,1,2,\ldots,d\) where \(d=4\), 6, 8 and 10. The Computational Procedure was applied for each of these values. There was a marked improvement in performance of the resulting 0.95 confidence interval when \(d\) was increased from 4 to 6. However, the improvement in performance of the resulting 0.95 confidence was negligible when \(d\) was increased from 6 to 8 and from 6 to 10. This suggests that increasing \(d\) beyond 6 will lead to a negligible improvement in performance of the confidence interval.
+
+Finally, consider \(d=6\), \(\lambda=0.2\) and two sets of evenly-spaced knots \(x_{i}\) at \(0,0.6,1.2,1.8,\ldots,6\) and \(0,0.5,1,1.5,\ldots,6\). The Computational Procedure was applied to both of these sets of knots. The improvements in performance of the resulting 0.95 confidence interval (compared to the performance for \(d=6\), \(\lambda=0.2\) and evenly-spaced knots \(x_{i}\) at \(0,1,2,\ldots,6\)) were practically negligible. This suggests that there will be a practically negligible improvement in performance if the spacing between the evenly-spaced knots is reduced to less than 1.
+
+Figure 2: Plots of the functions \(b\) and \(s\) for the new \(1-\alpha\) confidence interval in the context of a \(2\times 2\) factorial experiment with 20 replicates, parameter of interest the _simple_ effect \(\theta=2(\beta_{1}-\beta_{12})\) and \(1-\alpha=0.95\). These functions were obtained using \(d=6\), \(\lambda=0.2\) and the knots \(x_{i}\) at \(0,1,2,\ldots,6\).
+
+Figure 3: Plots of the coverage probability and \(e^{2}(\gamma;s)\), the squared scaled expected length, (as functions of \(\gamma=\beta_{12}/\sqrt{\text{var}(\hat{\beta}_{12})}\)) of the new 0.95 confidence interval for the _simple_ effect \(\theta=2(\beta_{1}-\beta_{12})\) for the \(2\times 2\) factorial experiment with 20 replicates. These functions were obtained using \(d=6\), \(\lambda=0.2\) and the knots \(x_{i}\) at \(0,1,2,\ldots,6\).
+
+**5. Discussion**
+
+Discussion 5.1 Our motivation for the weight function (5) is as follows. Suppose that the only restriction on the functions \(b:\mathbb{R}\rightarrow\mathbb{R}\) and \(s:[0,\infty)\rightarrow[0,\infty)\) is that \(b\) is an odd function. Consider the weight function \(\nu={\cal H}\), which corresponds to all of the weight being placed at \(\tau=0\). The minimization of (4), subject to \(P\big{(}\theta\in J(b,s)\big{)}\geq 1-\alpha\) for all \(\gamma\), leads to a \(1-\alpha\) confidence interval for \(\theta\) with the following properties. This interval has the smallest expected length when \(\tau=0\) (i.e. when the prior information is correct) of any \(1-\alpha\) confidence interval for \(\theta\). However, this confidence interval has the weakness that its expected length approaches infinity as \(|\gamma|\rightarrow\infty\) (Tuck, 2006). Now consider the weight function \(\nu=x\), which corresponds to a uniform weight over \(\mathbb{R}\). The minimization of (4), subject to \(P\big{(}\theta\in J(b,s)\big{)}\geq 1-\alpha\) for all \(\gamma\), leads to the standard \(1-\alpha\) confidence interval \(I\). Finally, consider the weight function (5), which is a mixture of the weight functions \({\cal H}\) and \(x\), for fixed \(\lambda>0\). This weight function puts a large amount of weight at \(\tau=0\), consistent with our desire that the confidence interval has relatively small expected length when the prior information is correct. Also, the \(x\) component of this weight function leads to a confidence interval whose expected length has a maximum value that is finite. In addition, the structure of the criterion (4) when \(\nu\) is given by (5) makes it highly plausible that the \(1-\alpha\) confidence interval resulting from the minimization of (4), subject to \(P\big{(}\theta\in J(b,s)\big{)}\geq 1-\alpha\) for all \(\gamma\), will have the desirable feature that it approaches the standard \(1-\alpha\) confidence interval \(I\) as the data increasingly contradict the prior information. Fortuitously, this property leads to the computational advantage described in Remark 3.1.
+
+Discussion 5.2 The new \(1-\alpha\) confidence interval is computed to satisfy the constraint that its minimum coverage probability is \(1-\alpha\). For the example described in Section 4, it is remarkable that the new \(1-\alpha\) confidence interval has coverage probability equal to \(1-\alpha\) throughout the parameter space. The new \(1-\alpha\) confidence interval has been computed for a wide range of values of \(1-\alpha\), \(\lambda\), \(\rho\), \(n-p\) (including the limiting case \(n-p\rightarrow\infty\)), \(d\) and knots \(x_{i}\). In each case, the new \(1-\alpha\) confidence interval has coverage probability equal to \(1-\alpha\) throughout the parameter space. This provides strong empirical evidence that the new \(1-\alpha\) confidence interval has the attractive property that its coverage probability is equal to \(1-\alpha\) throughout the parameter space.
+
+Discussion 5.3 The new \(1-\alpha\) confidence interval has been computed for a wide range of values of \(1-\alpha\), \(\lambda\), \(\rho\), \(n-p\) (including the limiting case \(n-p\rightarrow\infty\)), \(d\) and knots \(x_{i}\). For each of these values of \(1-\alpha\), \(\lambda\), \(d\) and knots \(x_{i}\), \(e^{2}(0;s)\) (which is the minimum value of \(e^{2}(\gamma;s)\)) decreases when \(|\rho|\) increases and/or \((n-p)\) decreases.
+
+Discussion 5.4 Consider the particular case that \(\rho=0\). In this case, we expect that any improvement in performance of the new \(1-\alpha\) confidence interval over the standard \(1-\alpha\) confidence interval \(I\) can only be due to improved estimation of the parameter \(\sigma\). Computations show that the new \(1-\alpha\) confidence interval performs well (in terms of utilizing the uncertain prior information) for small \(n-p\), when \(\lambda\) is chosen appropriately. However, the new \(1-\alpha\) confidence interval approaches the standard \(1-\alpha\) confidence interval \(I\) as \(n-p\rightarrow\infty\).
+
+Discussion 5.5 We briefly compare our frequentist approach with a Bayesian approach to the problem stated in the paper. A full discussion will be presented in a separate paper. For simplicity, suppose that \(\sigma^{2}\) is known and that
+
+\[\left[\begin{matrix}\hat{\Theta}\\ \hat{\tau}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta\\ \tau\end{matrix}\right],\left[\begin{matrix}1\quad\rho\\ \rho\quad 1\end{matrix}\right]\right).\]
+
+For the Bayesian approach, suppose that we choose independent prior pdf’s for \(\Theta\) and \(\tau\). Also suppose that for this approach (a) \(\Theta\) has an uniform improper prior pdf and (b) \(\tau\) has the prior pdf \(\xi\delta(\tau)+(1-\xi)\) where \(\delta\) denotes the delta function and \(\xi\) is a fixed number satisfying \(0\leq\xi\leq 1\). Contrasting features of the new frequentist \(1-\alpha\) confidence interval for \(\theta\) described in the present paper and the Bayesian \(1-\alpha\) highest probability density (HPD) regions for \(\Theta\) include the following:
+
+(a) Suppose that the only restriction on the functions \(b:\mathbb{R}\rightarrow\mathbb{R}\) and \(s:[0,\infty)\rightarrow[0,\infty)\) is that \(b\) is an odd function. Consider the weight function \(\nu={\cal H}\), which corresponds to all of the weight being placed at \(\tau=0\). The minimization of (4), subject to \(P\big{(}\theta\in J(b,s)\big{)}\geq 1-\alpha\) for all \(\gamma\), leads to a \(1-\alpha\) confidence interval with the smallest expected length when \(\tau=0\) of any \(1-\alpha\) confidence interval for \(\theta\). There is no Bayesian analogue of this confidence interval. If we choose \(\xi=1\) then the Bayesian \(1-\alpha\) HPD region for \(\Theta\) is equal to the usual \(1-\alpha\) confidence interval for \(\theta\) based on the assumption that \(\tau=0\). This confidence interval has coverage probability with infimum 0.
+
+(b) By the appropriate choices of \(1-\alpha\), \(\xi\), \(\rho\), \(\sigma\) and \(\hat{\tau}\), one can find Bayesian \(1-\alpha\) HPD regions for \(\Theta\) that consist of the union of two disjoint intervals. By contrast, the methodology of the present paper always produces a confidence interval.
+
+(c) By the appropriate choices of \(1-\alpha\), \(\xi\) where \(\xi<1\), \(\rho\) and \(\sigma\), one can find Bayesian \(1-\alpha\) HPD regions for \(\Theta\) that have frequentist minimum coverage probabilities far below \(1-\alpha\).
+
+Discussion 5.6 We briefly discuss the computation of the new confidence interval. A full discussion is provided by Giri (2008) and will be presented in a separate paper. Our first step has been to truncate the integrals with respect to \(w\) in (6), (7) and (S0.Ex6) and to find upper bounds on the truncation errors. The computational implementation of the constraints that \(c(\gamma;b,s,\rho)\geq 1-\alpha\) for all \(\gamma\geq 0\) is as follows. Restriction 3 implies that, for any reasonable choice of the functions \(b\) and \(s\), \(c(\gamma;b,s,\rho)\to 1-\alpha\) as \(\gamma\rightarrow\infty\). The constraints implemented in the computer programs are that \(c(\gamma;b,s,\rho)\geq 1-\alpha\) for each \(\gamma\in\{0,\Delta,2\Delta,\ldots,M\Delta\}\) where \(\Delta\) is sufficiently small and \(M\) is sufficiently large.
+
+Discussion 5.7 The new \(1-\alpha\) confidence interval for \(\theta\) is founded on the assumption that the random errors \(\varepsilon_{i}\) are independent and identically \(N(0,\sigma^{2})\) distributed. This confidence interval is based on the least squares estimator \(\hat{\Theta}\) of \(\theta\) and the estimator \(\hat{\sigma}\) of \(\sigma\). Consequently, it will display the same kind of lack of robustness to non-normality of the random errors as the standard \(1-\alpha\) confidence interval \(I\).
+
+Discussion 5.8 We illustrate our method with the following real data set. We extract a \(2\times 2\) factorial data set from the \(2^{3}\) factorial data set described in Table 7.5 of Box et al (1963) as follows. Define \(x_{1}=-1\) and \(x_{1}=1\) for “Time of addition of HNO₃” equal to 2 hours and 7 hours, respectively. Also define \(x_{2}=-1\) and \(x_{2}=1\) for “heel absent” and “heel present”, respectively. The observed responses are the following: \(y=87.2\) for \((x_{1},x_{2})=(-1,-1)\), \(y=88.4\) for \((x_{1},x_{2})=(1,-1)\), \(y=86.7\) for \((x_{1},x_{2})=(-1,1)\) and \(y=89.2\) for \((x_{1},x_{2})=(1,1)\). We use the model (9). The discussion on p.265 of Box et al (1963) implies that there is uncertain prior information that \(\beta_{12}=0\). The discussion on p.266 of Box et al (1963) implies that there is an estimator \(\hat{\sigma}^{2}\) of \(\sigma^{2}\), obtained from other related experiments, with the property that \(\hat{\sigma}^{2}/\sigma^{2}\sim Q/m\) where \(Q\sim\chi^{2}_{m}\) and \(m\) is effectively infinite. The observed value of \(\hat{\sigma}\) is 0.8. As in Section 4, define the parameter of interest \(\theta\) to be the simple effect (expected response when \(x_{1}=1\) and \(x_{2}=-1\)) \(-\) (expected response when \(x_{1}=-1\) and \(x_{2}=-1\)), so that \(\theta=2(\beta_{1}-\beta_{12})\). Thus
+
+\[\left[\begin{matrix}\hat{\Theta}\\ \hat{\beta}_{12}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta\\ \beta_{12}\end{matrix}\right],\sigma^{2}\left[\begin{matrix}\phantom{1}2\quad-1/2\\ -1/2\quad\ \ 1/4\end{matrix}\right]\right).\]
+
+The standard 0.95 confidence interval for \(\theta\) is \([-1.01745,3.41745]\). We have also computed the new 0.95 confidence interval for \(\theta\) using \(d=6\), \(\lambda=0.2\) and equally-spaced knots at \(0,6/8,\ldots,6\). This confidence interval is \([-0.81967,3.26345]\), which is substantially shorter than the standard 0.95 confidence interval.
+
+Discussion 5.9 Denote the the usual \(1-\alpha\) confidence interval for \(\theta\), based on the assumption that \(\tau=0\), by \(K\). The naive \(1-\alpha\) confidence interval described in Section 2 may be viewed as being obtained via a monotone discontinuous transition, based on the value of the test statistic \(|\hat{\tau}|/(\hat{\sigma}\sqrt{v_{22}})\), from the standard \(1-\alpha\) confidence interval \(I\) to \(K\). What are the properties of the confidence interval that results from replacing this monotone discontinuous transition by a monotone continuous transition?
+
+For simplicity, consider the case that \(n-p\) is large. Define the quantile \(z_{a}\) by \(P(Z\leq z_{a})=a\) for \(Z\sim N(0,1)\). In this case, \(I=\big{[}\hat{\Theta}-z_{1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma},\quad\hat{\Theta}+z_{1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma}\big{]}\) and \(K=\big{[}\hat{\Theta}-(\hat{\tau}/(\hat{\sigma}\sqrt{v_{22}}))\rho\sqrt{v_{11}}\hat{\sigma}\pm z_{1-\frac{\alpha}{2}}\sqrt{v_{11}}\hat{\sigma}\sqrt{1-\rho^{2}}\big{]}\). The naive \(1-\alpha\) confidence interval for \(\theta\) described in Section 2 may be expressed in the following form
+
+\[g\left(\frac{|\hat{\tau}|}{\hat{\sigma}\sqrt{v_{22}}}\right)I+\left(1-g\left(\frac{|\hat{\tau}|}{\hat{\sigma}\sqrt{v_{22}}}\right)\right)K\] (10)
+
+where \(g:[0,\infty)\rightarrow[0,1]\) is the step function defined by \(g(x)=0\) for all \(x\in[0,q]\) and \(g(x)=1\) for all \(x>q\).
+
+Now suppose that, instead, \(g\) is a continuous increasing function satisfying \(g(0)=0\) and \(g(x)\to 1\) as \(x\rightarrow\infty\). What are the properties of the confidence interval (10) in this case? It is straightforward to show that (10) can be expressed in the form (1) with \(b(x)=(1-g(|x|))\rho x\) for all \(x\in\mathbb{R}\) and \(s(x)=\left(g(x)\big{(}1-\sqrt{1-\rho^{2}}\big{)}+\sqrt{1-\rho^{2}}\right)z_{1-\frac{\alpha}{2}}\) for all \(x\geq 0\). In other words, the confidence interval (10) is of the form (1), but with very severe constraints on the functions \(b\) and \(s\). In particular, \(s(0)=\sqrt{1-\rho^{2}}z_{1-\frac{\alpha}{2}}\) and \(s(x)\) is a nondecreasing function that converges to \(z_{1-\frac{\alpha}{2}}\) as \(x\rightarrow\infty\). The new \(1-\alpha\) confidence interval described in Section 3 has been computed for a wide range of values of \(\rho>0\) and in every single case these very severe constraints are far from satisfied by \(s\). So, the confidence interval (10) does not provide a shortcut to finding the new confidence interval described in Section 3. Indeed, the strength of these constraints on the functions \(b\) and \(s\) implies that any confidence interval of the form (10) will be far inferior to the new confidence interval described in Section 3. The results of Joshi (1969) show that the confidence interval \(I\) is admissible, with the consequence that the minimum coverage probability of the confidence interval (10) must be less than \(1-\alpha\).
+
+**Appendix A. Invariance arguments**
+
+In this appendix we provide a motivation for considering a confidence interval for \(\theta\) of the form (1) where \(b:\mathbb{R}\rightarrow\mathbb{R}\) is constrained to be an odd function and \(s:[0,\infty)\rightarrow[0,\infty)\). We provide this motivation through the invariance arguments listed below. Traditional invariance arguments (see e.g. Casella and Berger (2002, section 6.4) do not include considerations of the available prior information. The novelty in the present appendix is that the invariance arguments need to take proper account of the prior information. Suppose that we have uncertain prior information that \(\tau=0\). Remember that the parameter of interest \(\theta\) is defined to be \(a^{T}\beta\).
+
+Our first step is to reduce the data to \((\hat{\Theta},\hat{\tau},\hat{\sigma})\). Note that \((\hat{\Theta},\hat{\tau})\) and \(\hat{\sigma}\) are independent random vectors with
+
+\[\left[\begin{matrix}\hat{\Theta}\\ \hat{\tau}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta\\ \tau\end{matrix}\right],\sigma^{2}V\right).\]
+
+and \((n-p)\hat{\sigma}^{2}/\sigma^{2}\sim\chi_{n-p}^{2}\). Consider a confidence interval
+
+\[\big{[}\ell(\hat{\Theta},\hat{\tau},\hat{\sigma}),u(\hat{\Theta},\hat{\tau},\hat{\sigma})\big{]}\] (A.1)
+
+for \(\theta\) where \(\ell:\mathbb{R}\times\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}\) and \(u:\mathbb{R}\times\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}\).
+
+**Invariance Argument 1**
+
+The model for the reduced data may be re-expressed
+
+\[\left[\begin{matrix}\hat{\Theta}^{{\dagger}}\\ \hat{\tau}^{{\dagger}}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta^{{\dagger}}\\ \tau^{{\dagger}}\end{matrix}\right],(\sigma^{{\dagger}})^{2}V\right).\]
+
+where \(\theta^{{\dagger}}=\theta+c\), \(\tau^{{\dagger}}=\tau\), \(\sigma^{{\dagger}}=\sigma\), \(\hat{\Theta}^{{\dagger}}=\hat{\Theta}+c\) and \(\hat{\tau}^{{\dagger}}=\hat{\tau}\). Also, let \(\hat{\sigma}^{{\dagger}}=\hat{\sigma}\). Note that \((\hat{\Theta}^{{\dagger}},\hat{\tau}^{{\dagger}})\) and \((\hat{\sigma}^{{\dagger}})^{2}\) are independent random vectors with \((n-p)(\hat{\sigma}^{{\dagger}})^{2}/(\sigma^{{\dagger}})^{2}\sim\chi_{n-p}^{2}\). The uncertain prior information may be re-expressed as \(\tau^{{\dagger}}=0\).
+
+This re-expressed model and prior information have the same form as the original model and prior information. Thus the confidence interval \(\big{[}\ell(\hat{\Theta}^{{\dagger}},\hat{\tau},\sigma),u(\hat{\Theta}^{{\dagger}},\hat{\tau},\sigma)\big{]}\) for \(\theta^{{\dagger}}\) must lead to a confidence interval for \(\theta\) that is identical to (A.1). This implies that \(\ell(\hat{\Theta},\hat{\tau},\hat{\sigma})=\hat{\Theta}+\tilde{\ell}(\hat{\tau},\hat{\sigma})\) and \(u(\hat{\Theta},\hat{\tau},\hat{\sigma})=\hat{\Theta}+\tilde{u}(\hat{\tau},\hat{\sigma})\), where \(\tilde{\ell}:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}\) and \(\tilde{u}:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}\).
+
+**Invariance Argument 2**
+
+Let \(c\) be a positive number. The model for the reduced data may be re-expressed
+
+\[\left[\begin{matrix}\hat{\Theta}^{{\dagger}}\\ \hat{\tau}^{{\dagger}}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta^{{\dagger}}\\ \tau^{{\dagger}}\end{matrix}\right],(\sigma^{{\dagger}})^{2}V\right).\]
+
+where \(\theta^{{\dagger}}=c\,\theta\), \(\tau^{{\dagger}}=c\,\tau\), \(\sigma^{{\dagger}}=c\,\sigma\), \(\hat{\Theta}^{{\dagger}}=c\,\hat{\Theta}\) and \(\hat{\tau}^{{\dagger}}=c\,\hat{\tau}\). Also, let \(\hat{\sigma}^{{\dagger}}=c\,\hat{\sigma}\). Note that \((\hat{\Theta}^{{\dagger}},\hat{\tau}^{{\dagger}})\) and \((\hat{\sigma}^{{\dagger}})^{2}\) are independent random vectors with \((n-p)(\hat{\sigma}^{{\dagger}})^{2}/(\sigma^{{\dagger}})^{2}\sim\chi_{n-p}^{2}\). The uncertain prior information may be re-expressed as \(\tau^{{\dagger}}=0\).
+
+This re-expressed model and prior information have the same form as the original model and prior information. Thus the confidence interval \(\big{[}\hat{\Theta}^{{\dagger}}+\tilde{\ell}(\hat{\tau}^{{\dagger}},\hat{\sigma}^{{\dagger}}),\hat{\Theta}^{{\dagger}}+\tilde{u}(\hat{\tau}^{{\dagger}},\hat{\sigma}^{{\dagger}})\big{]}\) for \(\theta^{{\dagger}}\) must lead to a confidence interval for \(\theta\) that is identical to \(\big{[}\hat{\Theta}+\tilde{\ell}(\hat{\tau},\hat{\sigma}),\hat{\Theta}+\tilde{u}(\hat{\tau},\hat{\sigma})\big{]}\) for \(\theta\). This implies that \(\ell(\hat{\Theta},\hat{\tau},\hat{\sigma})=\hat{\Theta}-\tilde{b}(\hat{\tau}/\hat{\sigma})\hat{\sigma}-\tilde{s}(\hat{\tau}/\hat{\sigma})\hat{\sigma}\) and \(u(\hat{\Theta},\hat{\tau},\hat{\sigma})=\hat{\Theta}-\tilde{b}(\hat{\tau}/\hat{\sigma})\hat{\sigma}+\tilde{s}(\hat{\tau}/\hat{\sigma})\hat{\sigma}\), where \(\tilde{b}:\mathbb{R}\rightarrow\mathbb{R}\) and \(\tilde{s}:\mathbb{R}\rightarrow[0,\infty)\).
+
+**Invariance Argument 3**
+
+The model for the reduced data may be re-expressed
+
+\[\left[\begin{matrix}\hat{\Theta}^{{\dagger}}\\ \hat{\tau}^{{\dagger}}\end{matrix}\right]\sim N\left(\left[\begin{matrix}\theta^{{\dagger}}\\ \tau^{{\dagger}}\end{matrix}\right],(\sigma^{{\dagger}})^{2}V\right).\]
+
+where \(\theta^{{\dagger}}=-\theta\), \(\tau^{{\dagger}}=-\tau\), \(\sigma^{{\dagger}}=\sigma\), \(\hat{\Theta}^{{\dagger}}=-\hat{\Theta}\) and \(\hat{\tau}^{{\dagger}}=-\hat{\tau}\). Also, let \(\hat{\sigma}^{{\dagger}}=\hat{\sigma}\). Note that \((\hat{\Theta}^{{\dagger}},\hat{\tau}^{{\dagger}})\) and \((\hat{\sigma}^{{\dagger}})^{2}\) are independent random vectors with \((n-p)(\hat{\sigma}^{{\dagger}})^{2}/(\sigma^{{\dagger}})^{2}\sim\chi_{n-p}^{2}\). The uncertain prior information may be re-expressed as \(\tau^{{\dagger}}=0\).
+
+This re-expressed model and prior information have the same form as the original model and prior information. Thus the confidence interval
+
+\[\left[\hat{\Theta}^{{\dagger}}-\tilde{b}\left(\frac{\hat{\tau}^{{\dagger}}}{\hat{\sigma}^{{\dagger}}}\right)\hat{\sigma}^{{\dagger}}-\tilde{s}\left(\frac{\hat{\tau}^{{\dagger}}}{\hat{\sigma}^{{\dagger}}}\right)\hat{\sigma}^{{\dagger}},\,\hat{\Theta}^{{\dagger}}-\tilde{b}\left(\frac{\hat{\tau}^{{\dagger}}}{\hat{\sigma}^{{\dagger}}}\right)\hat{\sigma}^{{\dagger}}+\tilde{s}\left(\frac{\hat{\tau}^{{\dagger}}}{\hat{\sigma}^{{\dagger}}}\right)\hat{\sigma}^{{\dagger}}\right]\]
+
+for \(\theta^{{\dagger}}\) must lead to a confidence interval for \(\theta\) that is identical to the confidence interval
+
+\[\left[\hat{\Theta}-\tilde{b}\left(\frac{\hat{\tau}}{\hat{\sigma}}\right)\hat{\sigma}-\tilde{s}\left(\frac{|\hat{\tau}|}{\hat{\sigma}}\right)\hat{\sigma},\,\hat{\Theta}-\tilde{b}\left(\frac{\hat{\tau}}{\hat{\sigma}}\right)\hat{\sigma}+\tilde{s}\left(\frac{|\hat{\tau}|}{\hat{\sigma}}\right)\hat{\sigma}\right]\]
+
+for \(\theta\). This implies that \(\tilde{b}\) is an odd function and \(\tilde{s}:[0,\infty)\rightarrow[0,\infty)\).
+
+Now define the functions \(b(x)=(1/\sqrt{v_{11}})\,\tilde{b}\big{(}\sqrt{v_{22}}x\big{)}\) for all \(x\in\mathbb{R}\) and \(s(x)=(1/\sqrt{v_{11}})\,\tilde{s}\big{(}\sqrt{v_{22}}x\big{)}\) for all \(x\geq 0\). Since \(\tilde{b}\) is constrained to be an odd function, \(b\) is also an odd function. Also, since \(\tilde{s}:[0,\infty)\rightarrow[0,\infty)\), \(s:[0,\infty)\rightarrow[0,\infty)\). The confidence interval (A.1) is therefore equal to \(J(b,s)\) where \(b:\mathbb{R}\rightarrow\mathbb{R}\) is an odd function and \(s:[0,\infty)\rightarrow[0,\infty)\).
+
+**Appendix B. Proof of Theorem 1**
+
+**Proof of part (a).**
+
+The random vectors \((G,H)\) and \(W\) are independent. It follows from (2) that the probability density function of \(H\), evaluated at \(h\), is \(\phi(h-\gamma)\). Thus
+
+\[c(\gamma;b,s,\rho)=\int_{0}^{\infty}\int_{-\infty}^{\infty}\int_{\ell(h,w)}^{u(h,w)}f_{G|H}(g|h)\,dg\,\phi(h-\gamma)\,dh\,f_{W}(w)\,dw\] (B.1)
+
+where \(f_{W}\) denotes the probability density function of \(W\) and \(f_{G|H}(g|h)\) denotes the probability density function of \(G\) conditional on \(H=h\), evaluated at \(g\). The probability distribution of \(G\) conditional on \(H=h\) is \(N\big{(}\rho(h-\gamma),1-\rho^{2}\big{)}\). Thus the right hand side of (B.1) is equal to
+
+\[\int_{0}^{\infty}\int_{-\infty}^{\infty}k(h,w,\gamma,\rho)\,\phi(h-\gamma)\,dh\,f_{W}(w)\,dw\] (B.2)
+
+The standard \(1-\alpha\) confidence interval \(I\) has coverage probability \(1-\alpha\). Hence
+
+\[1-\alpha=\int_{0}^{\infty}\int_{-\infty}^{\infty}k^{{\dagger}}(h,w,\gamma,\rho)\,\phi(h-\gamma)\,dh\,f_{W}(w)\,dw.\] (B.3)
+
+Subtracting (B.3) from (B.2) and noting that \(b(x)=0\) for all \(|x|\geq d\) and \(s(x)=t_{n-p,1-\frac{\alpha}{2}}\) for all \(x\geq d\), we find that
+
+\[c(\gamma;b,s,\rho)=(1-\alpha)+\int_{0}^{\infty}\int_{-dw}^{dw}\big{(}k(h,w,\gamma,\rho)-k^{{\dagger}}(h,w,\gamma,\rho)\big{)}\,\phi(h-\gamma)\,dh\,f_{W}(w)\,dw.\]
+
+Changing the variable of integration from \(h\) to \(x=h/w\) in the inner integral, we obtain (6). Using the fact that
+
+\[\left[\begin{matrix}-G\\ -H\end{matrix}\right]\sim N\left(\left[\begin{matrix}0\\ -\gamma\end{matrix}\right],\left[\begin{matrix}1\quad\rho\\ \rho\quad 1\end{matrix}\right]\right),\]
+
+it may be shown that \(P(\theta\in J(b,s))\) is an even function of \(\gamma\).
+
+**Proof of part (b).**
+
+The random variables \(H\) and \(W\) are independent. It follows from (2) that the probability density function of \(H\), evaluated at \(h\), is \(\phi(h-\gamma)\). Thus
+
+\[e(\gamma;s)=\frac{1}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int^{\infty}_{0}\int^{\infty}_{-\infty}s\left(\frac{|h|}{w}\right)\phi(h-\gamma)\,dh\,w\,f_{W}(w)\,dw\] (B.4)
+
+where \(f_{W}\) denotes the probability density function of \(W\). Obviously,
+
+\[1=\frac{1}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int^{\infty}_{0}\int^{\infty}_{-\infty}t_{n-p,1-\frac{\alpha}{2}}\,\phi(h-\gamma)\,dh\,w\,f_{W}(w)\,dw.\] (B.5)
+
+Note that \(s(x)=t_{n-p,1-\frac{\alpha}{2}}\) for all \(x\geq d\). Subtracting (B.5) from (B.4) we therefore obtain
+
+\[e(\gamma;s)=1+\frac{1}{t_{n-p,1-\frac{\alpha}{2}}\,E(W)}\int^{\infty}_{0}\int^{dw}_{-dw}\left(s\left(\frac{|h|}{w}\right)-t_{n-p,1-\frac{\alpha}{2}}\right)\phi(h-\gamma)\,dh\,w\,f_{W}(w)\,dw.\]
+
+Changing the variable of integration in the inner integral from \(h\) to \(x=h/w\), we obtain (7).
+
+**References**
+
+Bickel, P.J., 1983. Minimax estimation of the mean of a normal distribution subject to doing well at a point. In: Rizvi, M.H., Rustagi, J.S., Siegmund, D., (Eds), Recent Advances in Statistics, Academic Press, New York, 511–528.
+
+Bickel, P.J., 1984. Parametric robustness: small biases can be worthwhile. Annals of Statistics 12, 864–879.
+
+Bohrer, R., Sheft, J., 1979. Misclassification probabilities in \(2^{3}\) factorial experiments. Journal of Statistical Planning and Inference 3, 79–84.
+
+Box, G.E.P., Connor, L.R., Cousins, W.R., Davies, O.L., Hinsworth, F.R., Sillitto, G.P., 1963. The Design and Analysis of Industrial Experiments, 2nd edition, reprinted. Oliver and Boyd, London.
+
+Buring, J.E., Hennekens, C.H., 1990. Cost and efficiency in clinical trials: the U.S. Physicians’ Health Study. Statistics in Medicine 9, 29–33.
+
+Casella, G., Berger, R. L., 2002. Statistical Inference, 2nd ed. Duxbury, Pacific Grove, California.
+
+Dubé, M.A., Penlidis, A., Reilly, P.M., 1996. A systematic approach to the study of multicomponent polymerization kinetics: the butyl acrylate/methyl methacrylate/vinyl acetate example. IV. Optimal Bayesian design of emulsion terpolymerization experiments in a pilot plant reactor. Journal of Polymer Science: Part A: Polymer Chemistry 34, 811–831.
+
+Fabian, V., 1991. On the problem of interactions in the analysis of variance. Journal of the American Statistical Association 86, 362–373.
+
+Farchione, D., Kabaila, P., 2008. Confidence intervals for the normal mean utilizing prior information. Statistics & Probability Letters, 78, 1094–1100.
+
+Giri, K., 2008. Confidence intervals in regression utilizing prior information. Unpublished PhD thesis, August 2008, Department of Mathematics and Statistics, La Trobe University.
+
+Giri, K., Kabaila, P., 2008. The coverage probability of confidence intervals in \(2^{r}\) factorial experiments after preliminary hypothesis testing. Australian & New Zealand Journal of Statistics, 50, 69–79.
+
+Hodges, J.L., Lehmann, E.L., 1952. The use of previous experience in reaching statistical decisions. Annals of Mathematical Statistics 23, 396–407.
+
+Hung, H.M.J., Chi, G.Y.H., O’Neill, R.T., 1995. Efficiency evaluation of monotherapies in two-by-two factorial trials. Biometrics 51, 1483–1493.
+
+Joshi, V.M., 1969. Admissibility of the usual confidence sets for the mean of a univariate or bivariate normal population. Annals of Mathematical Statistics, 40, 1042–1067.
+
+Kabaila, P., 1998. Valid confidence intervals in regression after variable selection. Econometric Theory 14, 463–482.
+
+Kabaila, P., 2005a. On the coverage probability of confidence intervals in regression after variable selection. Australian & New Zealand Journal of Statistics 47, 549–562.
+
+Kabaila, P., 2005b. Assessment of a preliminary F-test solution to the Behrens-Fisher problem. Communications in Statistics - Theory and Methods 34, 291–302.
+
+Kabaila, P., Giri, K., 2007a. Large sample confidence intervals in regression utilizing prior information. Technical Report No. 2007-1, Jan 2007, Department of Mathematics and Statistics, La Trobe University.
+
+Kabaila, P., Giri, K., 2007b. Confidence intervals in regression utilizing prior information. arXiv:0711.3236v1.
+
+Kabaila, P., Giri, K., 2009a. Upper bounds on the minimum coverage probability of confidence intervals in regression after variable selection. To appear in Australian & New Zealand Journal of Statistics.
+
+Kabaila, P., Giri, K., 2009b. Large-sample confidence intervals for the treatment difference in a two-period crossover trial, utilizing prior information. Statistics & Probability Letters 79, 652–658.
+
+Kabaila, P., Leeb, H., 2006. On the large-sample minimum coverage probability of confidence intervals after model selection. Journal of the American Statistical Association 101, 619–629.
+
+Kabaila, P., Tuck, J., 2008. Confidence intervals utilizing prior information in the Behrens-Fisher problem. Australian & New Zealand Journal of Statistics 50, 309–328.
+
+Neyman, J., 1935. Comments on ‘Complex experiments’ by F. Yates. Supplement to the Journal of the Royal Statistical Society 2, 181–247.
+
+Ng, T-H., 1994. The impact of a preliminary test for interaction in a \(2\times 2\) factorial trial. Communications in Statistics: Theory and Methods 23, 435–450.
+
+Pratt, J.W., 1961. Length of confidence intervals. Journal of the American Statistical Association 56, 549–657.
+
+Shaffer, J.P., 1991. Probability of directional errors with disordinal (qualitative) interaction. Psychometrika 56, 29–38.
+
+Stampfer, M.J., Buring, J.E., Willett, W., Rosner, B., Eberlain, K., Hennekens, C.H., 1985. The \(2\times 2\) factorial design: its application to a randomized trial of aspirin and carotene in U.S. physicians. Statistics in Medicine 4, 111—116.
+
+Steering Committee of the Physicians’ Health Study Research Group: Belanger, C., Buring, J.E., Cook, N., Eberlain, K., Goldhaber, S.Z., Gordon, D., Hennekens, C.H. (chairman), Mayrent, S.L., Peto, R., Rosner, B., Stampfer, M.J., Stubblefield, F., Willett, W., 1988. Preliminary report: findings from the aspirin component of the ongoing Physicians’ Health Study. New England Journal of Medicine 318, 262–264.
+
+Traxler, R.H., 1976. A snag in the history of factorial experiments. In: Owen, D.B. (Ed), On the History of Statistics and Probability, Marcel Dekker, New York, 281–295.
+
+Tuck, J., 2006. Confidence intervals incorporating prior information. Unpublished PhD thesis, August 2006, Department of Mathematics and Statistics, La Trobe University.
+

arxiv_ground_truth/0802.0539.md

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+# Optical Absorption of Poly(thiophene vinylene) Conjugated Polymers. Experiment and First Principle Theory.
+
+A. V. Gavrilenko
+
+Center for Materials Research, Norfolk State University, 700 Park Ave, Norfolk, VA 23504
+
+T. D. Matos
+
+Center for Materials Research, Norfolk State University, 700 Park Ave, Norfolk, VA 23504
+
+C. E. Bonner
+
+Center for Materials Research, Norfolk State University, 700 Park Ave, Norfolk, VA 23504
+
+S.-S. Sun
+
+Center for Materials Research, Norfolk State University, 700 Park Ave, Norfolk, VA 23504
+
+C. Zhang
+
+Center for Materials Research, Norfolk State University, 700 Park Ave, Norfolk, VA 23504
+
+V. I. Gavrilenko
+
+Center for Materials Research, Norfolk State University, 700 Park Ave, Norfolk, VA 23504
+
+###### Abstract
+
+Optical absorption spectra of poly(thiophene vinylene) (PTV) conjugated polymers have been studied at room temperature in the spectral range of 450 to 800 nm. A dominant peak located at 577 nm and a prominent shoulder at 619 nm are observed. Another shoulder located at 685 nm is observed at high concentration and after additional treatment (heat, sonification) only. Equilibrium atomic geometries and optical absorption of PTV conjugated polymers have also been studied by first principles density functional theory (DFT). For PTV in solvent, the theoretical calculations predict two equilibrium geometries with different interchain distances. By comparative analysis of the experimental and theoretical data, it is demonstrated that the new measured long-wavelength optical absorption shoulder is consistent with new optical absorption peak predicted for most energetically favorable PTV phase in the solvent. This shoulder is interpreted as a direct indication of increased interchain interaction in the solvent which has caused additional electronic energy structure transformations.
+
+## I Introduction
+
+The search for inexpensive renewable energy sources has sparked considerable interest in the development of photovoltaics based on conjugated polymers and organic molecules Halls and Friend (2001); Gurau et al. (2007); Brown et al. (2003); Gregg (2003); Brabec et al. (1999). Compared to silicon-based devices, polymer solar cells are lightweight, inexpensive to fabricate, flexible, and have an ultra-fast opto-electronic response Skotheim (1997); McCullough (1998). They also exhibit a nearly continuous tunability of the energy levels and band gaps via molecular design and synthesis, versatile materials processing and device fabrication schemes, and low cost industrial manufacturing on a large scale Wu et al. (1995); Bouman and Meijer (1995). However, the energy conversion efficiency of existing organic and polymeric solar cells with donor/acceptor blends is still less than 5% Sun and Sariciftci (2005). Though both organic and conventional solar cells operate by absorbing light, a fundamental difference was recognized almost immediately.
+
+In organic materials, the light absorption results in the formation of excitons rather than the free electrons and holes directly produced in inorganic semiconductors, such as silicon Gurau et al. (2007); Brown et al. (2003); Matos (2007). The bandgaps of copolymers can be tailored by combining various repeat units of polymers with different bandgaps Fu et al. (1997). One important approach to modify bandgaps of polyaromatics systems is to incorporate vinylene linkages between the aromatic rings Patil et al. (1988); Eckhardt et al. (1989); Blohm et al. (1993). This has been demonstrated by the poly(p-phenylene vinylene) (PPV) and poly(thiophene vinylene) (PTV) systems, yielding bandgaps 0.3 eV lower than that of polyphenylene and polythiophene systems Eckhardt et al. (1989). The vinylene bond not only reduces the band gap of the polymer but also acts as a spacer to reduce steric hinderance on successive aromatic rings, therefore increasing the degree of coplanarity of the conjugated polymer backbone Fu et al. (1997). An advantage of PTV and its derivatives is their high absorption in the visible range of the spectrum, making them excellent candidates for energy alternatives Dhanabalan et al. (2001). For solar cell applications, poly(thiophene vinylene) (PTV) has already proven to be an interesting conjugated polymer with a high conductivity Fuchigami et al. (1993); Brown et al. (1997); Vandamme et al. (2002); Henckens et al. (2004). These polymers display high nonlinear optical responses and moderate charge mobilities Loewe and McCullough (2000). In addition, PTV has a band gap of about 1.7eV and, as a result, can be considered as a low-bandgap polymer Henckens et al. (2005).
+
+Optical absorption and emission spectra of conjugated polymers exhibit well pronounced peak attributed to the excitations of \(\pi-\pi^{*}\) electron transitions Gurau et al. (2007); Brown et al. (2003). Vibronic excitations in well-ordered polymers could be seen as additional fine structures Gurau et al. (2007). However, not all components of the fine structure in the optical absorption and emission spectra can be attributed to exciton-phonon coupling. It has been demonstrated earlier that the long-wavelength shoulder in optical absorption spectra of poly(3-hexylthiophene) has different nature and it could be interpreted as the effect of interchain interaction Brown et al. (2003).
+
+Due to the degree of difficulty in devising new synthesis schemes, it is important to rely on computational models which can predict the properties, since the characterization of polymers can only be done once the polymer has been successfully synthesized. A guided approach to the design of new organic materials is preferential. State-of-the-art first principle methods based on the Density Functional Theory (DFT) are very useful tools for providing a detailed understanding of structural, electronic, and optical processes Puschnig and Ambrosch-Draxl (2001); Rohlfing et al. (2001); Gavrilenko (2006). In this work we studied optical absorption of poly(thiophene vinylene) (PTV) conjugated polymers. Based on extensive first principle modeling of the ground state and optical absorption, we will demonstrate the possible coexistence of two stable geometrical phases with different interchain distances. Comparison with measured experimental data show that the observed long-wavelegth optical absorption shoulder is consistent with a predicted geometrical phase with a smaller interchain distance.
+
+Previous works on PPV Ruini et al. (2002); Ferretti et al. (2003) showed that crystalline arrangement crucially affects the optical properties of the polymer films and interchain interactions can be viewed as a tunable parameter for the design of efficient electronic devices based on organic materials. 3D arrangement is also a crucial element for the design of materials with efficient transport properties. In this work we studied effect of the aggregation in PTV which could be considered as an intermediate phase between liquid and solid. Our goal is to understand how aggregation affects optical absorption in PTV. Based on extensive first principle modeling combined with experiment we demonstrate that optical absorption spectra of PTV polymers indeed show specific features which could be interpreted as a result of interchain interaction. This paper is organized in the following way. First we describe fabrication procedure of PTV polymers and measurement conditions of optical absorption spectra. In the following section we describe theoretical methods used in this work for equilibrium geometry study and for optical calculations. In the final section we present comparative analysis of predicted and measured optical absorption spectra, supported by calculated Projected Density Of States (PDOS) and by equilibrium geometry studies in comparison with available data in literature.
+
+## II Experimental procedure and results
+
+Figure 1: Chemical structure of poly(3-dodecyl-2,5-thiophene vinylene). The PTV with n=20 was used for all UV-Vis measurements.
+
+The mono-end functionalized poly(3-dodecyl-2,5-thiophene vinylene) (PTV) conjugated polymer (see chemical structure in Fig. 1) used for UV-Vis measurement was synthesized by Horner-Emmons coupling of (3-dodecyl-thiophen-2-ylmethyl)-phosphonic acid diethyl ester with 3-dodecyl-5-formyl-thiophen-2-ylmethyl)-phosphonic acid diethyl ester. Samples were prepared according to the following procedure. In order to measure single chain optical response, a series of 5mM solutions were prepared for concentration dependance studies. The solid samples were mechanically grinded and dissolved in chloroform. The solutions were additionally heated and exposed to external mechanical vibration field (sonification) at 40 kHz for 60 minutes. From this mixture aliquots were extracted to prepare 1 mM solutions used for measurements.
+
+Figure 2: Optical absorption spectra dependance on heating and sonification treatment of PTV polymersat the concentration of 1mM. The solid (black) line is the absorption of the PTV heated for 1 min without sonification. The dashed (red) line and the dotted (blue) line show the absorption spectrum of the PTV heated for 1 min and 3 min, respectively, followed by sonification. Sonification was performed at a frequency of 40 kHz for 60 min.
+
+Optical absorption spectra were measured on a Varian Cary-5 spectrophotometer at room temperature in spectral range 400 to 800 nm. Fig. 2 presents UV-vis heating and sonification treatment studies of regioregular PTV solutions. Application of external treatment (heating and sonification) results in slight red shift stabilizing spectral location of optical absorption against further treatments (see Fig. 2). After the treatment of the solution we observed a dominant absorption peak located at \(\lambda_{max}=577nm\) accompanied by a prominent shoulder located at 619 nm (A-shoulder). This shoulder was observed on all samples studied and it shows relatively weak dependence on concentration and heat-sonification treatment.
+
+Another low intensity shoulder is observed around 685 nm (B-shoulder). Its strong dependence on external treatment is essentially different from A-shoulder as demonstrated in Fig. 2: the intensity of the B-shoulder decreases with more intense treatment untill becomming undetectable. The low heating caused slight red shift of optical absorption from its initial spectral location as shown in Fig. 2. Further external treatment does not affect the spectral location of the optical absorption.
+
+## III Theoretical method and results
+
+Density functional theory (DFT) has been shown for decades to be very successful in the ground state analysis of different materials Kohn et al. (1996). In order to be able to predict physical properties of a conjugated polymer system it is important to realistically describe both short (covalent) and long range (Coulomb and van der Waals, vdW) components of intermolecular interactions Ortmann et al. (2005). Local density (LDA) and generalized gradient approximations (GGA) are frequently used to account for the exchange and correlation (XC) interaction. The vdW interactions is not included in standard DFT. However, detailed analysis of organic molecule adsorption on solid surface demonstrates that around equilibrium the kinetic energy of valence electrons remains mainly repulsive, and XC effects are mostly responsible for the attraction Ortmann et al. (2005). It has been shown that equilibrium distance between organic molecule and solid (graphene) surface predicted by LDA is in good agreement to the value followed from explicit inclusion of the vdW into the interaction Hamiltonian Ortmann et al. (2005).
+
+(a)Unit cell
+
+(b)d2 = 3.75Å
+
+(c)d2 = 3.7Å
+
+Figure 3: 3D-view of the poly(thiophene vinylene) unit cell 3(a) and equilibrium geometries of straight 3(b) and tilted 3(c) chain configutaions. Optimized unit cell dimensions are shown by numbers.
+
+In this work we used _ab initio_ pseudopotential method within the DFT and the supercell (sc) formalism to study the effects of interchain interaction of the PTV. The system was modeled as an infinite chain as shown in Fig. 3(a). The geometry optimizations were performed using the LDA method Perdew and Wang (1992) employing norm-conserving pseudopotentials Hamann et al. (1979) using the DMol³Delley (2000) software package until all forces were below 0.05 eV/Å. An initial set of 28 irreducible k-points was used. We have explicitly checked that the structural and binding properties of our system are well converged for the double numerical plus polarization basis set used. The use of the exact DFT spherical atomic orbitals has several advantages. For one, the molecule can be dissociated exactly to its constituent atoms (within the DFT context). Because of the local character of these orbitals, basis set superposition effects Delley (1990) are minimized and good accuracy is achieved even for weak bonds. Solvent effects were taken into account during geometry optimization.
+
+In order to simulate the effects of a solvent the COnductor-like Screening MOdel (COSMO) Klamt and Schüürmann (1993); Delley (2006) is used within DMol³. COSMO is a continuum solvation model (CSM) Tomasi and Persico (1994) in which the solute molecule forms a cavity within the dielectric continuum of permittivity, \(\varepsilon\), that represents the solvent. The charge distribution of the solute polarizes the dielectric medium. The response of the dielectric medium is described by the generation of screening (or polarization) charges on the cavity surface. In contrast to other implementations of CSMs, COSMO calculates the screening charges using a boundary condition for a conductor. These charges are then scaled by a factor \(f(\varepsilon)=(\varepsilon-1)/(\varepsilon+1/2)\), to obtain a good approximation for the screening charges in a dielectric medium. In such a way, by using COSMO, we avoid complexity by specifying the actual structure of the solvent still incorporating into the theory the screening effect in the solution.
+
+Only Coulombic interaction providing most important contribution to the interchain interaction Ortmann et al. (2005) is included. The unit cell is replicated in all 3 dimensions, the height, _d3_, was chosen to be 15 Å in order to quench the interaction between the thiophene ring and the methyl group. We obtained very weak effect of the _d3_ on optics therefore this value was constrained. The equilibrium intermolecular distance, _d1_=6.55Å, was determined by cluster calculation of a single polymer chain of 3 units. The interchain distance _d2_ was varied between 10 Å and 3 Å in units of 0.1 Å.
+
+Figure 4: Total energy of the system versus PTV unit cell length _d2_. Dashed and solid lines correspond to straight and tilted chain geometies shown in Fig. 3(b) and Fig. 3(c), respectively.
+
+Optical absorption spectra of the PTV conjugated polymers are calculated within the independent particles picture (random phase approximation, RPA) employing ultrasoft pseudopotentials Hamann et al. (1979). For optical line shape analysis inclusion of excitonic effects in polymers is nessecary Rohlfing et al. (2001). This technically challenging theory of optical response in polymers is out of scope of present study. In this work we focus on relative changes of electron energy structure and optical response caused by atomic geometry modifications in different equilibrium polymer phases, which could be realistically described without inclusion of many-body efects in optics Gavrilenko (2006); Gavrilenko and Bechstedt (1997). Equilibrium geometry and self-consistently calculated eigen energies and eigen values are used as inputs for optics. Optical functions are obtained using CASTEP-GGA method with a planewave cutoff of 400 eV. More details of our approach can be found in Gavrilenko (2006). A blue-shift of 0.18 eV is applied to the calculated spectrum to match experimental data. It should be noted that the Kohn-Sham eigenvalues do not interpret as a quasiparticle energy requiring quasiparticle (QP) correction Godby (1992). The QP correction is a wave-vector dependent shift of the conduction band with respect to the valence band. This is attributed to a discontinuity in the exchange-correlation potential as the system goes from (_N_)-electrons to (_N+1_)-electrons during the excitation process. It has been demonstrated in Ruini et al. (2002); Ferretti et al. (2003), however, that in polymers exciton correction compensates QP shift resulting in smaller correction values than e.g. in semiconductors Godby (1992), as obtained in this work.
+
+### Equilibrium Geometry
+
+In order to model the PTV structure it is important to first determine the equilibrium configuration. As stated before the polymer was modeled as an infinite chain in a unit cell, see Fig. 3. Interchain interaction is an important aspect since the close proximity of neighboring chains will split the electronic states of the polymer, thereby reducing the bandgap and creating additional states. Only Coulombic interaction providing most important contribution to the interchain interaction Ortmann et al. (2005) is taken into account. The unit cell is replicated in all 3 dimensions, the height, _d3_, was chosen to be 15 Å in order to quench the interaction between the thiophene ring and the methyl group. We obtained very weak effect of the _d3_ on optics therefore this value was constrained. The equilibrium intermolecular distance, _d1_=6.55Å, was determined by cluster calculation of a single polymer chain of 3 units.
+
+One of the most important findings of this work is a very strong dependence of the PTV polymer ground state on the interchain distance (_d2_). This point is addressed here in detail. The equilibrium interchain distance is studied by a series of unit cell length _d2_ optimizations varying between 10 and 3 Å in steps of 0.05 Å. We report two equilibrium geometry configurations characterized by different total energy relaxation paths but having almost the same unit cell legth: _d2_=3.75Å and _d2_=3.7Å, see Fig. 4.
+
+It is important to note that accordingly to our finding the GGA method within CASTEP which employs _ab initio_ pseudopotentials does not predict a saddle point in total energy curve; as the layers come closer together the energy of the system increases without going through a minimum. In contrast, the LDA method does not contain gradient corrections to the charge density (as the GGA) correctly predicts equilibrium distances between polymer chains, even though it underestimates the energy of the system. This observation for molecular systems has been reported earlier Ortmann et al. (2005).
+
+### Optical Absorption
+
+Figure 5: Calculated optical absorption spectra for the PTV configurations shown in Figs. 3(b) and 3(c).
+
+Optical absorption spectra of the PTV conjugated polymers are calculated within the independent particles picture (random phase approximation, RPA) employing norm-conserving pseudopotentials Hamann et al. (1979). For optical line shape analysis inclusion of excitonic effects in polymers is nessecary Rohlfing et al. (2001). This technically challenging theory of optical response in polymers is out of scope of present study. In this work we focus on relative changes of electron energy structure and optical response caused by atomic geometry modifications in diferent equilibrium polymer phases, which could be realistically described without inclusion of many-body efects in optics Gavrilenko (2006); Gavrilenko and Bechstedt (1997). Equilibrium geometry and self-consistently calculated eigen energies and eigen values are used as inputs for optics. Optical functions are obtained using CASTEP-GGA method. More details of our approach can be found in Gavrilenko (2006). As discussed above the blue-shift of 0.18 eV is applied to the calculated spectrum in order to match the experimental bandgap.
+
+Figure 6: Comparison of experimental spectrum coresponding to low-heating and sonification treatment of the PTV polymer, solid (black) line, and the calculated optical absorption spectra, dashed (blue) line.
+
+In Fig. 5 we present calculated optical absorption spectra coresponding to the predicted tilted (dashed (back)) and straight (solid (blue)) PTV polymer chain geometries. The straight polymer chain configuration (see Fig. 3(b)) is characterised by the dominant optical absorption peak coresponding to the \(\pi-\pi^{*}\) electron excitations (see solid (blue) line in Fig. 5). Tilting and decrease of the interchain distance correspond to an energetically favorable geometrical phase causing dramatic modifications of the spectrum: main peak is blue shifted and in addition a new long-wavelength peak appears. Analysis of calculated PDOS spectra indicates that new optical absorption peak is a consequence of splitting of the bonding and antibonding \(\pi-\)electron states due to decrease of interchain distance, relative displacement of neighbouring polymer chain, that causes changes of \(\pi-\)orbitals overlap and strong increase of interchain interaction. Note that calculated electron energy structure does not incorporate any vibronic contributions normally presented in molecular systems. Therefore weak shoulders on calculated optical spectra relate to electronic states only.
+
+## IV Discussion
+
+The equilibrium dimensions of the super cell of the PTV polymer are determined from total energy minimization method. The effect of the interchain interactions on optical absorption spectra has been studied by varying the super cell length _d2_ perpendicular to the PTV chain direction, see Fig. 3. The predicted equilibrium value of _d2_=3.75 Å (that is equal to the interchain distance, _di_), and corresponds to absolute energy minimum of straight geometry (see Fig. 3(b)). However, our theoretical calculations predict another energetically favorable geometrical phase of the PTV polymer (see Fig. 4) with predicted equilibrium value of _d2_=3.70 Å and coresponding interchain distance value of _di_=3.29 Å see Fig. 3(c). Equilibrium PTV configuration in this case is characterized by substantial atomic reconstruction: the out of plane rotation of the thiophene ring by 27∘, relative displacement of neighbouring chains, and additional in-plane atomic distortions.
+
+Equilibrium geometry analysis of PTV polymers cleary predicts two geometrical phases with straight (Fig. 3(b)) and tilted (Fig. 3(c)) chain geometries. Accordingly to the total energy minimisation study the titlted geometry is the most favorable one (see Fig. 4). However, predicted energy difference of 0.16 eV between two phases minima allows for their coexistence at room temperature. For comparison with experiment in Fig. 6 we present theoretical absorption spectrum generated from the data of both predicted phases (see Fig. 5). The resulting spectrum is obtained assuming relative weight of the straight:tilted phases as 0.\(\overline{2}\):1.0. A recent study showed that in conjugated polymers there is a significant impact on the intrachain mobility by ring torsion Hultell and Stafström (2007). However, since the whole chain is rotated rather than the ring itself, no significant reduction of coplanarity is predicted and as a result charge mobility remains unaffected.
+
+It has been currently well understood that optical absorption (emission) spectra of molecular systems are characterized by the contributions of both electronic and vibronic excitations. Consequently in many cases vibronic fine structure is superimposed on electronic optical absorption (emission) spectrum Gurau et al. (2007); Brown et al. (2003). Our experimental data seem to confirm this issue: the measured A-shoulder (located at 619 nm) weekly depent on treatment, supporting vibronic character of A-shoulder (see Fig. 2). On the other hand our calculations predicts a relative spectral shift of the main optical absorption peak between tilted and straight phases (Fig. 5) which is very close to the measured spectral shift between main peak position and A-shoulder. This suggest that straight phase may also contribute to A-shouder (see Fig. 6), in particular, if the coexistence of two phases will be enhanced by external conditions.
+
+Our results suggest substantially different interpretation of B-shoulder located at 685 nm (see Fig. 2). Relative intensity of B-shoulder is strongly dependent on treatment. Spectral position of B-shoulder agrees well with that of new optical absorption peak predicted for the tilted phase (Fig. 6). According to the above analysis, the B-shoulder appears as the result of splitting of the bonding and antibonding \(\pi-\)electron states due to decrease of interchain distance, relative displacement of neighboring polymer chain, that causes changes of \(\pi-\)orbitals overlap and strong increase of interchain interaction. This interpretation is in the alley of the complex optical absorption and emission study Brown et al. (2003) of poly(3-hexyl thiophene) polymers, where the measured lowest energy feature in the \(\pi-\pi^{*}\) region of the optical absorption spectrum was associated with an interchain absorption, the intensity of which was shown to be correlated with the degree of order in the polymer.
+
+Results of this work highlight further studies of polymer optics. In particular, our conclusions about flexibilities of aggregation structures made from the comparison between theory and experiment are based on few assumsions. Agreement between calculated and measured optical absorption spectra shown in Fig. 5 is achieved by assumtion of the coexistance in solution of aggregated and non-aggregated phases with relative weight of 0.\(\overline{2}\):1.0. Note that our analysis is currently limitted to optical absorption attributed to the \(\pi-\pi^{*}\) delocalized electron excitations. Analysis of theoretical predictions of the local atomic structure modifications in polymer chains in solution obtained in this work remains out of the scope of present paper since additional experimental data are required. Infrared and Raman spectroscopy methods are sensitive to the local vibrations and could therefore provide with more direct information about atomic structure modifications. In combination with modeling this additional results will be very important for detailed understanding of ground state and optics of polymers as well as for polymer materials engineering, which is a subject of future studies in the field.
+
+## V Conclusions
+
+The optical absorption spectra of PTV conjugated polymers have been studied both experimentally and theoretically. At equilibrium, the PTV chains are characterized by two different geometric configurations: straight and tilted, with the tilted configuration more energetically favorable. These phases are characterized with different interchain distances and chain distortions. Measured optical absorption spectra indicate appearance of long-wavelength shoulder strongly dependent on polymer treatment (heating and sonification). Based on the first principle modeling of the ground state and optical absorption, the measured long-wavelength shoulder in PTV polymers is interpreted as an indication of increased interchain interaction in tilted phase.
+
+## VI Acknowledgment
+
+This work is supported by STC MDITR NSF DMR-0120967, NSF PREM DRM-0611430, NSF NCN EEC-0228390, and NASA CREAM NCC3-1035.
+
+## References
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+

arxiv_ground_truth/0802.0733.md

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+# Optimal boarding method for airline passengers
+
+Jason H. Steffen
+
+jsteffen_AT_fnal.gov
+MS 127, PO Box 500
+Batavia, IL 60510
+
+###### Abstract
+
+Using a Markov Chain Monte Carlo optimization algorithm and a computer simulation, I find the passenger ordering which minimizes the time required to board the passengers onto an airplane. The model that I employ assumes that the time that a passenger requires to load his or her luggage is the dominant contribution to the time needed to completely fill the aircraft. The optimal boarding strategy may reduce the time required to board and airplane by over a factor of four and possibly more depending upon the dimensions of the aircraft. In addition, knowledge of the optimal boarding procedure can inform decisions regarding changes to methods that are employed by a particular carrier. I explore some of the salient features of the optimal boarding method and discuss practical modifications to the optimal. Finally, I mention some of the benefits that could come from implementing an improved passenger boarding scheme.
+
+†
+
+Footnote †: thanks: Brinson Postdoctoral Fellow, Fermilab Center for Particle Astrophysics
+
+## 1 Introduction
+
+Several passenger boarding schemes are used by the airline industry in effort to quickly load passengers and their luggage onto an airplane. Since the passenger boarding time often takes longer than refueling and restocking the airplane its reduction could constitute a significant savings to a particular carrier, especially for airplanes which make several trips in a day.
+
+Conventional wisdom would suggest that boarding from the front to the back is the worst case but that boarding from the back to the front is optimal or nearly so. Indeed, this is the strategy that is often employed, boarding passengers in blocks from the rear of the plane to the front. In this case, conventional wisdom only provides an answer that is half right. The worst boarding method is, indeed, to board the plane from front to back. As I will show however, boarding the airplane from the back to the front is very likely the _second worst_ method.
+
+Previous studies of the airplane boarding used a variety of approaches. Pioneering work done by Van Landeghem & Beuselinck (2002) used computer simulations of several boarding schemes and found there is much room for improvement over traditional back-to-front boarding. Of particular interest is that they point out that random boarding is superior to many traditional methods. A later computer simulation study (van den Briel et al., 2005) confirmed that traditional methods are not optimal, even in light of different assumptions regarding the primary cause of delay in the boarding process. Analytic work by Bachmat et al. (2006), which modeled optimal airplane boarding as an extremal path in a two-dimensional, Lorentzian geometry, broadly confirmed the findings of Van Landeghem & Beuselinck (2002) and was able to interpret those findings in the context of their model—thus providing an explanation for why the different boarding strategies perform as they do relative to each other.
+
+Each of these studies, particularly the computational works, focused on different components of delay. Consequently, while all found traditional methods lacking, none agree on the best overall approach. A portion of their disagreement arises from differing assumptions. Another point of discrepancy stems from the fact that their focus was largely on finding the best practical approach or the best of a set of approaches. This limitation on the scope of previous works meant that they didn’t always study the same boarding methods.
+
+Thus, the question remains, “what method, practical or otherwise, gives the fastest boarding time?” It is this question that I address here. The answer to this question, whether directly applicable or not, is valuable as it can inform decisions regarding the worth changing an existing policy since it indicates how much room there is for improvement. Moreover, once the optimal boarding method is identified one can explore the reasons why it is successful and incorporate those characteristics into a more practical scheme.
+
+In this work I infer from the results of a computer simulation and an associated optimization algorithm the optimal passenger boarding method. The fundamental assumption that I make is that the bulk of the time to load the airplane is consumed by time that it takes the passengers to load their luggage. All other effects, such as the time used by passengers who stand up to retrieve an item, who sit in the wrong seat, or who must pass by someone that is already seated in their row are initially treated as negligible. A more sophisticated model could include such detail; or, many of their effects could be folded into the model via some of the available parameters or components (such as the distribution from which the passenger luggage loading times are selected).
+
+With the stated assumption, I find that the boarding time for the optimal scheme can be significantly faster than the boarding time of the worst case—between a factor of 4 and 10 faster depending upon the length of the airplane and other model parameters. In this article I describe the techniques used to find the optimal method, I interpret the results and use that interpretation to discuss the merits of some schemes that are employed by the industry, finally I give some concluding remarks. Note that I generally use the term “boarding” to refer to the boarding process itself and the term “loading” to refer to the passengers loading their luggage. Thus, boarding time and loading time are the times required to fill the aircraft and the time required to load one’s luggage respectively.
+
+## 2 Analysis Approach
+
+### Airplane and Passenger Models
+
+The nominal airplane model that I use seats 120 passengers with six passengers per row and 20 rows. Since the focus is on the general boarding procedures, there is no first-class cabin, no priority seating, and each flight is completely full. I discuss the effects of changes to this airplane model in section 3.
+
+The passengers are each assigned a seat and the number of time steps that they need to load their luggage, a random number between 0 and 100 unless otherwise stated. Other, human nature assumptions include: 1) that a person will not move unless there is enough space between them and the person in front of them—two steps in this case, 2) that if they are moving, then they will occupy any empty space in front of them prior to stopping (this results in passengers bunching-up again as they come to a halt), 3) that they require one space either in front of or behind them in order to load their luggage (this is related to the congestion parameter \(k\) introduced by Bachmat et al. (2006)), and 4) that they only load their luggage into the bins above their assigned row. In section 3 I discuss the effects of changing any of these parameters or the distribution from which the luggage loading times are assigned.
+
+This model does not include the effects of aisle vs. window seats, the clustering of passengers into companions or families, and other effects of human nature. While adding these features might improve the accuracy of the results, these effects are not likely to be the primary issue and consequently should not be the fundamental concern when finding the general strategy for a passenger boarding scheme. Moreover, many of their effects can be accounted for once the optimal boarding method, based upon the stated assumptions, is identified. I discuss these issues in sections 4 and 5.
+
+### Optimization Algorithm
+
+The algorithm that I use to find the optimal loading order is based upon a Markov Chain Monte Carlo (MCMC) algorithm and is similar to the METROPOLIS algorithm (Metropolis et al., 1953). Starting with an initial passenger order I load the airplane and record the loading time. Then, starting with that initial order, the positions of two random passengers are exchanged and the airplane is loaded again. If the airplane boards as fast or faster than the previous iteration, then I accept the current passenger order, swap the positions of two additional random passengers, and repeat the process. If the current configuration loads more slowly than the previous, then I reject the change, return to the previous configuration, and repeat the process beginning there. I stop after \(\sim 10,000\) iterations since adding additional steps does not significantly change the results.
+
+Unlike a traditional MCMC, I do not allow any configuration which loads more slowly to be accepted. Technically, this means that the stated algorithm only finds a local minimum. However, when I include this aspect the results are unchanged while the time needed to converge increases. Thus, the choice to neglect that aspect of an MCMC analysis should not affect the results stated here.
+
+That being said, it is possible for many configurations to board in the same time or to be near enough that the differences in boading time are not important. This shows that a class of configurations that are effectively equivalent is more important than a single, optimum order. For example, there is no difference in the loading time if the two aisle-seat passengers in the same row are swapped, and there is little difference in swapping two random passengers. To identify the class of optimal configurations I tabulate the differences in seat number between adjacent passengers. It is this distribution of seat number differences, the “seating distribution”, that remains effectively constant from one optimization run to the next. This fact illustrates the important point that the actual seat numbers of two adjacent passengers is less important than how far apart they sit from each other.
+
+## 3 Results
+
+The results of applying the above analysis gives the seating distribution shown in Figure 1. These results are from 100 realizations of the boarding scenario where the passengers are reassigned their luggage loading time for each of the realizations. The largest feature is the peak near a seat difference of 12. That difference corresponds to two rows, or the distance that I assume neighboring passengers require to load their luggage. Other features of the peak, aside from its location, are its shape, its height relative to the rest of the distribution, and its width. All four of these aspects depend upon the passenger and airplane model parameters and each of them could be calibrated with data as I describe below.
+
+Figure 1: Example of the resulting seating distribution obtained from 100 realizations of the luggage loading time distribution.
+
+As stated, the location of the peak corresponds to the distance required by a passenger to load his luggage. If passengers need only the space that corresponds to their assigned row, then the location of the peak would shift to a value of 6 or one row of difference. If passengers require three rows of loading space (including their assigned row), then the peak shifts to 18. This effect can be seen in Figure 2 where I make these changes while leaving all other model parameters fixed.
+
+Figure 2: Changes in the seating distribution as a function of the required “personal space” of the passengers. This shows the distribution if no space is required (black), a single row is required (gray), and two rows are required (white). Each distribution is calculated from 100 realizations of the luggage loading times.
+
+The peak of the seating distribution is symmetrical near its apex. This is because the passengers can load their luggage using either the space in front of them or the space behind them. The width of the peak is related to the number of seats per row. If there are only four seats per row, then the width of the peak is more narrow. If there are eight seats per row (still with only one aisle), then it is more broad.
+
+The height of the peak depends upon the time that the passengers take to load their luggage. If passengers load their luggage instantaneously, then the peak disappears altogether. As the luggage loading time increases the penalty for having someone out of order increases and the algorithm forces more passengers to be separated by amounts nearer the minimum row separation required to load their luggage (here between 7 and 18 seats). The height of the peak ultimately saturates when the average luggage loading time approaches the time to walk the length of the airplane. This means that if it takes significantly longer for a typical passenger to walk the length of the airplane than it does for him to load his luggage, then the passenger ordering is much less important.
+
+### The Optimal Loading Method
+
+The optimal boarding scheme is found by extrapolating from the results given above, using the insight that they provide. The reason that these seating distributions load faster than the worst case is that they allow multiple passengers to load their luggage at the same time. The peak occurs at a distance that corresponds exactly to the space needed by adjacent passengers to do so. Taking this to an extreme, we wish to find the configuration that allows the maximum number of passengers to load their luggage at all times during the boarding process. That is the case where all adjacent passengers are assigned seats that are separated by exactly two rows such that the person at the front of the line is assigned a seat on the back row. The second-order effect of windows vs. aisle seats can be incorporated here by having the passengers in the window seats enter the airplane first, then the middle seats, then the aisle seats. The resulting seat ordering could be one of those shown in Figure 3 though there are many equivalent orderings. This ordering scheme provides nearly a five-fold reduction in the time that it takes over the worst case—an improvement that gets larger with a larger aircraft.
+
+Figure 3: Examples of the optimal passenger ordering—there are other permutations which would give identical results. The seating would proceed following the patterns illustrated here. The shading only indicates the passengers that would be inside the airplane at the same time.
+
+A look at the optimal boarding method shows why loading from the back of the plane to the front does not provide any benefit. If the back two rows of passengers were to board the airplane first, they would occupy roughly 12 rows of the aisle. All but the first few would be putting their luggage away while the others waited their turn; the passengers load their luggage serially. The optimal boarding strategy uses this aisle space more efficiently since each member of the first group of passengers who enter the airplane (10 passengers in the fiducial model) can put their luggage away; they load their luggage in parallel. In this manner the aisle is not used as a passive extension of the waiting area, but rather as a place for passengers to actively situate themselves. Ideally, all of the passengers that are inside the aircraft should either be seated or be loading their luggage; with none waiting.
+
+One question that arises from this is whether or not it is practical to implement the optimal boarding scheme, where each passenger enters the airplane in a particular order. Such a scenario may well be possible since Southwest Air has recently implemented a similar policy, at least to some extent. Given that, however, there will always be some fraction of the passengers who are out of order; there will always be families or other groups who board together regardless of their assignments. These same issues will affect other boarding methods in similar ways. In the next section I test the robustness of the optimal boarding method under these circumstances. Section 5 is a comparison of a few practical boarding methods where the passengers board in groups but are ordered randomly within those groups.
+
+## 4 Robustness of the Optimal
+
+To test the robustness of the optimal boarding scheme I conducted two experiments. The first is to change the distribution from which I select the passenger’s loading time. The second test is to make random changes to the passenger ordering. These changes include swapping the locations of several random pairs of passengers and shifting the entire line by some random number (moving people at the end of the line to the front).
+
+### Changes to Loading Time Distribution
+
+To test the effect of a different distribution of luggage loading time, I ran my minimization software on 100 realizations of each of several distributions. These distributions include a uniform distribution with a given mean, a normal distribution with the same mean and with a variance equal to that mean (essentially a Poisson distribution), and an exponential distribution with the same mean. For each of these cases the resulting passenger seating distributions were statistically indistinguishable as shown by a Kolmogorov-Smirnov test (Press et al., 2002). Moreover, the time required to load the entire plane is not affected by these different distributions; it depends primarily upon the mean luggage loading time.
+
+The fact that the results of this analysis does not depend upon the distribution from which the luggage loading times were chosen indicates that the optimal boarding method does not depend upon the presence of non-Gaussian outlier disturbances (as it should not). Thus, if a particular passenger takes an unusually long time to load his luggage, the optimal method of passenger boarding is still, on average, optimal. Moreover, since one cannot generally identify which passenger—if it is a passenger—will be the cause of delay and since that passenger’s seat will be at a random location within the aircraft, the best that one can robustly expect to do is to board optimally both before and after the disturbance.
+
+### Random Shifts and Swaps
+
+Randomly shifting the line does not significantly affect the boarding time. This is because it only changes the starting point of the boarding process. All of the passengers keep their 12 seat spacing from their neighbors and so the advantage of the optimal boarding scheme is preserved.
+
+If pairs of passengers are swapped, which effectively randomizes portions of the line, then the time to board the airplane can change significantly. Indeed, a 20% increase in the boarding time results from randomly swapping only 10% of the passengers—that is 6 pairs for the case of 120 passengers. However, there is an upper limit to the adverse results of swapping passengers. Once they are completely randomized additional swaps preserve the random nature of the passenger ordering and it doesn’t get any worse. Interestingly, random boarding takes much less than half the time of the worst case boarding; indicating that randomization is not catastrophic, a finding also identified by Van Landeghem & Beuselinck (2002). Indeed, we will see in the next section that random boarding compares favorably with traditional boarding techniques. The effects of swapping pairs of passengers is shown in Figure 4.
+
+Figure 4: Boarding times for 100 realizations of passengers when pairs of passengers are swapped. The black is optimal with no swapping, then there are the histograms which correspond to 10% (dark gray), 20% (medium gray), 40% (light gray), and 60% (white) swapping. 10% means that 6 pairs of passengers are exchanged out of 120 passengers. The mean boarding times for these scenarios are 1312, 1585, 1795, 2084, and 2311 counts respectively.
+
+## 5 Practical Comparison
+
+While the optimal scheme would produce the fastest boarding times, there are issues of practicality to consider. It may be challenging to arrange all of the passengers in the proper order—though, as mentioned, at least one airline has implemented this policy. Regardless, most airlines board the airplane in groups, presumably out of convenience and in effort to reduce confusion. In this section I introduce a few practical modifications to the optimal boarding scheme and choose one to compare with existing methods.
+
+### Practical Modification to the Optimal
+
+The advantage of the optimal boarding method comes from the fact that neighboring passengers do not sit near each other and consequently can load their luggage simultaneously. One way to accomplish a similar effect while allowing blocks of passengers to board is to have each block contain passengers from widely separated rows. After trying several possibilities I chose to use, as a modification to the optimal method, blocks of three consecutive seats separated by 12.
+
+This scenario has four boarding groups and is equivalent to calling all passengers that are in even rows and from one side of the airplane. The three remaining groups are for the other side of the airplane in the same row, then the two sides of the odd rows. The loading time that results from this scheme is not as fast as the optimum, indeed it took about twice as long to board, but it was more than a factor of two faster than the worst case. I call this boarding method the “modified optimal” method. Similar boarding methods, arrived at by very different means, are suggested by Van Landeghem & Beuselinck (2002) and Bachmat et al. (2006) and these methods also prove successful in the respective studies.
+
+### Comparison with Other Methods
+
+I selected two group boarding strategies to compare with the optimal, the modified optimal, the worst case, and the second worst case (boarding from the back to the front with the passengers in order). These are: 1) ordered blocks, where a fourth of the cabin is loaded at a time starting in the back and moving to the front and 2) a scheme where the windows are boarded first, then the middle, then the aisle seats. Within each of these groups the travelers were randomly distributed. Tests which used unordered blocks (blocks of 5 rows but not in order from back to front) gave similar results to the ordered blocks and were therefore not included.
+
+The ordered block scenario reduced the boarding time to 74% of the worst case. Since the main contributor to fast boarding is having multiple passengers load their luggage at once, this method suffers from the fact that only passengers within a small portion of the airplane are boarding at a given time. Within the boarding block there is the possibility of multiple people loading their luggage at once, but it is relatively small since at most three passengers (out of 30) could be simultaneously loading their luggage.
+
+The windows-middle-aisle approach reduced the boarding time to 43% of the worst case. The advantage here is that passengers from anywhere in the cabin are allowed to board. Thus, many people can load their luggage at once and the probability having two passengers from adjacent rows near each other is small. Moreover, this probability decreases as the length of the airplane increases. Another advantage is that the second-order effect of getting past the person in the aisle is eliminated. One drawback of this approach is that many people travel in groups and would likely not adhere strictly to this particular boarding policy. In general, however, this approach did very well (as also found by van den Briel et al. (2005) though for different reasons).
+
+The modified optimal approach performed slightly better than, though almost identically to, the windows-middle-aisle approach. This method does not depend as strongly on the length of the airplane. So, for shorter airplanes it compares better while on longer planes it compares less well. An advantage that the modified optimal approach has over boarding window-middle-aisle is that it allows passengers who are travelling together and sitting side-by-side to board at the same time without boarding out of order.
+
+Random boarding, where the passengers positions in line are completely uncorrelated with their seat assignment, shares the advantage of the windows-middle-aisle method of spreading passengers throughout the length of the airplane. Moreover, it is not disadvantaged, in implementation at least, by traveling groups. The random method performed the similar to, but slightly worst than, the windows-middle-aisle method and the modified optimal method. This demonstrates that the optimal approach, even with a significant fraction of people out of order can do at least as well as or better than the most efficient of the methods that employ boarding groups.
+
+Figure 5 shows a histogram of the loading times for 100 realizations of seven different boarding schemes, each realization having different selections for the time it takes to load the luggage and different passenger ordering where applicable. These include: 1) optimal boarding, 2) the modified optimal approach, 3) the window-middle-aisle method, 4) random boarding, 5) ordered blocks from back to front, 6) back to front with all passengers in order, and 7) the worst-case (front to back with all passengers in order). We see from this figure that optimal boarding has, by far, the best improvment in loading times, nearly a factor of five faster than the worst case, more than a factor of three better than the ordered blocks, and it is more than a factor of two faster than the modified optimal, random, and windows-middle-aisle methods. This improvement grows with the length of the airplane such that an airplane that seats 240 passengers (40 rows) will board over seven times faster than the worst case!
+
+Figure 5: Histogram of the loading times for 100 realizations of seven different boarding schemes. The luggage loading time for each realization is drawn from a uniform distribution with a mean of 50 counts. These are: 1) optimal boarding (mean boarding time: 1312 counts), 2) the modified optimal approach (2670), 3) the window-middle-aisle method (2750), 4) random boarding (2846), 5) ordered blocks from back to front (4727), 6) back to front with all passengers in order (6276), and 7) the worst-case—front to back with all passengers in order (6373).
+
+## 6 Conclusion
+
+The results of this study, based upon the assumption that a passenger loading his luggage consumes the bulk of the time that it takes for him to be seated, identify the primary cause for delay in the boarding process as well as the best means to overcome these delays. By boading passengers in a manner that allows several passengers to load their luggage simultaneously the boarding time can be dramatically reduced. This result contradicts conventional wisdom and practice that loads passengers from the back of the airplane to the front. Indeed, it shows that loading from the back to the front is hardly better than the worst case scenario. The goal of an optimized boarding strategy should focus on spreading the passengers who are loading their luggage throughout the length of the airplane instead of concentrating them in a particular portion of the cabin.
+
+By boarding in groups where passengers whose seats are separated by a particular number of rows, by boarding from the windows to the aisle, or by allowing passengers to board in random order one can reduce the time to board by better than half of the worst case and by a significant amount over conventional back-to-front blocks—which, while better than the worst case performed worse than all other block loading schemes. The primary drawbacks for any of these methods is likely to be psychological instead of practical. Groups of passengers who wish to board together would be an issue to investigate from both a customer satisfaction point of view and as a component in a more detailed model.
+
+If a workable method to have passengers line up in an assigned order could be found—and it likely may be employed already, then there is the potential for a substantial savings in time. Such a savings would most likely benefit flights between nearby cities where a particular airplane would make several trips in a given day since it might allow one or two additional flights. Or, it might allow an airline to reduce the number of gates that it requires to meet its obligations since each gate would be cleared more rapidly.
+
+While the generic features of this model are well understood, a real application of it would require some data so that it can be properly calibrated. In particular, the distributions of luggage loading times, the fraction of people traveling in groups, the queueing habits of passengers, and empirical measurements of “personal space” are all pieces of information that are necessary to state these results in terms of actual times and distances instead of arbitrary time steps and lengths that are used by a computer model.
+
+Regardless of the ultimate application of this technique, the establishement of a firm lower bound can be used to inform a decision maker of the worth of further improvements to a particular boarding strategy. If an improvement could provide only a marginal gain while costing significant amounts money and time to implement then it is not likely to be worth the investment. On the other hand, if a particular strategy is clearly failing to meet the demands of competition and customer satisfaction, then knowing just how much room there is for improvement could expedite changes.
+
+In the end, the time that it takes to load passengers into the airplane affect not only the airline company and the airport, it also affects the passengers. Few enjoy standing in line longer than necessary and fewer still enjoy sitting in an airplane longer than needed. Faster boarding would be a significant improvement for all involved parties.
+
+**Acknowledgements:** J. Steffen acknowledges the generous support of the Brinson Foundation. Fermilab is supported by the U.S. Department of Energy under contract No. DE-AC02-07CH11359. A record of invention of this software and its results are on file at Fermilab.
+
+## References
+
+* Bachmat et al. (2006) E. Bachmat, D. Berend, L. Sapir, S. Skiena, and N. Stolyarov, Journal of Physics A, 39:L453-459, (2006).
+* Metropolis et al. (1953) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller., Journal of Chemical Physics, 21(6):1087-1092, (1953).
+* Press et al. (2002) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C”, Cambridge University Press, USA, (2002).
+* van den Briel et al. (2005) M. H. L. van den Briel, J. R. Villalobos, G. L. Hogg, T. Lindemann, and A. Mulé, Interfaces, 35(3):191-201, (2005).
+* Van Landeghem & Beuselinck (2002) H. Van Landeghem and A. Beuselinck, European Journal of Operational Research, 142:294-308, (2002).
+

arxiv_ground_truth/0802.2620.md

@@ -0,0 +1,171 @@
+SPIN-08/09, ITP-UU-08/09, DCPT/08/09
+
+**Dynamics of icosahedral viruses: what does Viral Tiling Theory teach us?**
+(Contribution to the proceedings of the ‘Second Mathematical Virology Workshop’, Edinburgh (6-10 August 2007)
+**Kasper Peeters ¹ and Anne Taormina ²**
+
+Footnote 1: E-mail: kasper.peeters@aei.mpg.de
+
+Footnote 2: E-mail: anne.taormina@durham.ac.uk
+
+¹_Institute for Theoretical Physics Utrecht University P.O. Box 80195, 3508 TD Utrecht, The Netherlands. ²Department of Mathematical Sciences University of Durham Durham DH1 3LE, U.K._
+
+###### Abstract
+
+We present a top-down approach to the study of the dynamics of icosahedral virus capsids, in which each protein is approximated by a point mass. Although this represents a rather crude coarse-graining, we argue that it highlights several generic features of vibrational spectra which have been overlooked so far. We furthermore discuss the consequences of approximate inversion symmetry as well as the role played by Viral Tiling Theory in the study of virus capsid vibrations.
+
+## 1 Introduction
+
+It has been experimentally observed that viruses can alter their shape to fulfill specific functions. In particular, they may swell during maturation [1, 2, 3, 4, 5, 6], twist to release their genetic material during infection, or morph during assembly. Such large scale conformational changes are consistent with the widespread hypothesis that viruses do vibrate, and it is therefore of interest to study their dynamics with the help of mathematical and computational techniques which have been tried and tested in the context of biomacromolecule vibrations (see [7] for a review).
+
+Normal mode analysis is one such method [8, 9, 10, 11], which has been successfully applied to the study of proteins and a variety of viruses to date [12, 13, 14, 15]. A major challenge is the huge number of degrees of freedom involved in such systems. Several degrees of coarse-graining, as well as group theoretical methods (inspired by their successful application to small molecules and fullerenes [16, 17, 18]), have been implemented in computer simulations in order to extract information on the low-frequency modes of vibration which are thought to be relevant for protein and virus function [19, 20]. Although such theoretical data become increasingly available for icosahedral systems [15, 14, 21] thanks to advances in computer power, a clear and insightful vibrational pattern across icosahedral viruses has not emerged yet. The art of coarse-graining is a delicate one, as it is often argued that excessive coarse-graining produces a dynamical picture that has little to do with reality. We actually need a hierarchy of coarse-grained calculations, which hopefully reveal complementary aspects of the dynamical jigsaw.
+
+We argue here that even the crudest approximation, where each capsid protein is treated as a point mass located at its centre of mass, is helpful in highlighting dynamical features that are present in more sophisticated normal mode analyses, but have been overlooked so far. Our initial mathematical motivation was to assess to which extent Viral Tiling Theory, a recently proposed model for icosahedral viral capsids which solves a classification puzzle in the Caspar-Klug nomenclature [22, 23], provides a new insight in the dynamics of viruses. In particular, we ask whether there is a correlation between the vibrational patterns of viruses with a given number of coat proteins and their viral tiling.
+
+The paper is organised as follows. In Section 2, we briefly describe Viral Tiling Theory in the context of the viral capsids RYMV (\(T=3\)), HK97 (\(T=7\ell\)) and SV40 (pseudo \(T=7d\)), with emphasis on how the underlying icosahedral symmetry manifests itself in different subtle ways for these three cases. In particular, it has implications for the group theoretical analysis of normal modes of vibrations. An expanded version of these remarks, applicable to viruses and phages of all \(T\) numbers, is available in [24]. Section 3 provides a simple normal mode analysis for the three capsids above, where group theoretical techniques reminiscent of those used in calculations of vibrational modes of small molecules are implemented. This paves the way for the more extensive study performed in [25], which reveals an intriguing universal pattern of low frequency normal modes. We conclude with some open questions prompted by our investigations.
+
+## 2 Tilings of Rice Yellow Mottle, Hong-Kong 97 and Simian Virus 40
+
+Viral tiling theory provides an elegant way of encoding the icosahedral symmetry of viral capsids by keeping track of the location of coat proteins and the orientation of capsomers on the viral shell, while also keying in the dominant³ bond structure between those proteins.
+
+Footnote 3: In some cases, there exist stronger bonds between proteins pertaining to different tiles; RYMV is an example.
+
+The Rice Yellow Mottle Virus (RYMV) belongs to the Sobemovirus genus. It is classified as a \(T=3\) virus in the Caspar-Klug labelling system [26], and its icosahedral capsid accommodates 180 coat proteins or subunits which are clustered in 12 pentamers around the 5-fold axes and 20 hexamers about the 3-fold global symmetry axes of the icosahedron. The location of the proteins are consistent with a triangular tiling à la Caspar-Klug, and each triangular tile encodes trimer interactions between coat proteins, as represented in Fig. 1.
+
+Figure 1: _In the above picture, the location of the \(T=3\) RYMV capsid proteins coincide with the location of the centre of mass of each of them, calculated from the experimental data collected in the file 1f2n.vdb. A triangular tiling (dashed lines) is superimposed on the icosahedral structure, and the colour coding is faithful to that of the VIPER website: the A chain is blue, the B chain, red and the C chain, green. The grey shaded triangular prototiles highlight trimer interactions between capsid proteins, while the blue shaded region corresponds to the fundamental domain of the proper rotation subgroup \({\cal I}\) of the full icosahedral group \(H_{3}\) ._
+
+The HK97 bacteriophage on the other hand has a \(T=7\ell\) capsid made of 420 proteins arranged in 12 pentamers and 60 hexamers, with four types of dimer interactions modelled by rhomb prototiles; see Fig. 2. The SV40 virus is a member of the Polyomaviridae family and has a pseudo \(T=7d\) capsid which accommodates 360 coat proteins organised in pentamers through two types of spherical prototiles, namely rhombs, encoding two types of dimer interactions, and kites encoding trimer interactions, as represented in Figure 2 of reference [27]. SV40 is an example of an all-pentamer capsid, for which the Caspar-Klug classification is not applicable, and whose symmetries are captured by Viral Tiling Theory.
+
+Figure 2: _The location of the \(T=7\ell\) HK97 capsid proteins coincide with the location of the centre of mass of each of them, calculated from the experimental data collected in the file 2fte.vdb. A rhomb tiling (dashed lines) is superimposed on the icosahedral structure, and the colour coding is faithful to that of the VIPER website: A chain (blue), B chain (red), C chain (green), D chain (yellow), E chain (cyan), F chain (purple) and G chain (pink). The rhomb prototiles highlight four types of dimer interactions between capsid proteins, while the blue shaded region corresponds to the fundamental domain of the proper rotation subgroup \({\cal I}\) of the full icosahedral group \(H_{3}\) ._
+
+In order to extract qualitative features of vibrational patterns from viral capsids, we restrict ourselves to a coarse-grained approximation where each capsid protein is replaced by a point mass whose location coincides with the centre of mass of the protein considered. This centre of mass is calculated by taking into consideration all crystallographically identified atoms of the protein, according to data stored in the Protein Data Bank or equivalently the VIPER website. We then assess how much deviation there is between the above distribution of point masses and a theoretical distribution exhibiting a centre of inversion. On the basis of the experimental data available to us, we argued in [24] that the SV40 capsid has an approximate centre of inversion, while RYMV ⁴ and HK97 do not. This has subtle consequences for the group-theoretical properties of normal modes of vibrations: when the capsid exhibits an effective centre of inversion, the group involved is the full icosahedral group \(H_{3}\) with 120 elements (usually called \({\cal I}_{h}\) in the science literature), while it is reduced to its subgroup \({\cal I}\) of 60 proper rotations in the absence of a centre of inversion.
+
+Footnote 4: The argument for RYMV is similar to the argument given for TBSV in [24].
+
+A viral capsid with \(N\) ‘point mass’ coat proteins has \(3N\) degrees of freedom, and hence \(3N\) modes of vibrations, of which 6 are associated with 3 rotations and 3 translations of the capsid as a whole. These are therefore not genuine normal modes of vibration. Group theory accounts for the degeneracies of these vibrational modes, and provides a mean to organize the normal mode spectrum of a given capsid [28]. A key ingredient in this exercise is the construction of the displacement representation of the given capsid, which is a reducible representation of \(H_{3}\) or \({\cal I}\) according to whether the distribution of capsid proteins exhibits a centre of inversion or not. Such representation consists of 120 (resp. 60) matrices Γ3⁢Ndispl(g),g∈H3(resp.ℐ) of size \(3N\times 3N\), which encode how proteins are interchanged under the action of each element \(g\), as well as how the displacements of each protein from the equilibrium position are rotated under the action of \(g\). The latter information is gathered in \(3\times 3\) rotation matrices \(R(g)\) which form an irreducible representation of \(H_{3}\) (resp. ℐ), while the former is encoded in permutation matrices \(P(g)\) of size \(N\times N\), so that we have
+
+Γ3⁢Ndispl(g)=P(g)⊗R(g),∀g∈H3(resp.ℐ). (2.1)
+
+The permutation matrices \(P(g)\) act on vectors whose components are the vector positions r→i  0,i=1,..,N of the N proteins at equilibrium. The entry \(P_{ij}(g)\) of the permutation matrix is 1 if \(\vec{r}^{\,\,0}_{j}\) is mapped on \(\vec{r}^{\,\,0}_{i}\) by \(g\), and is zero otherwise.
+
+Once the displacement representation is constructed, it remains to invoke the well-known property that it can be written in block diagonal form with the help of a (\(3N\times 3N\)) matrix \(U\)⁵ with
+
+Footnote 5: The explicit form of the matrix \(U\) is not needed at this stage, but rather when the force matrix is partially diagonalised to obtain the frequencies of vibration.
+
+\[U\Gamma^{\text{displ}}_{3N}(g)U^{-1}=\Gamma^{\text{displ}\,\,^{\prime}}_{3N}(g)=\oplus_{p}n_{p}\Gamma^{p}(g),\] (2.2)
+
+where the multiplicities \(n_{p}\) are obtained via the following character formula
+
+\[n_{p}=\frac{1}{{\rm dim}\,H_{3}}\sum_{g\in H_{3}}\chi^{\text{displ}}(g)^{*}\,\chi^{p}(g)\qquad{\rm or}\qquad n_{p}=\frac{1}{{\rm dim\,{\cal I}}}\sum_{g\in{\cal I}}\chi^{\text{displ}}(g)^{*}\,\chi^{p}(g).\] (2.3)
+
+The characters \(\chi^{p}(g)\) of irreducible representations of the icosahedral group can be found in [24], while the characters of the displacement representations \(\chi^{\text{displ}}(g)\) are obtained by inspection of the displacement representation considered. Note that, in view of the very definition of the permutation matrices \(P(g)\) given in the previous subsection, and the fact that the characters of a representation are the traces of its constituent matrices, one has
+
+\[\chi^{\text{displ}}(g)={\rm Tr}\,(P(g))\,{\rm Tr}\,(R(g))=\pm({\rm number\,of\,proteins\,unmoved\,by\,}g)\cdot(1+2\cos\theta),\] (2.4)
+
+where \(\theta\) is the angle of the proper rotation associated with \(g\), and the minus sign is taken when \(g\in H_{3}\setminus{\cal I}\). So \(\chi^{\text{displ}}(g)\) is zero when \(\theta=\frac{2\pi}{3}\) or whenever \(g\) is such that no protein of a given capsid is kept fixed under its action.
+
+The decomposition of the displacement representation of a given capsid boils down to the knowledge of the coefficients \(n_{p}\) in (2.3) which, in view of the expression (2.4), are non zero whenever at least one capsid protein is unmoved under the action of an element \(g\) (and \(\theta\neq\frac{2\pi}{3}\)).
+
+It can be shown that distributions of capsid proteins with no centre of inversion are such that the only group element which keeps any ‘point mass’ protein unmoved is the identity element \(g=e\) (and under its action, all \(N\) proteins are obviously fixed). The second expression in (2.3) thus yields
+
+\[n_{p}=\frac{1}{{\rm dim}\,{\cal I}}\chi^{\text{displ}}(e)^{*}\,\chi^{p}(e)=\frac{3N}{60}\,\chi^{p}(e),\] (2.5)
+
+where we used \(\rm{dim}\,{\cal I}=60\) and \(\chi^{\text{displ}}(e)=3N\) (taking the plus sign and \(\theta=0\) in (2.4)). Recalling that \(\chi^{p}(e)=p\), we arrive at the following decomposition formula,
+
+\[\Gamma^{\text{displ}\,^{\prime}}_{3N}=\frac{3N}{60}\left\{\Gamma^{1}_{+}+3\Gamma^{3}_{+}+3\Gamma^{3^{\prime}}_{+}+4\Gamma^{4}_{+}+5\Gamma^{5}_{+}\right\}.\] (2.6)
+
+The number \(N\) of capsid proteins is always a multiple of sixty, \(N=60k\). In the many cases where the proteins are organised in 12 pentamers and a number of hexamers, \(k\) is the \(T\)-number of the Caspar-Klug nomenclature. Then, the number of non-degenerate normal modes in the singlet (symmetric) representation \(\Gamma^{1}_{+}\) is \(3T\), while the number of \(p\)-fold degenerate normal modes (corresponding to the \(p\)-dimensional representation \(\Gamma^{p}_{+}\)) is \(3p^{2}T\), for \(p=3,4\) and \(5\). In particular, \(N=180\) for RYMV, and the displacement representation decomposes into
+
+\[\Gamma^{\text{displ}\,^{\prime}}_{540,\text{RYMV}}=9\Gamma^{1}_{+}+27\Gamma^{3}_{+}+27\Gamma^{3^{\prime}}_{+}+36\Gamma^{4}_{+}+45\Gamma^{5}_{+}.\] (2.7)
+
+The 6 non-genuine modes belong to two copies of the \(\Gamma^{3}_{+}\) irreducible representation. There are nine non-degenerate and forty-five 5-fold degenerate Raman active modes, as well as twenty-five 3-fold degenerate infrared active modes.
+
+Since \(N=420\) for HK97, the displacement representation decomposes into
+
+\[\Gamma^{\text{displ}\,^{\prime}}_{1260,\text{HK97}}=21\Gamma^{1}_{+}+63\Gamma^{3}_{+}+63\Gamma^{3^{\prime}}_{+}+84\Gamma^{4}_{+}+105\Gamma^{5}_{+},\] (2.8)
+
+and by the same argument as above, one arrives at twenty-one non-degenerate and one hundred and five 5-fold degenerate Raman active modes, as well as sixty-one 3-fold degenerate infrared active modes.
+
+The normal modes of the SV40 capsid would be organised according to the decomposition (2.6) with \(N=360\) if we were not taking into account that the protein distribution on the capsid exhibits an approximate centre of inversion. We would have
+
+\[\Gamma^{\text{displ}\,^{\prime}}_{1080,\text{SV40}}=18\Gamma^{1}_{+}+54\Gamma^{3}_{+}+54\Gamma^{3^{\prime}}_{+}+72\Gamma^{4}_{+}+90\Gamma^{5}_{+}.\] (2.9)
+
+Instead, we use the first expression in (2.3) and note that the distribution of ‘point-mass’ proteins on the capsid is such that, besides the identity element \(g=e\) in \(H_{3}\) which leaves all \(N\) proteins unmoved, the fifteen rotations g2(i),i=1,..,15 about the 2-fold axes of the icosahedron, when combined with the inversion \(g_{0}\), produce 15 further elements \(g_{0}g^{(i)}_{2}\) which altogether leave 24 capsid proteins unmoved. Those fifteen group elements are in the same conjugacy class and therefore have the same character \(\chi^{p}(g_{0}g^{(i)}_{2})=1\) for \(p=1,5\), \(\chi^{p}(g_{0}g^{(i)}_{2})=-1\) for \(p=3,3^{\prime}\) and \(\chi^{4}(g_{0}g^{(i)}_{2})=0\). Taking into account that for these group elements, \(Tr(R(g_{0}g^{(i)}_{2}))=-(1+2\cos\pi)=1\), we arrive at the following decomposition of the displacement representation,
+
+\[\Gamma^{\text{displ}\,^{\prime}}_{1080,\text{SV40}}=12\Gamma^{1}_{+}+24\Gamma^{3}_{+}+24\Gamma^{3^{\prime}}_{+}+36\Gamma^{4}_{+}+48\Gamma^{5}_{+}+6\Gamma^{1}_{-}+30\Gamma^{3}_{-}+30\Gamma^{3^{\prime}}_{-}+36\Gamma^{4}_{-}+42\Gamma^{5}_{-}.\] (2.10)
+
+The six non-genuine modes of vibrations are confined to one copy of the 3-dimensional irreducible representation \(\Gamma^{3}_{+}\), and one copy of the 3-dimensional irreducible representation \(\Gamma^{3}_{-}\). There are twelve non-degenerate and forty-eight 5-fold degenerate Raman active modes, and fifty-two 3-fold degenerate infrared active modes .
+
+## 3 Low frequency modes
+
+Our calculation of the low frequency normal modes is based on a spring-mass model, where the \(N\) ‘point-mass’ proteins are connected by a network of elastic forces described by a harmonic potential which is manifestly rotation and translation invariant,
+
+\[V={\sum_{\begin{subarray}{c}m<n\\ m,n=1\end{subarray}}^{N}\,\frac{1}{2}\kappa_{mn}\,(|\vec{r}_{m}-\vec{r}_{n}|-|\vec{r}^{\,0}_{m}-\vec{r}^{\,0}_{n}|)^{2}}.\] (3.11)
+
+The associated force matrix or ‘Hessian’ is given by
+
+\[F^{ij}_{mn}=\frac{\partial^{2}V}{\partial r_{m}^{i}\partial r_{n}^{j}}\Bigg{|}_{x=x_{0}}=\left\{\begin{array}[]{ll}{\sum_{p\neq m}\kappa_{mp}\frac{(r_{m}-r_{p})^{i}\,(r_{m}-r_{p})^{j}}{(r_{m}-r_{p})^{2}}\Bigg{|}_{r=r_{0}}}&{\rm if}\,m=n,\\ {-\kappa_{mn}\frac{(r_{m}-r_{n})^{i}\,(r_{m}-r_{n})^{j}}{(r_{m}-r_{n})^{2}}\Bigg{|}_{r=r_{0}}}&{\rm otherwise.}\end{array}\right.\] (3.12)
+
+In the above formulae, the vector \(\vec{r}^{\,0}_{m}\) refers to the equilibrium position of protein \(m\) and the vector \(\vec{r}_{m}\) of components \(r_{m}^{i}\), to its position after elastic displacement, all vectors originating at the centre of the capsid. The masses of the proteins are all set to unity (reflecting the fact that the various protein chains in a capsid have masses which are too good approximation identical), and \(\kappa_{mn}\) is the spring constant of the spring connecting protein \(m\) to protein \(n\). The set of non-zero spring constants we choose, i.e. the topology of the elastic network we adopt, is dictated by the information derived from the association energies listed in VIPER for RYMV (1f2n.vdb), HK97 (2fte.vdb) and SV40 (1sva.vdb). Fig. 3 encodes the bonds provided by VIPER _before_ acting on them with the icosahedral group in order to generate the complete spring network.
+
+Figure 3: _Inter-protein bonds are given for (a) RYMV, (b) HK97 and (c) SV40. The relative strengths are represented by line segments of varying thickness, from the strongest bonds (thick lines) to the weakest (thin lines). The above diagrammes should be read in conjunction with the figures of Section 2._
+
+We have used the relative values of these energies, and therefore we are left with one parameter \(\kappa\) in the force matrix, which sets the overall scale of the vibration frequencies. We are not aware of any experimental measurements of association energies between capsid proteins for the viruses and phages we are considering, and the _absolute_ theoretical values calculated in [29] must be taken with extreme caution.
+
+The force matrices \(F^{ij}_{mn}\) we consider here have size \(3N\times 3N\) with \(N=180\) for RYMV, \(N=420\) for HK97 and \(N=360\) for SV40. Although computers can handle a brute force diagonalisation of such matrices, and provide eigenvalues which are the square of the sought frequencies of vibration of normal modes, a group theoretical approach reduces considerably the size of the matrices to be diagonalized and above all, yields information on the distribution of normal modes within irreducible representations of the icosahedral group. This proves to be useful in an analysis of universal features of such vibrations.
+
+We have calculated the lowest frequency modes of vibration for the RYMV, HK97 and SV40 capsids using well-known group theoretical methods. The association energies listed in Viper for RYMV allow for a stable capsid. Crucial to the stability are the C-arms linking together distant proteins of the C chain in Fig. 3a. The spectrum of the first 40 low frequency modes is presented in Fig. 4a. Apart from the six zero modes associated with the rotations and translations of the capsid as a whole, and which belong to two copies of the irreducible representation \(\Gamma_{+}^{3}\) of \({\cal I}\), one notices a cluster of 24 normal modes of very low and similar frequencies organized in a sum of irreducible representations according to \(\Gamma_{+}^{5}+\Gamma_{+}^{3^{\prime}}+\Gamma_{+}^{5}+\Gamma_{+}^{4}+\Gamma_{+}^{3^{\prime}}+\Gamma_{+}^{4}\). This low plateau is disrupted by a significant jump in wave number.
+
+Figure 4: _Spectrum of low frequency normal modes for the RYMV capsid (a) and the HK97 capsid (b). The triangular-shape modes \(\rhd\) belong to 3-dimensional irreducible representations \(\Gamma_{+}^{3}\) of the icosahedral group, while the triangular-shape modes \(\triangle\) belong to 3-dimensional irreducible representations \(\Gamma_{+}^{3^{\prime}}\). Accordingly, the diamond-shape modes belong to 4-dimensional irreducible representations and the pentagon-shape modes to 5-dimensional irreducible representations. The \(x\)-axis labels the normal modes while the \(y\)-axis gives the wave numbers in cm-1 (up to an overall normalisation which cannot be fixed from Viper data)._
+
+A similar analysis was performed for HK97. The association energies listed in Viper for HK97 allow for a nearly-stable capsid, with nine strictly zero modes instead of the six expected. The 21 subsequent modes have similar frequencies, as can be seen from Fig. 4b. They are organized in the following sum of irreducible representations of \({\cal I}\): \(\Gamma_{+}^{4}+\Gamma_{+}^{4}+\Gamma_{+}^{5}+\Gamma_{+}^{3^{\prime}}+\Gamma_{+}^{5}\). If the spurious triplet of zero modes were lifted by the addition of extra bonds in the spring-mass model of HK97, one would again observe a cluster of 24 low frequency normal modes forming a plateau disrupted by a jump of the same scale as that appearing in RYMV. We have observed this phenomenon in a large number of viral capsids, and we will detail our findings in [25].
+
+The SV40 case is particularly interesting because it does not quite fit with the above observations. As mentioned in Section 2, the viral capsid has a near centre of inversion, and one might want to explore the implications of treating the normal mode analysis with a symmetry-corrected ‘point-mass’ protein distribution. This, however, destabilizes the capsid, as the vertices of some triangular cells in the network become collinear. We will therefore refrain from considering a capsid with a centre of inversion, and perform the normal mode analysis as in the two previous cases (RYMV and HK97). Once more, we have plotted the low frequency spectrum in Fig. 5.
+
+Figure 5: _Low frequency normal modes for the SV40 capsid. Symbol conventions as in Fig. 4._
+
+Apart from the six zero modes associated with the rotations and translations of the capsid as a whole, one could argue that the next 23 modes should be considered as a cluster since their frequencies are very similar. However the plateau in this case is not disrupted by a spectacular jump in frequency, as the seven subsequent frequencies are roughly 1.6 larger than the first 23 non-zero modes. Early comparison with Murine Polyomavirus vibrational patterns does not shed light on the significance of these all-pentamer viral capsids spectra, and more investigations are needed.
+
+## 4 Conclusion
+
+We have discussed the vibration spectrum of icosahedral virus capsids, obtained from a coarse-grained model in which protein chains and their interactions are replaced by a spring-mass model. The goal of this programme is to understand, in a top-down approach, how properties of the capsid structure, such as an approximate inversion symmetry or a particular tiling type, reflect themselves in the vibrational spectrum. We believe this a useful complement to existing bottom-up approaches, which are rooted in all-atom computations.
+
+A comparison of our results with the spectra obtained in earlier all-atom computations reveals some interesting similarities. The most striking one is the existence of a low-frequency plateau of 24 modes, separated by a rather large gap from the remainder of the spectrum. This plateau is present for RYM as well as HK’97 and a large number of other virus capsids. It has been seen before in isolated examples [13, 14, 30], but the simplicity of our model offers a better chance to understand the general reason behind its existence (more details will be provided in [25]).
+
+While Viral Tiling Theory provides a beautiful classification of the structure of virus capsids, its role in understanding the vibration spectra is at present less clear. Besides the bonds which bind together proteins on the same tile, many other bonds are required in order to obtain a stable capsid. These other, inter-tile bonds are often of a similar strength as the bonds on a single tile. In fact, it is an interesting mathematical problem to understand the best network topology (in terms of the optimal number of bonds) required for stability of a capsid.
+
+The present analysis focuses exclusively on the viral capsid, ignoring in particular the interaction of the virion with its environment and the presence of matter within the shell, which are undoubtedly worth considering in more elaborated models. Large-scale simulations have revealed that some virus capsids are unstable without RNA content [21]. It would be interesting to understand this instability, as well as the effect of RNA content, for larger classes of capsids.
+
+## References
+
+* Canady et al. [2000] M. A. Canady, M. Tihova, T. N. Hanzlik, J. E. Johnson, and M. Yeager, “Large conformational changes in the maturation of a simple RNA virus, _Nudaurelia capensis_ omega virus (N omega V)”, _J. Mol. Biol._**299** (2000) 573–584.
+* Conway et al. [2001] J. F. Conway, W. R. Wikoff, N. Cheng, R. L. Duda, R. W. Hendrix, J. E. Johnson, and A. C. Steven, “Virus maturation involving large subunit rotations and local refolding”, _Science_**292** (2001) 744–748.
+* Wikoff et al. [2000] W. Wikoff, L. Liljas, R. Duda, H. Tsuruta, R. Hendrix, and J. Johnson, “Topologically linked protein rings in the bacteriophage HK97 capsid”, _Science_**289** (2000) 2129–2133.
+* Heymann et al. [2003] J. B. Heymann, N. Q. Cheng, W. W. Newcomb, B. L. Trus, J. C. Brown, and A. C. Steven, “Dynamics of herpes simplex virus capsid maturation visualized by time-lapse cryo-electron microscopy”, _Nature Struct. Biol._**10** (2003) 334–341.
+* Kuhn et al. [2002] R. J. Kuhn, W. Zhang, M. G. Rossmann, S. V. Pletnev, J. Corver, and E. Lenches, “Structure of dengue virus: implications for flavivirus organization, maturation and fusion”, _Cell_**108** (2002) 717–725.
+* Jiang et al. [2003] W. Jiang, Z. L. Li, Z. X. Zhang, M. L. Baker, P. E. Prevelige, and W. Chiu, “Coat protein fold and maturation transition of bacteriophage P22 seen at subnanometer resolutions”, _Nature Struct. Biol._**10** (2003) 131–135.
+* Chou [1988] K. C. Chou, “Low-frequency collective motion in biomacromolecules and its biological functions”, _Biophys. Chem._**30** (1988) 3–48.
+* Noguti and Go [1982] T. Noguti and N. Go, “Collective variable description of small-amplitude conformational fluctuations in a globular protein”, **296** (1982) 776.
+* Go and Noguti [1983] N. Go and T. Noguti, T. Nishikawa, “Dynamics of a small globular protein in terms of low-frequency vibrational modes”, _Proc. Nat. Acad. Sci._**80** (1983) 3696.
+* Brooks and Karplus [1983] B. Brooks and M. Karplus, “Harmonic dynamics of proteins: normal modes and fluctuations in bovine pancreatic trypsin inhibitor”, _Proc. Nat. Acad. Sci._**80** (1983) 6571.
+* Levy et al. [1984] R. Levy _et al._, “Quasi-harmonic method for studying very low frequency modes in proteins”, _Biopolymers_**23** (1984) 1099.
+* Kurkcuoglu et al. [????] O. Kurkcuoglu, R. Jernigan, and P. Doruker, “Mixed levels of coarse-graining of large proteins using elastic network model succeeds in extracting the slowest motion”, _Polymer_**45** 649–657.
+* Tama and Brooks [2002] F. Tama and C. L. Brooks, “The mechanism and pathway of pH induced swelling in cowpea chlorotic mottle virus”, _J. Mol. Biol._**318** (2002) 733.
+* Tama and Brooks [2005] F. Tama and C. L. Brooks, “Diversity and identity of mechanical properties of icosahedral viral capsids studied with elastic network normal mode analysis”, _J. Mol. Biol._**345** (2005) 299.
+* Vlijmen and Karplus [2005] H. W. T. v. Vlijmen and M. Karplus, “Normal mode calculations of icosahedral viruses with full dihedral flexibility by use of molecular symmetry”, _J. Mol. Biol._**350** (2005) 528–542.
+* Wilson [1934] E. Wilson, “The normal modes and frequencies of vibration of the regular plane hexagon model of the benzene molecule”, _Phys. Rev._**45** (1934) 706–714.
+* Weeks and Harter [1989] D. E. Weeks and W. G. Harter, “Rotation-vibration spectra of icosahedral molecules. II. Icosahedral symmetry, vibrational eigenfrequencies, and normal modes of buckminsterfullerene”, _J. Chem. Phys._**90** (1989) 4744–4771.
+* Wu et al. [1987] Z. C. Wu, D. A. Jelski, and T. F. George, “Vibrational modes of Buckminster fullerenes”, _Chem. Phys. Lett._**137** (1987) 291–294.
+* Simonson and Perahia [1992] T. Simonson and D. Perahia, “Normal modes of symmetric protein assemblies”, _Biophys. J._**61** (1992) 427.
+* Vlijmen and Karplus [2001] H. W. T. v. Vlijmen and M. Karplus, “Normal mode analysis of large systems with icosahedral symmetry: Application to (Dialanine)₆₀ in full and reduced basis set implementations”, _J. Comput. Phys._**115** (2001) 691–698.
+* Freddolino et al. [2006] P. Freddolino, A. Arkhipov, S. Larson, A. McPherson, and K. Schulten, “Molecular dynamics simulations of the complete satellite tobacco mosaic virus”, _Structure_**14** (2006) 437–449.
+* Twarock [2004] R. Twarock, “A tiling approach to virus capsid assembly explaining a structural puzzle in virology”, _J. Theor. Biol._**226** (2004) 477.
+* Twarock [2005] R. Twarock, “The architecture of viral capsids based on tiling theory”, _J. Theor. Medicine_**6** (2005) 87–90.
+* ElSawy et al. [????] K. M. ElSawy, A. Taormina, R. Twarock, and L. Vaughan, “Dynamical implications of viral tiling theory”, arXiv:0711.0541 [q-bio.BM], to appear in J. Theor. Biol.
+* Peeters and Taormina [????] K. Peeters and A. Taormina, “Viral tilings, group theory and the dynamics of viral capsids”, in preparation.
+* Caspar and Klug [1962] D. Caspar and A. Klug, “Physical principles in the construction of regular viruses”, _Cold Spring Harbor Sympos. Quant. Biol._**27** (1962) 1–24.
+* Keef et al. [2005] T. Keef, A. Taormina, and R. Twarock, “Assembly models of papovaviridae based on tiling theory”, _Phys. Biol._**2** (2005) 175–188, q-bio/0508031.
+* Cornwell [1984] J. F. Cornwell, “Group theory in physics, vol. 1”, Academic Press, 1984.
+* Reddy et al. [1998] V. Reddy, H. Giesing, R. Morton, A. Kumar, C. Post, C. r. Brooks, and J. Johnson, “Energetics of quasiequivalence: computational analysis of protein-protein interactions in icosahedral viruses”, _Biophys._**74** (1998) 546.
+* Rader et al. [2005] A. Rader, D. Vlad, and I. Bahar, “Maturation dynamics of bacteriophage HK97”, _Structure_**13** (2005) 413–421.
+

arxiv_ground_truth/0802.3310.md

@@ -0,0 +1,497 @@
+# A characterization of quadric constant mean curvature hypersurfaces of spheres
+
+Luis J. Alías
+
+Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, E-30100 Espinardo, Murcia, Spain
+ljalias@um.es
+
+Aldir Brasil Jr
+
+Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza-Ce, Brazil
+aldir@mat.ufc.br
+
+Oscar Perdomo
+
+Department of Mathematical Sciences, Central Connecticut State University, New Britain, CT 06050, USA
+perdomoosm@ccsu.edu
+Dedicated to the memory of Professor Luis J. Alías-Pérez
+
+###### Abstract.
+
+Let \(\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}\) be an immersion of a complete \(n\)-dimensional oriented manifold. For any \(v\in\mathbb{R}^{n+2}\), let us denote by \(\ell_{v}:M\to\mathbb{R}\) the function given by \(\ell_{v}(x)={\langle}\phi(x),v{\rangle}\) and by \(f_{v}:M\to\mathbb{R}\), the function given by \(f_{v}(x)={\langle}\nu(x),v{\rangle}\), where \(\nu:M\to\mathbb{S}^{n}\) is a Gauss map. We will prove that if \(M\) has constant mean curvature, and, for some \(v\neq{\bf 0}\) and some real number \(\lambda\), we have that \(\ell_{v}=\lambda f_{v}\), then, \(\phi(M)\) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface \(M^{n}\) in \(\mathbb{S}^{n+1}\) which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to \(2n+4\).
+
+Key words and phrases: constant mean curvature, Clifford hypersurface, stability operator, first eigenvalue 2000 Mathematics Subject Classification: Primary 53C42, Secondary 53A10 L.J. Alías was partially supported by MEC project MTM2007-64504, and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010). A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.
+
+## 1. Introduction
+
+Let \(\phi:M\to\mbox{$\mathbb{S}^{n+1}$}\subset\mbox{$\mathbb{R}^{n+2}$}\) be an immersion of a complete \(n\)-dimensional oriented manifold. For every \(x\in M\) we will denote by \(T_{x}M\) the tangent space of \(M\) at \(x\). Sometimes, specially when we are dealing with local aspects of \(M\), we will identify \(M\) with the set \(\phi(M)\subset\mbox{$\mathbb{R}^{n+2}$}\), and the space \(T_{x}M\) with the linear subspace \(d\phi_{x}(T_{x}M)\) of \(\mathbb{R}^{n+2}\). Let us denote by \(\nu:M\to\mbox{$\mathbb{S}^{n+1}$}\subset\mbox{$\mathbb{R}^{n+2}$}\), a normal unit vector field along \(M\), i.e., for every \(x\in M\), \(\nu(x)\) is perpendicular to the vector \(x\) and to the vector space \(T_{x}M\). The shape operator \(A_{x}:T_{x}M\to T_{x}M\), is given by \(A_{x}(v)=-d\nu_{x}(v)=-\beta^{\prime}(0)\) where \(\beta(t)=\nu(\alpha(t))\) and \(\alpha(t)\) is any smooth curve in \(M\) such that \(\alpha(0)=x\) and \(\alpha^{\prime}(0)=v\). It can be shown that the linear map \(A_{x}:T_{x}M\to T_{x}M\) is symmetric, therefore it has \(n\) real eigenvalues \(\kappa_{1}(x),\ldots,\kappa_{n}(x)\). These eigenvalues are known as the principal curvatures of \(M\) at \(x\). The mean curvature of \(M\) at \(x\) is the average of the principal curvatures,
+
+\[H(x)=\frac{\kappa_{1}(x)+\cdots+\kappa_{n}(x)}{n},\]
+
+and the norm square of the shape operator is defined by the equation
+
+\[\|A\|^{2}(x)=\mathrm{trace}(A_{x}^{2})=\kappa_{1}^{2}(x)+\cdots+\kappa_{n}^{2}(x).\]
+
+### Examples: Totally umbilical spheres and Clifford hypersurfaces
+
+In this section we will describe two families of examples that are related with the main result of this paper.
+
+**Example 1****.**: Let \(v\in\mbox{$\mathbb{R}^{n+2}$}\) be a fixed unit vector and \(c\) a real number with \(|c|<1\). Let us define
+
+\[\mbox{$\mathbb{S}^{n}$}(v,c)=\{x\in\mbox{$\mathbb{S}^{n+1}$}:{\langle}x,v{\rangle}=c\}.\]
+
+Clearly, \(\mbox{$\mathbb{S}^{n}$}(v,c)\) is a hypersurface of \(\mathbb{S}^{n+1}\). In this case the map \(\nu:\mbox{$\mathbb{S}^{n}$}(v,c)\to\mbox{$\mathbb{S}^{n+1}$}\) given by
+
+\[\nu(x)=\frac{1}{\sqrt{1-c^{2}}}\left(v-cx\right)\]
+
+is a normal unit vector field along \(\mbox{$\mathbb{S}^{n}$}(v,c)\). Therefore, for every \(x\in\mbox{$\mathbb{S}^{n}$}(v,c)\) the shape operator \(A_{x}\) is the map \(c(1-c^{2})^{-{1\over 2}}I\), where \(I\) is the identity map, and
+
+\[\kappa_{1}(x)=\cdots=\kappa_{n}(x)=\frac{c}{\sqrt{1-c^{2}}}\]
+
+for all \(x\in\mbox{$\mathbb{S}^{n}$}(v,c)\). It is not difficult to show that these examples are the only totally umbilical complete hypersurfaces of \(\mathbb{S}^{n+1}\). In this case
+
+\[H=\frac{c}{\sqrt{1-c^{2}}}\quad\hbox{and}\quad\|A\|^{2}=\frac{nc^{2}}{1-c^{2}}\]
+
+are both constant on \(\mbox{$\mathbb{S}^{n}$}(v,c)\).
+
+**Example 2****.**: Given any integer \(k\in\{1,\ldots,n-1\}\) and any real number \(r\in(0,1)\), let us define \(\ell=n-k\) and
+
+\[M_{k}(r) = \{(x,y)\in\mbox{$\mathbb{R}^{k+1}$}\times\mbox{$\mathbb{R}^{\ell+1}$}:\|x\|^{2}=r^{2}\textrm{ and }\|y\|^{2}=1-r^{2}\}\]
+\[= \mbox{$\mathbb{S}^{k}$}(r)\times\mbox{$\mathbb{S}^{n-k}$}(\sqrt{1-r^{2}})\subset\mbox{$\mathbb{S}^{n+1}$}.\]
+
+It is not difficult to see that for any \((x,y)\in M_{k}(r)\) one gets
+
+\[T_{(x,y)}M_{k}(r)=\{(v,w)\in\mbox{$\mathbb{R}^{k+1}$}\times\mbox{$\mathbb{R}^{{\ell+1}}$}:{\langle}x,v{\rangle}=0\quad\hbox{and}\quad{\langle}w,y{\rangle}=0\}\]
+
+Therefore, the map \(\nu:M_{k}(r)\to\mbox{$\mathbb{S}^{n+1}$}\) given by
+
+\[\nu(x,y)=({\sqrt{1-r^{2}}\over r}x,-{r\over\sqrt{1-r^{2}}}y)\]
+
+defines a normal unit vector field along \(M_{k}(r)\), i.e. it is a Gauss map on \(M_{k}(r)\). Notice that the vectors in \(T_{(x,y)}M_{k}(r)\) of the form \((v,{\bf 0})\) define a \(k\) dimensional space. A direct computation, using the expression for \(\nu\), gives us that if \((v,{\bf 0})\in T_{(x,y)}M_{k}(r)\), then,
+
+\[A_{(x,y)}(v,{\bf 0})=-\frac{\sqrt{1-r^{2}}}{r}(v,{\bf 0}).\]
+
+Therefore \(-\sqrt{1-r^{2}}/r\) is an eigenvalue of \(A_{(x,y)}\) with multiplicity \(k\). In the same way we can show that \(r/\sqrt{1-r^{2}}\) is an eigenvalue of \(A_{(x,y)}\) with multiplicity \(\ell\). Therefore, the principal curvatures of \(M_{k}(r)\) are given by
+
+\[\kappa_{1}(x,y)=\cdots=\kappa_{k}(x,y)=-\frac{\sqrt{1-r^{2}}}{r},\quad\kappa_{k+1}(x,y)=\cdots=\kappa_{n}(x,y)=\frac{r}{\sqrt{1-r^{2}}},\]
+
+and we also have that
+
+\[H=\frac{nr^{2}-k}{nr\sqrt{1-r^{2}}}\quad\hbox{and}\quad\|A\|^{2}=\frac{k}{r^{2}}+\frac{n-k}{1-r^{2}}-n\]
+
+are both constant. Hypersurfaces that, up to a rigid motion, are equal to \(M_{k}(r)\) for some \(k\) and \(r\), are called Clifford hypersurfaces.
+
+### Two families of geometric functions on hypersurfaces in spheres
+
+Given a fixed vector \(v\in\mbox{$\mathbb{R}^{n+2}$}\), let us define the functions \(\ell_{v}:M\to\mbox{$\mathbb{R}$}\) and \(f_{v}:M\to\mbox{$\mathbb{R}$}\) by \(\ell_{v}(x)={\langle}\phi(x),v{\rangle}\) and \(f_{v}(x)={\langle}\nu(x),v{\rangle}\), where \(\nu:M\to\mbox{$\mathbb{S}^{n+1}$}\) is a Gauss map. When we consider all possible \(v\in\mbox{$\mathbb{R}^{n+2}$}\) we obtain the families
+
+\[V_{1}=\{\ell_{v}:v\in\mbox{$\mathbb{R}^{n+2}$}\}\quad\hbox{and}\quad V_{2}=\{f_{v}:v\in\mbox{$\mathbb{R}^{n+2}$}\}.\]
+
+These two families are very useful in the study of the spectrum of important elliptic operators defined on \(M\) like the Laplacian an the stability operator. For example, in [10] and [11], Solomon computed the whole spectrum for the Laplace operator of every minimal isoparametric hypersurface of degree 3 in spheres using these two families of functions. For the totally umbilical spheres \(\mbox{$\mathbb{S}^{{n}}$}(v,c)\) we have that if \(c=0\), then dim\((V_{1})=n+1\) and dim\((V_{2})=1\). Indeed, it is not difficult to prove that if for some compact hypersurface \(M^{n}\) in \(\mathbb{S}^{n+1}\), we have that either \(\hbox{dim}(V_{1})<n+2\) or \(\dim(V_{2})<n+2\), then \(M=\mbox{$\mathbb{S}^{{n}}$}(v,0)\) for some unit vector \(v\in\mbox{$\mathbb{R}^{n+2}$}\), [8, Lemma 3.1].
+
+If we take \(c\neq 0\), and we consider the example \(\mbox{$\mathbb{S}^{n}$}(v,c)\) we observe that if \(w\in\mbox{$\mathbb{R}^{n+2}$}\) is a vector perpendicular to the vector \(v\), then
+
+\[f_{w}=-\frac{c}{\sqrt{1-c^{2}}}\ell_{w}.\]
+
+We also have this kind of relation between the function \(f_{w}\) and the function \(\ell_{w}\) in the Clifford hypersurfaces; more precisely, if we consider the example \(M_{k}(r)\) and we take \(w=(w_{1},\dots,w_{k+1},0,\dots,0)\in\mbox{$\mathbb{R}^{n+2}$}\) then we have that
+
+\[f_{w}=\frac{\sqrt{1-r^{2}}}{r}\ell_{w}.\]
+
+Also, if we take \(w=(0,\dots,0,w_{k+2},\dots,w_{n+2})\in\mbox{$\mathbb{R}^{n+2}$}\), then, we have that
+
+\[f_{w}=-\frac{r}{\sqrt{1-r^{2}}}\ell_{w}.\]
+
+In this paper we will prove that these two examples are the only hypersurfaces with constant mean curvature in \(\mathbb{S}^{n+1}\) where the relation \(f_{w}=\lambda\ell_{w}\), for some non-zero vector \(w\in\mbox{$\mathbb{R}^{n+2}$}\), is possible. More precisely, we will prove the following result.
+
+**Theorem 3****.**: _Let \(\phi:M\to\mbox{$\mathbb{S}^{n+1}$}\subset\mbox{$\mathbb{R}^{n+2}$}\) be an immersion with constant mean curvature of a complete \(n\)-dimensional oriented manifold. If for some non-zero vector \(v\neq{\bf 0}\) and some real number \(\lambda\), we have that \(\ell_{v}=\lambda f_{v}\), then, \(\phi(M)\) is either a totally umbilical sphere or a Clifford hypersurface._
+
+Recall that constant mean curvature hypersurfaces in \(\mathbb{S}^{n+1}\) are characterized as critical points of the area functional restricted to variations that preserve a certain volume function. As is well-known, the Jacobi operator of this variational problem is given by \(J=\Delta+\|A\|^{2}+n\), with associated quadratic form given by
+
+\[Q(f)=-\int_{M}fJf\]
+
+and acting on the space
+
+\[\mathcal{C}_{T}^{\infty}(M)=\{f\in\mathcal{C}^{\infty}(M):\mbox{$\int_{M}f=0$}\}.\]
+
+Precisely, the restriction \(\int_{M}f=0\) means that the variation associated to \(f\) is volume preserving.
+
+In contrast to the case of minimal hypersurfaces, in the case of hypersurfaces with constant mean curvature one can consider two different eigenvalue problems: the usual Dirichlet problem, associated with the quadratic form \(Q\) acting on the whole space of smooth functions on \(M^{n}\), and the so called _twisted_ Dirichlet problem, associated with the same quadratic form \(Q\), but restricted to the subspace of smooth functions satisfying the additional condition \(\int_{M}f=0\). Similarly, there are two different notions of stability and index, the _strong stability_ and _strong index_, denoted by \(\mathrm{Ind}(M)\) and associated to the usual Dirichlet problem, and the _weak stability_ and _weak index_, denoted by \(\mathrm{Ind}_{T}(M)\) and associated to the twisted Dirichlet problem. Specifically, the strong index of the hypersurface is characterized as
+
+\[\mathrm{Ind}(M)=\max\{\mathrm{dim}V:V\leqslant\mathcal{C}^{\infty}(M),\quad Q(f)<0\quad\mbox{for every }f\in V\},\]
+
+and \(M\) is called strongly stable if and only if \(\mathrm{Ind}(M)=0\). On the other hand, the weak stability index of \(M^{n}\) is characterized by
+
+\[\mathrm{Ind}_{T}(M)=\max\{\mathrm{dim}V:V\leqslant\mathcal{C}_{T}^{\infty}(M),\quad Q(f)<0\quad\mbox{for every }f\in V\},\]
+
+and \(M\) is called weakly stable if and only if \(\mathrm{Ind}_{T}(M)=0\). From a geometrical point of view, the weak index is more natural than the strong index. However, from an analytical point of view, the strong index is more natural and easier to use (for further details, see [1]).
+
+As an application of our Theorem 3, we will prove that the weak stability index of a compact constant mean curvature hypersurface \(M^{n}\) in \(\mathbb{S}^{n+1}\) with constant scalar curvature must be greater than or equal to \(2n+4\) whenever \(M^{n}\) is neither a totally umbilical sphere nor a Clifford hypersurface (see Theorem 9). This result complements the one obtained in [2] where the authors showed that the weak index of a compact constant mean curvature hypersurface \(M^{n}\) in \(\mathbb{S}^{n+1}\) which is not totally umbilical and has constant scalar curvature is greater than or equal to \(n+2\), with equality if and only if \(M^{n}\) is a Clifford hypersurface \(M_{k}(r)=\mathbb{S}^{k}(r)\times\mathbb{S}^{n-k}(\sqrt{1-r^{2}})\) with radius \(\sqrt{k/(n+2)}\leqslant r\leqslant\sqrt{(k+2)/(n+2)}\). At this respect, it is worth pointing out that the weak stability index of the Clifford hypersurfaces \(M_{k}(r)\) depends on \(r\), reaching its minimum value \(n+2\) when \(\sqrt{k/(n+2)}\leqslant r\leqslant\sqrt{(k+2)/(n+2)}\), and converging to \(+\infty\) as \(r\) converges either to \(0\) or \(1\) (see [2, Section 3] for further details).
+
+## 2. Preliminaries and auxiliary results
+
+Let us start this section by computing the gradient of the functions \(\ell_{v}\) and \(f_{v}\). For any fixed vector in \(\mathbb{R}^{n+2}\), let us define the tangent vector field \(v^{\top}:M\to\mbox{$\mathbb{R}^{n+2}$}\) by
+
+\[v^{\top}(x)=v-\ell_{v}(x)x-f_{v}(x)\nu(x)\qquad\hbox{for all $x\in M$},\]
+
+where, as in the previous section, \(\nu:M\to\mbox{$\mathbb{R}^{n+2}$}\) is a Gauss map. Clearly, \(v^{\top}\) is a tangent vector field on \(M\) because \({\langle}v^{\top}(x),x{\rangle}=0\) and \({\langle}v^{\top}(x),\nu(x){\rangle}=0\) for every \(x\in M\). More precisely, \(v^{\top}(x)\) is the orthogonal projection of the vector \(v\) on \(T_{x}M\).
+
+**Proposition 4****.**: _If \(M^{n}\) is a smooth hypersurface of \(\mathbb{S}^{n+1}\) and \(A\) denotes its shape operator with respect to the unit normal vector field \(\nu:M\to\mbox{$\mathbb{R}^{n+2}$}\) then, the gradient of the functions \(\ell_{v}\) and \(f_{v}\) are given by:_
+
+\[\nabla\ell_{v}=v^{\top},\quad\quad\nabla f_{v}=-A(v^{\top}).\]
+
+Proof.: For any vector \(w\in T_{x}M\), let \(\alpha:(-\varepsilon,\varepsilon)\to M\) be a curve such that \(\alpha(0)=x\) and \(\alpha^{\prime}(0)=w\). Notice that
+
+\[d\ell_{v}(w)={d\ell_{v}(\alpha(t))\over dt}\big{|}_{t=0}={d{\langle}\alpha(t),v{\rangle}\over dt}\big{|}_{t=0}={\langle}\alpha^{\prime}(0),v{\rangle}={\langle}w,v^{\top}(x){\rangle}.\]
+
+Since the equality above holds true for every \(w\in T_{x}M\) and \(v^{\top}(x)\in T_{x}M\), then, \(\nabla\ell_{v}(x)=v^{\top}(x)\). For the function \(f_{v}\), we have
+
+\[df_{v}(w) = {df_{v}(\alpha(t))\over dt}\big{|}_{t=0}={{\langle}\nu(\alpha(t)),v{\rangle}\over dt}\big{|}_{t=0}={\langle}d\nu(\alpha^{\prime}(0)),v{\rangle}\]
+\[= -{\langle}A(w),v^{\top}(x){\rangle}={\langle}w,-A(v^{\top}(x)){\rangle}.\]
+
+Therefore, \(\nabla f_{v}(x)=-A(v^{\top}(x))\). ∎
+
+We also have the following expressions for the Laplacian of the functions \(\ell_{v}\) and \(f_{v}\).
+
+**Proposition 5****.**: _If \(M^{n}\) is a smooth hypersurface of \(\mathbb{S}^{n+1}\) with constant mean curvature \(H\), and \(A\) denotes the shape operator with respect to the unit normal vector field \(\nu:M\to\mbox{$\mathbb{R}^{n+2}$}\) then, the Laplacian of the functions \(\ell_{v}\) and \(f_{v}\) are given by:_
+
+\[\Delta\ell_{v}=-n\ell_{v}+nHf_{v},\quad\quad\Delta f_{v}=-\|A\|^{2}f_{v}+nH\ell_{v}.\]
+
+Proof.: For any vector \(w\in T_{x}M\), we have
+
+\[\nabla_{w}\nabla\ell_{v}=\nabla_{w}v^{\top}=-\ell_{v}(x)w+f_{v}(x)A_{x}(w),\]
+
+where \(\nabla\) denotes here the intrinsic derivative on \(M\). Let \(\{e_{1},\ldots,e_{n}\}\) be an orthonormal basis of \(T_{x}M\). Then, the Laplacian of \(\ell_{v}\) at the point \(x\) is given by
+
+\[\Delta\ell_{v}(x)=\sum_{i=1}^{n}\mbox{$\langle\nabla_{e_{i}}\nabla\ell_{v},e_{i}\rangle$}=-n\ell_{v}(x)+\mathrm{tr}(A_{x})f_{v}(x)=-n\ell_{v}(x)+nHf_{v}(x).\]
+
+On the other hand, using Codazzi equation we also have that
+
+\[\nabla_{w}\nabla f_{v} = -\nabla_{w}(A(v^{\top}))=-(\nabla_{w}A)(v^{\top}(x))-A_{x}(\nabla_{w}v^{\top})\]
+\[= -(\nabla_{v^{\top}(x)}A)(w)+\ell_{v}(x)A_{x}(w)-f_{v}(x)A_{x}^{2}(w).\]
+
+Therefore
+
+\[\Delta f_{v}(x) = \sum_{i=1}^{n}\mbox{$\langle\nabla_{e_{i}}\nabla f_{v},e_{i}\rangle$}\]
+\[= -\sum_{i=1}^{n}\mbox{$\langle(\nabla_{v^{\top}(x)}A)(e_{i}),e_{i}\rangle$}+nH\ell_{v}(x)-\|A\|^{2}(x)f_{v}(x)\]
+\[= -n\mbox{$\langle v^{\top}(x),\nabla H(x)\rangle$}+nH\ell_{v}(x)-\|A\|^{2}(x)f_{v}(x)\]
+\[= nH\ell_{v}(x)-\|A\|^{2}(x)f_{v}(x),\]
+
+since the mean curvature \(H\) is constant. ∎
+
+The following two lemmas will be used in the proof of our main theorem. The first one is an elementary geometric lemma whose proof is left to the reader.
+
+**Lemma 6****.**: _Let \(M^{n}\) be a smooth hypersurface of \(\mathbb{S}^{n+1}\) and let \(\alpha:I\subset\mbox{$\mathbb{R}$}\to M\) be a regular curve such that_
+
+\[\alpha^{\prime\prime}(t)=f(t)\alpha^{\prime}(t)+\eta(t)\]
+
+_where \(f:I\to\mbox{$\mathbb{R}$}\) is a smooth function and \(\eta:I\to\mbox{$\mathbb{R}^{n+2}$}\) is a normal vector field along \(\alpha\), i.e. \(\eta(t)\) is orthogonal to \(T_{\alpha(t)}M\). If \(s=s(t)\) is the arc-length parameter for \(\alpha\), then \(\beta(s)=\alpha(t(s))\) satisfies that \(\beta^{\prime\prime}(s)\) is a normal vector field along \(\beta\), i.e. \(\beta\) is a geodesic in \(M\)._
+
+The other one is an algebraic lemma.
+
+**Lemma 7****.**: _If \(p_{1}(X)=b_{1}X+c_{1},\ldots,p_{k}(X)=b_{k}X+c_{k}\) are \(k\) polynomials of degree 1, \(k\geq 2\), with the property that \(c_{i}/b_{i}\neq c_{j}/b_{j}\) whenever \(i\neq j\), then, the polynomials_
+
+\[q_{i}=\Pi_{j=1,j\neq i}^{k}p_{j}\]
+
+_are linearly independent. Moreover, an equation of the form_
+
+\[{a_{1}\over p_{1}(X)}+\cdots+{a_{k}\over p_{k}(X)}=d\]
+
+_with \(a_{i}\) and \(d\) real numbers, can not hold true unless all the \(a_{i}\)’s and \(d\) are zero._
+
+Proof.: By the condition on the numbers \(c_{j}/b_{j}\) we have that at \(X_{i}=-c_{i}/b_{i}\) every polynomial \(q_{j}\), except the polynomial \(q_{i}\), vanishes. Therefore, if there exists constants \(\alpha_{i}\) such that
+
+\[\alpha_{1}q_{1}(X)+\cdots+\alpha_{k}q_{k}(X)=0\]
+
+then, taking \(X=X_{i}\) we get that \(\alpha_{i}=0\) for every \(i\). Therefore, the polynomials \(q_{i}\)’s are linearly independent. On the other hand, notice that the second equation in the lemma can be written as
+
+\[a_{1}q_{1}(X)+\cdots+a_{k}q_{k}(X)=dR(X)\]
+
+where \(R\) is a polynomial of degree \(k\). Since the expression on the left of the last equation is a polynomial of degree \(k-1\), we obtain that the constant on the right hand side must be zero. Then the second part of the lemma follows by the independence of the polynomials \(q_{i}\)’s. ∎
+
+## 3. Proof of Theorem 3
+
+We are now ready to give our main argument and prove Theorem 3. Since most of the arguments are local and the thesis of the theorem is on \(\phi(M)\) and not on \(M\), we will identify \(M\) with \(\phi(M)\) and \(T_{x}M\) with \(T_{\phi(x)}M\). By multiplying the equation \(\ell_{v}=\lambda f_{v}\) by an appropriated constant we may assume that \(|v|=1\). We will also assume that \(\ell_{v}\) is not constant, otherwise \(\phi(M)\subset\mbox{$\mathbb{S}^{n}$}(v,c)\) for some \(c\), which implies, using the completeness of \(M\), that \(\phi(M)=\mbox{$\mathbb{S}^{n}$}(v,c)\).
+
+Notice that, since \(\ell_{v}\) is not constant, then \(\lambda\neq 0\). Taking the gradient in both sides of the expression \(\ell_{v}=\lambda f_{v}\) we obtain that
+
+(1) \[A(v^{\top}(x))=-\lambda^{-1}v^{\top}(x)\]
+
+at every point \(x\in M\).
+
+**Step 1: The integral curves of \(v^{\top}\) in \(M\) are Euclidean circles.** Let us take a point \(x\in M\) such that \(\nabla\ell_{v}(x)=v^{\top}(x)\) does not vanish. Let \(\alpha_{x}(t)\) be the integral curve of the vector field \(v^{\top}\) such that \(\alpha_{x}(0)=x\). Since
+
+\[\alpha_{x}^{\prime}(t) = v^{\top}(\alpha_{x}(t))=v-\ell_{v}(\alpha_{x}(t))\alpha_{x}(t)-f_{v}(\alpha_{x}(t))\nu(\alpha_{x}(t))\]
+\[= v-\ell_{v}(\alpha_{x}(t))\big{(}\alpha_{x}(t)+\lambda^{-1}\nu(\alpha_{x}(t))\big{)}\]
+
+then,
+
+\[\alpha_{x}^{\prime\prime}(t) = -{\langle}\nabla\ell_{v}(\alpha_{x}(t)),\alpha_{x}^{\prime}(t){\rangle}\big{(}\alpha_{x}(t)+\lambda^{-1}\nu(\alpha_{x}(t))\big{)}\]
+\[-\ell_{v}(\alpha_{x}(t))\big{(}\alpha_{x}^{\prime}(t)-\lambda^{-1}A(\alpha_{x}^{\prime}(t))\big{)}\]
+\[= -|\alpha_{x}^{\prime}(t)|^{2}\big{(}\alpha_{x}(t)+\lambda^{-1}\nu(\alpha_{x}(t))\big{)}-\ell_{v}(\alpha_{x}(t))\big{(}\alpha_{x}^{\prime}(t)-\lambda^{-1}A(v^{\top}(\alpha_{x}(t)))\big{)}\]
+\[= -|\alpha_{x}^{\prime}(t)|^{2}\big{(}\alpha_{x}(t)+\lambda^{-1}\nu(\alpha_{x}(t))\big{)}-\ell_{v}(\alpha_{x}(t))\big{(}\alpha_{x}^{\prime}(t)+\lambda^{-2}v^{\top}(\alpha_{x}(t))\big{)}\]
+\[= f(t)\alpha_{x}^{\prime}(t)+\eta(t).\]
+
+Here
+
+(3) \[\eta(t)=-|\alpha_{x}^{\prime}(t)|^{2}\big{(}\alpha_{x}(t)+\lambda^{-1}\nu(\alpha_{x}(t))\big{)}\]
+
+is a normal vector field along \(\alpha_{x}\) and
+
+\[f(t)=-\big{(}1+\lambda^{-2}\big{)}\ell_{v}(\alpha_{x}(t)).\]
+
+Therefore if \(s=s(t)\) is the arc-length parameter for the curve \(\alpha_{x}\) with \(s(0)=0\), and \(t=t(s)\) is the inverse of the function \(s=s(t)\), we have, by Lemma 6, that \(\beta_{x}(s)=\alpha_{x}(t(s))\) is a geodesic in \(M\). Moreover, from (3) and (3) we also get that
+
+(4) \[\beta_{x}^{\prime\prime}(s)=\frac{1}{|\alpha_{x}^{\prime}(t(s))|^{2}}\eta(t(s))=-\beta_{x}(s)-\lambda^{-1}\nu(\beta_{x}(s)).\]
+
+If we differentiate (4), we get that the function \(\beta_{x}^{\prime}\) moves along a circle because it satisfies the equation
+
+\[(\beta_{x}^{\prime})^{\prime\prime}(s)+(1+\lambda^{-2})\beta_{x}^{\prime}(s)=0.\]
+
+More precisely, if we define \(w=\sqrt{1+\lambda^{-2}}>0\), then
+
+\[\beta_{x}^{\prime}(s)=\beta_{x}^{\prime}(0)\cos{(ws)}+w^{-1}\beta_{x}^{\prime\prime}(0)\sin{(ws)},\]
+
+and
+
+(5) \[\beta_{x}(s)=w^{-1}\beta_{x}^{\prime}(0)\sin{(ws)}-w^{-2}\beta_{x}^{\prime\prime}(0)\cos{(ws)}+\beta_{x}(0)+w^{-2}\beta_{x}^{\prime\prime}(0).\]
+
+**Step 2: The intersection \(N=M\cap\mathbb{S}^{n}(v,0)\) is non-empty.** Let us compute \(\beta_{x}^{\prime}(0)\) and \(\beta_{x}^{\prime\prime}(0)\) in order to obtain an explicit expression for \(\beta_{x}^{\prime}(s)\). From the definition of \(\beta_{x}\) we have that \(\beta_{x}(0)=x\) and
+
+(6) \[\beta_{x}^{\prime}(0)=\frac{\alpha_{x}^{\prime}(0)}{|\alpha_{x}^{\prime}(0)|}=\frac{v^{\top}(x)}{|v^{\top}(x)|}.\]
+
+Notice that
+
+\[v^{\top}(y)=v-\ell_{v}(y)y-f_{v}(y)\nu(y)=v-\ell_{v}(y)y-\lambda^{-1}\ell_{v}(y)\nu(y)\]
+
+at every point \(y\in M\). Therefore
+
+\[|v^{\top}(y)|^{2}=1-\ell_{v}(y)^{2}-\lambda^{-2}\ell_{v}(y)^{2}=1-w^{2}\ell_{v}(y)^{2}.\]
+
+From this last expression we obtain that \(-w^{-1}\leq\ell_{v}(y)\leq w^{-1}\), at every \(y\in M\), and
+
+(7) \[v^{\top}(y)={\bf 0}\mbox{ if and only if }\ell_{v}(y)=\pm w^{-1}.\]
+
+Let us define \(a=\ell_{v}(x)\), and \(b=\sqrt{w^{-2}-a^{2}}\). By (7) we have that \(b>0\), because \(\nabla\ell_{v}(x)=v^{\top}(x)\neq{\bf 0}\). With this notation, we obtain that \(|v^{\top}(x)|^{2}=1-w^{2}a^{2}=w^{2}b^{2}\), and
+
+\[{\langle}\beta_{x}(0),v{\rangle} = \ell_{v}(x)=a\]
+\[{\langle}\beta_{x}^{\prime}(0),v{\rangle} = {\langle}{v^{\top}(x)\over|v^{\top}(x)|},v{\rangle}={\langle}{v^{\top}(x)\over|v^{\top}(x)|},v^{\top}(x){\rangle}=|v^{\top}(x)|=\sqrt{1-w^{2}a^{2}}=wb\]
+\[{\langle}\beta_{x}^{\prime\prime}(0),v{\rangle} = {\langle}-\beta_{x}(0)-\lambda^{-1}\nu(\beta_{x}(0)),v{\rangle}=-a-\lambda^{-2}a=-w^{2}a,\]
+
+where we have used (4) to derive the last equation. Now, using these equations jointly with (5) we get that
+
+\[\ell_{v}(\beta_{x}(s))={\langle}\beta_{x}(s),v{\rangle}=a\cos{(ws)}+b\sin{(ws)}.\]
+
+Notice that \((wa)^{2}+(wb)^{2}=1\) with \(wb>0\). Therefore for some \(s_{1}\in(-{\pi\over 2w},{\pi\over 2w})\) we have
+
+\[-wa=\sin{(ws_{1})}\quad\hbox{and}\quad wb=\cos{(ws_{1})},\]
+
+so that
+
+\[\ell_{v}(\beta_{x}(s))=a\cos{(ws)}+b\sin{(ws)}=w^{-1}\sin{(ws-ws_{1})}.\]
+
+Notice that when \(s\) moves from \(0\) to \(s_{1}\), we have that \(\ell_{v}(\beta_{x}(s))\) never reaches the values \(\pm w^{-1}\), therefore by (7)\(v^{\top}(\beta_{x}(s))\neq{\bf 0}\) and all these \(\beta_{x}(s)\) belong to the integral curve of the vector field \(v^{\top}\). In particular, \(\ell_{v}(\beta_{x}(s_{1}))=0\) and \(v^{\top}(\beta_{x}(s_{1}))=v\neq{\bf 0}\). This argument shows that
+
+\[N=\ell_{v}^{-1}(0)=\{y\in M:\ell_{v}(y)=0\}\]
+
+is not empty. Observe that if we were assuming that \(M\) were compact instead of complete, the fact that \(N=\ell_{v}^{-1}(0)\) is not empty would have followed from the fact that the function \(\ell_{v}\) must reach its maximum value and a minimum value on \(M\), and the fact that necessarily these values must be \(\pm w^{-1}\), since \(\nabla\ell_{v}=v^{\top}\) must vanish at its critical points. From now on we will assume that the \(x\) that we were considering before is an element in \(N\), i.e, we will assume that \(a=0\), and therefore \(b=w^{-1}\) and \(s_{1}=0\).
+
+**Step 3: The intersection \(N=M\cap\mathbb{S}^{n}(v,0)\) as a hypersurface of \(M\) and as a hypersurface of \(\mathbb{S}^{n}(v,0)\).** Clearly the set \(N\subset M^{n}\) is an \((n-1)\)-dimensional manifold because \(0\) is a regular value of the function \(\ell_{v}\) on \(M\). Moreover, for every \(x\in N\) we have that \(\nabla\ell_{v}(x)=v^{\top}(x)=v\) is a constant vector, and therefore \(N\) is a totally geodesic hypersurface of \(M\). Notice that for every \(x\in N\) we have that \(v\in T_{x}M\) and \(A_{x}(v)=-\lambda^{-1}v\). Therefore we can take vectors \(v_{1},\dots,v_{n-1}\) in \(T_{x}M\), all of them orthogonal to \(v\), such that \(A_{x}(v_{i})=\lambda_{i}(x)v_{i}\). Since the vectors \(v_{i}\)’s are perpendicular to \(v=\nabla\ell_{v}(x)\), they form a basis for \(T_{x}N\). On the other hand, notice that \(N\) is also a hypersurface of the unit \(n\)-dimensional sphere \(\mbox{$\mathbb{S}^{n}$}(v,0)\), and that for every \(x\in N\), \(\nu(x)\) gives a unit vector field normal to \(N\) in \(\mbox{$\mathbb{S}^{n}$}(v,0)\) (see Figure 1).
+
+Taking into account that \(N\) is totally geodesic in \(M^{n}\) and that \(\mbox{$\mathbb{S}^{n}$}(v,0)\) is totally geodesic in \(\mathbb{S}^{n+1}\), it follows from the fact that \(\nu\) is both normal to \(M^{n}\) in \(\mathbb{S}^{n+1}\) and normal to \(N\) in \(\mbox{$\mathbb{S}^{n}$}(v,0)\) that, for every \(x\in N\), \(\lambda_{1}(x),\ldots,\lambda_{n-1}(x)\) are the principal curvatures of \(N\) as a hypersurface of \(\mbox{$\mathbb{S}^{n}$}(v,0)\) with respect to \(\nu\)
+
+**Step 4: Computation of the principal curvatures of \(M\) along the integral curves of \(v^{\top}\).** Under the assumption that \(x\in N\), we obtain from (6) that
+
+\[\beta_{x}^{\prime}(0)=v.\]
+
+Therefore, from (4) and (5) we get the following expression for \(\beta_{x}(s)\),
+
+(8) \[\beta_{x}(s)=w^{-1}\sin{(ws)}v+w^{-2}(\cos{(ws)}-1)(x+\lambda^{-1}\nu(x))+x.\]
+
+By differentiating two times this equation, and using the equation (4), we obtain the following expression,
+
+(9) \[\nu(\beta_{x}(s))=\lambda w\sin(ws)v+\lambda\cos(ws)(x+\lambda^{-1}\nu(x))-\lambda\beta_{x}(s).\]
+
+Recall that, if \(s\in(-{\pi\over 2w},{\pi\over 2w})\), then
+
+(10) \[|\ell_{v}(\beta_{x}(s))|<w^{-1}\quad\textrm{ and }\quad v^{\top}(\beta_{x}(s))\neq{\bf 0}.\]
+
+Observe that if \(\gamma(t)\) is a smooth curve in \(N\) such that \(\gamma(0)=x\) and \(\gamma^{\prime}(0)=v_{i}\), then by (8), we have that the curve
+
+\[\gamma_{s}(t)=\beta_{\gamma(t)}(s)=w^{-1}\sin{(ws)}v+w^{-2}(\cos{(ws)}-1)(\gamma(t)+\lambda^{-1}\nu(\gamma(t)))+\gamma(t)\]
+
+is a curve on \(M\) such that \(\gamma_{s}(0)=\beta_{x}(s)\). A direct computation shows that
+
+(11) \[\gamma_{s}^{\prime}(0)=w^{-2}(\cos{(ws)}-1)(v_{i}-\lambda^{-1}\lambda_{i}(x)v_{i})+v_{i}=\mu_{i}(x)v_{i},\]
+
+where
+
+\[\mu_{i}(x)=\frac{\lambda(\lambda-\lambda_{i}(x))\cos(ws)+(1+\lambda\lambda_{i}(x))}{1+\lambda^{2}}.\]
+
+The computation above shows us that the vectors \(v_{i}\)’s are also elements in \(T_{\beta_{x}(s)}M\) for every \(s\in(-{\pi\over 2w},{\pi\over 2w})\). Actually, it follows directly from (11) that if \(\mu_{i}(x)\neq 0\) then \(v_{i}=\gamma_{s}^{\prime}(0)/\mu_{i}(x)\in T_{\beta_{x}(s)}M\); hence by a continuity argument, since the equation \(\mu_{i}(x)=0\) has finitely many solutions on \((-{\pi\over 2w},{\pi\over 2w})\), we conclude that \(v_{i}\in T_{\beta_{x}(s)}M\) for every \(s\in(-{\pi\over 2w},{\pi\over 2w})\).
+
+Recall that, by (10) and (1), \(-\lambda^{-1}\) is a principal curvature at the point \(\beta_{x}(s)\), for every \(s\in(-{\pi\over 2w},{\pi\over 2w})\), with associated principal direction in the direction of \(v^{\top}(\beta_{x}(s))\neq\mathbf{0}\). Let us compute now the other \(n-1\) principal curvatures of \(M\) at the point \(\beta_{x}(s)\). Since \(\gamma(t)\in N\) for every \(t\), then the expression (9) holds true when replacing \(x\) by \(\gamma(t)\) and then we have that
+
+\[\nu(\gamma_{s}(t))=\nu(\beta_{\gamma(t)}(s))=\lambda w\sin(ws)v+\lambda\cos(ws)(\gamma(t)+\lambda^{-1}\nu(\gamma(t)))-\lambda\gamma_{s}(t).\]
+
+Differentiating this equation with respect to \(t\) at \(t=0\) and using (11), we get that
+
+\[A_{\beta_{x}(s)}(\gamma^{\prime}_{s}(0))=\mu_{i}(x)A_{\beta_{x}(s)}(v_{i})=-d\nu(\gamma_{s}^{\prime}(0))=(\lambda_{i}(x)-\lambda)\cos(ws)v_{i}+\lambda\mu_{i}(x)v_{i}.\]
+
+That is,
+
+\[A_{\beta_{x}(s)}(v_{i})=\left(\lambda+\frac{(\lambda_{i}(x)-\lambda)(1+\lambda^{2})\cos(ws)}{\lambda(\lambda-\lambda_{i}(x))\cos(ws)+(1+\lambda\lambda_{i}(x))}\right)v_{i}\]
+
+Therefore, we get the following expression for the other \(n-1\) principal curvatures at \(\beta_{x}(s)\),
+
+(12) \[\lambda_{i}(\beta_{x}(s)) = \lambda+{(\lambda_{i}(x)-\lambda)(1+\lambda^{2})\cos(ws)\over\lambda(\lambda-\lambda_{i}(x))\cos(ws)+(1+\lambda\lambda_{i}(x))}\]
+\[= -\lambda^{-1}+{(1+\lambda^{2})(\lambda^{-1}+\lambda_{i}(x))\over\lambda(\lambda-\lambda_{i}(x))\cos{(ws)}+(1+\lambda\lambda_{i}(x))}.\]
+
+Notice that, as it is supposed to be, when \(s=0\), i.e at the point \(x\), the expression (12) above reduces to \(\lambda_{i}(x)\). Also notice that if \(\lambda_{i}(x)=-\lambda^{-1}\) then, the expression (12) reduces to \(-\lambda^{-1}\) for every \(s\).
+
+**Step 5: \(M\) is isoparametric with at most two distinct principal curvatures.** Now, we will use the hypothesis on the mean curvature of \(M\). By (12), for every point \(x\in N\) and every \(s\in(-{\pi\over 2w},{\pi\over 2w})\) we have that
+
+\[nH = nH(\beta_{x}(s))=-\lambda^{-1}+\sum_{i=1}^{n-1}\lambda_{i}(\beta_{x}(s))\]
+\[= -n\lambda^{-1}+(1+\lambda^{2})\sum_{i=1}^{n-1}\frac{\lambda^{-1}+\lambda_{i}(x)}{\lambda(\lambda-\lambda_{i}(x))\cos{(ws)}+(1+\lambda\lambda_{i}(x))}.\]
+
+That is,
+
+(13) \[\sum_{i=1}^{n-1}\frac{\lambda^{-1}+\lambda_{i}(x)}{\lambda(\lambda-\lambda_{i}(x))\cos{(ws)}+(1+\lambda\lambda_{i}(x))}=\frac{n(H+\lambda^{-1})}{1+\lambda^{2}}.\]
+
+For every \(x\in N\), let
+
+\[I_{1}(x) = \{i\in\{1,\ldots,n-1\}:\lambda_{i}(x)=-\lambda^{-1}\},\]
+\[I_{2}(x) = \{i\in\{1,\ldots,n-1\}:\lambda_{i}(x)=\lambda\},\]
+\[I_{3}(x) = \{1,\ldots,n-1\}\setminus(I_{1}(x)\cup I_{2}(x)).\]
+
+Then (13) can be written as
+
+(14) \[\sum_{i\in I_{3}(x)}\frac{\lambda^{-1}+\lambda_{i}(x)}{\lambda(\lambda-\lambda_{i}(x))\cos{(ws)}+(1+\lambda\lambda_{i}(x))}=d(x)\]
+
+where
+
+\[d(x)=\frac{n(H+\lambda^{-1})-n_{2}(x)(\lambda+\lambda^{-1})}{1+\lambda^{2}}.\]
+
+and \(n_{i}(x)=\mathrm{card}(I_{i}(x))\). We claim that \(I_{3}(x)=\emptyset\). Otherwise, for every \(i\in I_{3}(x)\) let \(a_{i}(x)=\lambda^{-1}+\lambda_{i}(x)\neq 0\), \(b_{i}(x)=\lambda(\lambda-\lambda_{i}(x))\neq 0\), and \(c_{i}(x)=1+\lambda\lambda_{i}(x)\neq 0\). Thus, equation (14) means that, for every \(s\in(-{\pi\over 2w},{\pi\over 2w})\), \(\cos{(ws)}\) is a root of the polynomial equation on \(X\)
+
+(15) \[\sum_{i\in I_{3}(x)}\frac{a_{i}(x)}{b_{i}(x)X+c_{i}(x)}=d(x).\]
+
+If \(\lambda_{i}(x)=\lambda_{j}(x)\) for every \(i,j\in I_{3}(x)\) (in particular, if \(n_{3}(x)=1\)), then (15) becomes
+
+\[\frac{n_{3}(x)a_{i}(x)}{b_{i}(x)X+c_{i}(x)}=d(x),\]
+
+which can hold only if \(a_{i}(x)=d(x)=0\). But this is a contradiction because \(a_{i}(x)\neq 0\). Therefore, we can decompose
+
+\[I_{3}(x)=\bigcup_{i=1}^{k}J_{i}(x),\quad k\geq 2,\]
+
+with \(\lambda_{j_{1}}(x)=\lambda_{j_{2}}(x)\) if and only if \(j_{1},j_{2}\in J_{i}(x)\) for some \(i\). In that case, let \(\lambda_{i}(x)=\lambda_{j}(x)\) for every \(j\in J_{i}(x)\), and (15) becomes
+
+(16) \[\sum_{i=1}^{k}\frac{m_{i}(x)a_{i}(x)}{b_{i}(x)X+c_{i}(x)}=d(x)\]
+
+with \(m_{i}(x)=\mathrm{card}(J_{i}(x))>0\), \(m_{i}(x)a_{i}(x)\neq 0\). But this contradicts our Lemma 7, because
+
+\[\frac{c_{i}(x)}{b_{i}(x)}=\frac{1+\lambda\lambda_{i}(x)}{\lambda(\lambda-\lambda_{i}(x))}\neq\frac{1+\lambda\lambda_{j}(x)}{\lambda(\lambda-\lambda_{j}(x))}=\frac{c_{j}(x)}{b_{j}(x)}\]
+
+for every \(i\neq j\), \(1\leq i,j\leq k\).
+
+Summing up, \(I_{3}(x)=\emptyset\) for every \(x\in N\), which means that all the principal curvatures of \(M\) at the points of \(N\) are constant and they are equal to either \(-\lambda^{-1}\) or \(\lambda\). From the expression (12), the same happens along the geodesics \(\beta_{x}(s)\) for every \(s\in(-{\pi\over 2w},{\pi\over 2w})\). Taking into account that every point of \(M\) which is not a critical point of \(\ell_{v}\) can be reached through a geodesic \(\beta_{x}(s)\), we conclude that the principal curvatures of \(M\) are constant on the whole \(M\) and they are equal to either \(-\lambda^{-1}\) or \(\lambda\). That is, \(M\) is a complete isoparametric hypersurface of \(\mathbb{S}^{n+1}\) with at most two distinct principal curvatures, and from the well known rigidity result by Cartan [4] (see also [6, Chaper 3]) we conclude that \(M\) is either a totally umbilical sphere (in the case that all its principal curvatures are equal to \(-\lambda^{-1}\)) or it is either Clifford hypersurface of the form \(M_{k}(r)=\mathbb{S}^{k}(r)\times\mathbb{S}^{n-k}(\sqrt{1-r^{2}})\) with radius \(0<r<1\) (in the case that the principal curvatures take both values).
+
+This finishes the proof of Theorem 3.
+
+Let us exhibit an example that shows that the condition on the mean curvature to be constant is necessary in the previous result.
+
+**Example 8****.**: Let \(e_{1}=(1,0,\dots,0)\in\mbox{$\mathbb{R}^{n+1}$}\) and \(c=4/5\). From Example 1 we know that the principal curvatures of \(\mbox{$\mathbb{S}^{n-1}$}(e_{1},c)\subset\mbox{$\mathbb{S}^{n}$}\) are all equal to \(4/3\). By perturbing \(\mbox{$\mathbb{S}^{n-1}$}(e_{1},c)\) we can find a hypersurface \(N\subset\mbox{$\mathbb{S}^{n}$}\) whose mean curvature is not constant and such that all its principal curvatures \(\lambda_{i}\) satisfy that
+
+(17) \[1<\lambda_{i}(x)<2\quad\hbox{for every $x\in N$ and $i=1,\dots,n-1$}\]
+
+Let \(M^{n}=\mbox{$\mathbb{S}^{1}$}\times N\) and \(\phi:M\to\mbox{$\mathbb{S}^{n+1}$}\subset\mbox{$\mathbb{R}^{n+2}$}\) the map given by
+
+\[\phi((\cos s,\sin s),x)=({1\over\sqrt{2}}\sin(\sqrt{2}s),{1\over 2}(x+\nu(x))\cos(\sqrt{2}s)+{1\over 2}(x-\nu(x))),\]
+
+where \(x\in N\subset\mbox{$\mathbb{S}^{n}$}\subset\mbox{$\mathbb{R}^{n+1}$}\) denotes the points in \(N\) and \(\nu:N\to\mbox{$\mathbb{S}^{n}$}\subset\mbox{$\mathbb{R}^{n+1}$}\) is a Gauss map of \(N\). In particular, \(\mbox{$\langle x,\nu(x)\rangle$}=0\).
+
+Let \({\partial\over\partial s}=(-\sin s,\cos s)\) and let \(v_{1},\dots,v_{n-1}\) be a basis of \(T_{x}N\) such that \(-d\nu_{x}(v_{i})=\lambda_{i}(x)v_{i}\). Notice that \(\bar{{\partial\over\partial s}}=((-\sin s,\cos s),{\bf 0})\in\mbox{$\mathbb{R}^{{n+3}}$}\) and \(\bar{v}_{1}=(0,0,v_{1}),\dots,\bar{v}_{n-2}=(0,0,v_{n-2})\) form a basis for the tangent space of \(M\) at \(p=((\cos s,\sin s),x)\). A direct computation shows that
+
+\[d\phi_{p}(\bar{{\partial\over\partial s}})=(\cos(\sqrt{2}s),-{1\over\sqrt{2}}(x+\nu(x))\sin(\sqrt{2}s))\]
+
+and
+
+\[d\phi_{p}(\bar{v}_{i})={1\over 2}(0,((1-\lambda_{i}(x))\cos(\sqrt{2}s)+1+\lambda_{i}(x))v_{i}).\]
+
+By (17), the expression \((1-\lambda_{i}(x))\cos(\sqrt{2}s)+(1+\lambda_{i}(x))\) never vanishes, therefore \(\phi\) is an immersion. Moreover, it is easy to check that \(\tilde{\nu}:M\to\mbox{$\mathbb{S}^{n+1}$}\subset\mbox{$\mathbb{R}^{n+2}$}\) given by
+
+\[\tilde{\nu}(p)=({1\over\sqrt{2}}\sin(\sqrt{2}s),{1\over 2}(x+\nu(x))\cos(\sqrt{2}s)-{1\over 2}(x-\nu(x)))\]
+
+is a Gauss map on \(M\). Using the expression for \(\phi\) and for \(\tilde{\nu}\) we get that \(\ell_{v}=f_{v}\) for \(v=(1,0,\dots,0)\in\mbox{$\mathbb{R}^{n+2}$}\).
+
+## 4. Stability index of hypersurfaces with constant mean curvature
+
+In this section, and as an application of our Theorem 3, we will prove that the weak stability index of a compact constant mean curvature hypersurface \(M^{n}\) in \(\mathbb{S}^{n+1}\) with constant scalar curvature must be greater than or equal to \(2n+4\) whenever \(M^{n}\) is neither a totally umbilical sphere nor a Clifford hypersurface. Recall that constant mean curvature hypersurfaces in \(\mathbb{S}^{n+1}\) are critical points of the area functional restricted to variations that preserve a certain volume function. The Jacobi operator of this variational problem is given by \(J=\Delta+\|A\|^{2}+n\), with associated quadratic form given by
+
+\[Q(f)=-\int_{M}fJf\]
+
+and acting on the space
+
+\[\mathcal{C}_{T}^{\infty}(M)=\{f\in\mathcal{C}^{\infty}(M):\mbox{$\int_{M}f=0$}\}.\]
+
+Precisely, the restriction \(\int_{M}f=0\) means that the variation associated to \(f\) is volume preserving. The weak stability index of the hypersurface, denoted here by \(\mathrm{Ind}_{T}(M)\), is characterized by
+
+\[\mathrm{Ind}_{T}(M)=\max\{\mathrm{dim}V:V\leqslant\mathcal{C}_{T}^{\infty}(M),\quad Q(f)<0\quad\mbox{for every }f\in V\},\]
+
+and \(M\) is called weakly stable if and only if \(\mathrm{Ind}_{T}(M)=0\) (see [1] for further details).
+
+In [3], Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only compact weakly stable constant mean curvature hypersurfaces in \(\mathbb{S}^{n+1}\). In [2] the authors have recently showed that the weak index of a compact constant mean curvature hypersurface \(M^{n}\) in \(\mathbb{S}^{n+1}\) which is not totally umbilical and has constant scalar curvature is greater than or equal to \(n+2\), with equality if and only if \(M^{n}\) is a Clifford hypersurface \(M_{k}(r)=\mathbb{S}^{k}(r)\times\mathbb{S}^{n-k}(\sqrt{1-r^{2}})\) with radius \(\sqrt{k/(n+2)}\leqslant r\leqslant\sqrt{(k+2)/(n+2)}\). Here we will complement this result by showing the following.
+
+**Theorem 9****.**: _Let \(M^{n}\) be a compact orientable hypersurface immersed into the Euclidean sphere \(\mathbb{S}^{n+1}\) with constant mean curvature. If \(M\) has constant scalar curvature and \(M\) is neither a Clifford nor an umbilical hypersurface, then the weak stability index of \(M\) is greater than or equal to \(2n+4\)._
+
+Proof.: The condition on the scalar curvature implies that, \(\|A\|^{2}\) is constant. Let us first consider the case where \(H=0\). Since \(M^{n}\) is not totally umbilical (i.e., totally geodesic), then \(\|A\|^{2}>0\). Even more, since \(M\) is not a minimal Clifford hypersurface we have that \(\|A\|^{2}>n\), by a classical result due to [9] and [5, 7] (see [1, Theorem 6]). By Proposition 5 we have that the functions \(\ell_{v}\) and \(f_{v}\) are eigenfunctions of the Laplacian with positive eigenvalues \(n\) and \(\|A\|^{2}>n\), respectively (observe that with our criterion, a real number \(\lambda\) is an eigenvalue of \(\Delta\) if and only if \(\Delta u+\lambda u=0\) for some smooth function \(u\in\mathcal{C}^{\infty}(M)\), \(u\not\equiv 0\)). In particular, the functions \(\ell_{v}\) and \(f_{v}\) satisfy the condition \(\int_{M}f=0\), and they also satisfy \(J(\ell_{v})=\|A\|^{2}\ell_{v}\) and \(Jf_{v}=nf_{v}\). That is, they are also eigenfunctions of \(J\) with negative eigenvalues \(-\|A\|^{2}\) and \(-n\), respectively. Let
+
+\[V_{1}=\{\ell_{v}:v\in\mbox{$\mathbb{R}^{n+2}$}\}\quad\hbox{and}\quad V_{2}=\{f_{v}:v\in\mbox{$\mathbb{R}^{n+2}$}\}.\]
+
+Then,
+
+(18) \[\mathrm{Ind}_{T}(M)\geq\mathrm{dim}(V_{1}\oplus V_{2})=\mathrm{dim}V_{1}+\mathrm{dim}V_{2},\]
+
+where the last equality is due to the fact that \(V_{1}\) and \(V_{2}\) are \(L^{2}\)-orthogonal subspaces, because they are eigenspaces of \(\Delta\) associated to different eigenvalues. Finally, as pointed out in Subsection 1.2, we also know that if either \(\mathrm{dim}V_{1}<n+2\) or \(\mathrm{dim}V_{2}<n+2\), then \(M\) must be a totally geodesic sphere (see [8, Lemma 3.1]). Therefore, in our case we have \(\mathrm{dim}V_{1}=\mathrm{dim}V_{2}=n+2\), and by (18) we conclude that \(\mathrm{Ind}_{T}(M)\geq 2n+4\).
+
+We will now consider the case \(H\neq 0\). By Cauchy-Schwarz inequality we have that \(\|A\|^{2}\geq nH^{2}\), and equality only occurs if \(M\) is totally umbilical. In this case, following our ideas in [2], we will work with test functions of the form \(\ell_{v}-\alpha_{{\pm}}f_{v}\), where
+
+\[\alpha_{\pm}=\frac{\|A\|^{2}-n\pm\sqrt{D}}{2nH}\quad\hbox{with}\quad D=(\|A\|^{2}-n)^{2}+4n^{2}H^{2}>0.\]
+
+Let
+
+\[U_{+}=\{\ell_{v}-\alpha_{+}f_{v}:v\in\mbox{$\mathbb{R}^{n+2}$}\}\quad\hbox{and}\quad U_{-}=\{\ell_{v}-\alpha_{-}f_{v}:v\in\mbox{$\mathbb{R}^{n+2}$}\}.\]
+
+Then, by Proposition 5 we have that \(\Delta u+\mu_{\pm}u=0\) for every \(u\in U_{\pm}\), where
+
+\[0<\mu_{-}=\frac{n+\|A\|^{2}-\sqrt{D}}{2}<\mu_{+}=\frac{n+\|A\|^{2}+\sqrt{D}}{2},\]
+
+and, therefore, \(Ju+\lambda_{\pm}u=0\) for every \(u\in U_{\pm}\), with
+
+\[\lambda_{-}=\frac{-(n+\|A\|^{2})-\sqrt{D}}{2}<\lambda_{+}=\frac{-(n+\|A\|^{2})+\sqrt{D}}{2}<0\]
+
+(for the details, see [2, Section 4]). In particular, functions belonging to \(U_{\pm}\) also satisfy the condition \(\int_{M}f=0\), and
+
+(19) \[\mathrm{Ind}_{T}(M)\geq\mathrm{dim}(U_{+}\oplus U_{-})=\mathrm{dim}U_{+}+\mathrm{dim}U_{-}.\]
+
+Finally, since \(M\) is neither a totally umbilical sphere nor a Clifford hypersurface, our Theorem 3 implies that \(\mathrm{dim}U_{+}=\mathrm{dim}U_{-}=n+2\), and by (19) we conclude that \(\mathrm{Ind}_{T}(M)\geq 2n+4\).
+
+∎
+
+## Acknowledgements
+
+The authors would like to thank to the referee for valuable suggestions which improved the paper.
+
+## References
+
+* [1] L.J. Alías, On the stability index of minimal and constant mean curvature hypersurfaces in spheres, _Rev. Un. Mat. Argentina_**47** (2006), 39–61 (2007).
+* [2] L.J. Alías, A. Brasil Jr. & O. Perdomo, On the stability index of hypersurfaces with constant mean curvature in spheres, _Proc. Amer. Math. Soc._**135** (2007) 3685–3693.
+* [3] J.L. Barbosa, M. do Carmo & J. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds, _Math. Z._**197** (1988), 123–138.
+* [4] E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, _Annali di Mat._**17** (1938), 177–191.
+* [5] S.S. Chern, M. do Carmo, & S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length. 1970 _Functional Analysis and Related Fields_ (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) pp. 59–75 Springer, New York.
+* [6] Cecyl, T.E. & Ryan, P.J. _Tight and taut immersions of manifolds._ Research Notes in Mathematics, 107. Pitman (Advanced Publising Program), Boston, MA, 1985.
+* [7] H.B. Lawson Jr., Local rigidity theorems for minimal hypersurfaces, _Ann. of Math. (2)_**89** (1969), 187–197.
+* [8] O. Perdomo, Low index minimal hypersurfaces of spheres, _Asian J. Math._**5** (2001), 741–749.
+* [9] J. Simons, Minimal varieties in Riemannian manifolds, _Ann. of Math. (2)_, **88** (1968) 62–105.
+* [10] B. Solomon, The harmonic analysis of cubic isopatametric minimal hypersurfaces I: dimensions 3 and 6. _Amer. J. Math._**112** (1990), 157–203.
+* [11] B. Solomon, The harmonic analysis of cubic isoparametric minimal hypersurfaces II: dimensions 12 and 24. _Amer. J. Math._**112** (1990), 205–241.
+

arxiv_ground_truth/0802.3949.md

@@ -0,0 +1,145 @@
+# Adaptation dynamics of the quasispecies model
+
+Kavita Jain
+
+Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064,India ¹
+
+Footnote 1: Also at Evolutionary and Organismal Biology Unit
+
+(July 5, 2024)
+
+###### Abstract
+
+We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen’s model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a _quasispecies_ which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or step-like) pattern in which a population jumps from a low fitness peak to a higher one, stays there for a considerable time before shifting the peak again and eventually reaches the global maximum of the fitness landscape. We calculate exactly several properties of this dynamical process within a simplified version of the quasispecies model.
+
+## I Introduction
+
+Consider a maladapted population such as a bacterial colony in a glucose-limited environment, or a viral population in a vaccinated animal cell. In such harsh environments, the less fit members of the population are likely to perish and only the highly fit ones can survive to the next generation. In this manner, the fitness of the population increases with time and the initially maladapted population evolves to a well-adapted state. In the last century, there has been a concerted effort to put this verbal theory of Darwin Darwin (1859) on a solid quantitative footing by performing long-term experiments on microbial populations and studying theoretical models of biological evolution.
+
+One of the questions in evolutionary biology concerns the mode of evolution. In the experiments on microbes, it is found that the fitness of the maladapted population can increase with time in either a smooth continuous manner Novella et al. (1995) or sudden jumps Elena and Lenski (2003). The latter mode is consistent with evolution on a fitness landscape defined on genotypic space with many local peaks separated by fitness valleys. On such a rugged fitness landscape, a low fitness population initially climbs a fitness peak until it encounters a local peak where it gets trapped since a better peak lies some mutational distance away. In a population of realistic size, it takes a finite time for an adaptive mutation to arise and the fitness stays constant during this time (stasis). Once some beneficial mutants become available, the fitness increases quickly as the population moves to a higher peak where it can again get stuck. Such dynamics alternating between stasis and rapid changes in fitness go on until the population reaches the global maximum.
+
+This punctuated behavior of fitness is also seen in deterministic models that assume infinite population size. An example of such a step-like pattern for average fitness is shown in Fig. 1. A neat and unambiguous way of defining a step is by considering the fitness of the most populated genotype also shown in Fig. 1. Since large but finite populations evolve deterministically at short times Jain and Krug (2007a), it is worthwhile to study the punctuated evolution in models with infinite number of individuals. In this article, we will briefly describe some exact results concerning the dynamics of an infinitely large population on rugged fitness landscapes Jain and Krug (2005); Jain (2007a). We will find that the mechanism producing the step-like behavior is not due to “valley crossing” as in finite populations but when a fitter population “overtakes” the less fit one as described in the subsequent sections.
+
+Figure 1: (Color online)Punctuated change in the average population fitness (dotted line) and the fitness of the most populated genotype (solid line) for an infinite population evolving on a maximally rugged fitness landscape. Here genome length \(L=15\) and mutation probability \(\mu=10^{-4}\).
+
+## II Quasispecies model and its steady state
+
+We consider an infinitely large population reproducing asexually via the elementary processes of selection and mutation. Each individual in the population carries a binary string \(\sigma=\{\sigma_{1},...,\sigma_{L}\}\) of length \(L\) where \(\sigma_{i}=0\) or \(1\). The \(2^{L}\) sequences are arranged on the multi-dimensional Hamming space. The information about the environment is encoded in fitness landscape defined as a map from the sequence space into the real numbers and is generated by associating a non-negative real number \(W(\sigma)\) to each sequence \(\sigma\). Fitness landscapes can be simple possessing some symmetry properties such as permutation invariance, or complex devoid of any such symmetries Gavrilets (2004); Jain and Krug (2007b). Fitness functions with single peak are an example of simple fitness landscapes while rugged landscapes with many hills and valleys belong to the latter class.
+
+The average population fraction \({\cal X}(\sigma,t)\) with sequence \(\sigma\) at time \(t\) follows mutation-selection dynamics described by the following discrete time equation Eigen (1971); Jain and Krug (2007b)
+
+\[{\cal X}(\sigma,t+1)=\frac{\sum_{\sigma^{\prime}}p_{\sigma\leftarrow\sigma^{\prime}}W(\sigma^{\prime}){\cal X}(\sigma^{\prime},t)}{\sum_{\sigma^{\prime}}W(\sigma^{\prime}){\cal X}(\sigma^{\prime},t)}~{}.\] (1)
+
+The last two factors in the numerator of the above equation give the population fraction when a sequence \(\sigma^{\prime}\) copies itself with replication probability \(W(\sigma^{\prime})\) since fitness is defined as the average number of offspring produced per generation. After the reproduction process, point mutations are introduced independently at each locus of the sequence \(\sigma^{\prime}\) with probability \(\mu\) per generation. Thus, a sequence \(\sigma\) is obtained via mutations in \(\sigma^{\prime}\) with probability
+
+\[p_{\sigma\leftarrow\sigma^{\prime}}=\mu^{d(\sigma,\sigma^{\prime})}(1-\mu)^{L-d(\sigma,\sigma^{\prime})}\] (2)
+
+where the Hamming distance \(d(\sigma,\sigma^{\prime})\) is the number of point mutations in which the sequences \(\sigma\) and \(\sigma^{\prime}\) differ. The denominator of (1) is the average fitness of the population at time \(t\) which ensures that the density \({\cal X}(\sigma,t)\) is conserved.
+
+The stationary state of the quasispecies equation (1) has been studied extensively in the last two decades for various fitness landscapes. These numerical and analytical studies have shown that for most landscapes, there exists a critical mutation rate \(\mu_{c}\) below which the population forms a quasispecies consisting of fittest genotype and its closely related mutants while above it, the population delocalises over the whole sequence space. This _error threshold_ phenomenon can be easily demonstrated for a single peak fitness landscape defined as
+
+\[W(\sigma)=W_{0}\delta_{\sigma,\sigma_{0}}+(1-\delta_{\sigma,\sigma_{0}})~{},~{}W_{0}>1\] (3)
+
+where \(\sigma_{0}\) is the fittest sequence. In the limit \(\mu\to 0,L\to\infty\) keeping \(U=\mu L\) fixed, the frequency of the fittest sequence in the steady state of (1) is given by
+
+\[{\cal X}(\sigma_{0})=\frac{W_{0}e^{-U}-1}{W_{0}-1}\] (4)
+
+which is an acceptable solution provided \(U\leq U_{c}=\ln W_{0}\). For \(U>U_{c}\), selection is unable to counter the delocalising effects of mutation and the population can not be maintained at the fitness peak. For a discussion of error threshold phenomenon on other fitness landscapes and generalisations of the basic quasispecies equation (1), we refer the reader to Jain and Krug (2007b).
+
+## III Quasispecies dynamics on rugged fitness landscapes
+
+We now turn our attention to the dynamical evolution of \({\cal X}(\sigma,t)\) on rugged fitness landscapes. We consider maximally rugged fitness landscapes for which the fitness \(W(\sigma)\) is a random variable chosen independently from a common distribution. It is useful to introduce the unnormalised population defined as
+
+\[{\cal Z}(\sigma,t)={\cal X}(\sigma,t)\prod_{\tau=0}^{t-1}\sum_{\sigma^{\prime}}W(\sigma^{\prime}){\cal X}(\sigma^{\prime},t)\] (5)
+
+in terms of which the nonlinear evolution (1) reduces to the following linear iteration
+
+\[{\cal Z}(\sigma,t+1)=\sum_{\sigma^{{}^{\prime}}}p_{\sigma\leftarrow\sigma^{{}^{\prime}}}W(\sigma^{\prime}){\cal Z}(\sigma^{\prime},t)~{}.\] (6)
+
+Since at the beginning of the adaptation process the population finds itself at a low fitness genotype, we start with the initial condition \({\cal X}(\sigma,0)={\cal Z}(\sigma,0)=\delta_{\sigma,\sigma^{(0)}}\) where \(\sigma^{(0)}\) is a randomly chosen sequence. For mutation probability \(\mu\to 0\), after one iteration we have
+
+\[{\cal Z}(\sigma,1)\sim\mu^{d(\sigma,\sigma^{(0)})}W(\sigma^{(0)})~{}.\] (7)
+
+Thus in an infinite population model, each sequence gets populated in one generation obviating the need for “valley crossing” which is required for finite populations. Although an exact solution of (6) for \(t>1\) is not available, it is possible to obtain several asymptotically exact results concerning the most populated genotype using a simplified version of the quasispecies dynamics. Numerical simulations of Krug and Karl (2003) showed that dynamical properties involving the most populated genotype are well described by a simplified model which approximates the population \({\cal Z}(\sigma,t)\) in (6) by
+
+\[{\cal Z}(\sigma,t)\sim\mu^{d(\sigma,\sigma^{(0)})}W^{t}(\sigma)~{},~{}t>1~{}.\] (8)
+
+This model ignores mutations once each sequence has been populated and allows the population at each sequence to grow with its own fitness. However, a recent perturbative analysis in the small parameter \(\mu\) shows that this approximation holds for highly fit sequences and at short times Jain (2007a).
+
+Writing \(W(\sigma)=e^{F(\sigma)}\) and rescaling time by \(|\ln\mu|\) in (8), we find that the logarithmic population \(E(\sigma,t)\) obeys the following linear equation:
+
+\[E(\sigma,t)=-d(\sigma,\sigma^{(0)})+F(\sigma)~{}t~{}.\] (9)
+
+The linear evolution of the (logarithmic) population of \(2^{L}\) sequences for \(L=4\) is shown in Fig. 2a. Since the initial population fraction given by (7) is same for all the sequences at constant Hamming distance \(d(\sigma,\sigma^{(0)})\) from \(\sigma^{(0)}\), \({L\choose d}\) lines are seen to emanate from the same intercept. However as the genotype with the largest slope (fitness) at constant intercept has the potential to become the most populated sequence, we arrive at the model in Fig. 2b in which \(L+1\) genotypes are retained, each of whose fitness \(F(k),k=0,...,L\) is an independent but non-identically distributed variable Krug and Karl (2003); Jain and Krug (2005).
+
+Figure 2: (a) Evolutionary trajectories \(E(\sigma,t)\) defined by (9) for \(L=4\). The bold lines have the largest fitness amongst the \({L\choose k}\) fitnesses at distance \(k\) from the origin. (b) Evolutionary race: The sequence at distance \(3\) is the most populated sequence (winner) while the one at distance \(2\) is a record (contender).
+
+In a sequence \(\{F(k)\}\) of random variables, a _record_ is said to occur at \(m\) if \(F(m)>F(k)\) for all \(k<m\). In Fig. 2b, the sequences at distance \(k=0,2\) and \(3\) from the initial sequence are records but the sequence at \(k=2\) does not become a most populated genotype. In order to qualify as a _jump_, it is not sufficient to have a record fitness; the population should also be able to overtake the current winner in minimum time. Due to the overtaking time minimization constraint, the records and jumps have different statistical properties which we describe briefly in the next subsections.
+
+### Statistics of records
+
+Although the record statistics for independent and identically distributed (i.i.d.) random variables is well studied, much less is known when the variables are not i.i.d.Nevzorov (2001). Here we have a situation in which \(F(k)\) is a maximum of \(\alpha_{k}={L\choose k}\) i.i.d. random variables. However, since the \(k\)th record fitness \(F(k)\) is the largest amongst \(\sum_{j=0}^{k}\alpha_{j}\) i.i.d. variables and there are \(\alpha_{k}\) ways of choosing it, the probability \(\tilde{P}_{k}\) that the \(k\)th fitness is a record is given by Jain and Krug (2005); Krug and Jain (2005)
+
+\[\tilde{P}_{k}=\frac{{L\choose k}}{\sum_{j=0}^{k}{L\choose j}}\approx\frac{L-2k}{L-k}~{},~{}k<L/2~{}.\] (10)
+
+The meaning of the above distribution is intuitively clear: as it is easier to break records in the beginning, the probability to find a record is near unity for \(k\ll L\) and it vanishes beyond \(L/2\) because the global maximum typically occurs at this distance. The average number \({\cal{R}}\) of records can be obtained by simply integrating \(\tilde{P}(k)\) over \(k\) to yield \({\cal{R}}\approx(1-\ln 2)L\). It is also possible to find the typical spacing \(\tilde{\Delta}(j)\) between the \(j\)th and \((j+1)\)th record where we have labeled the last record (i.e. global maximum) as \(j=1\). A straightforward calculation shows that the typical inter-record spacing falls as a power law given by Jain and Krug (2005)
+
+\[\tilde{\Delta}(j)\approx\sqrt{\frac{L}{4\pi j}}\;\;,\;\;j\gg 1\;\;.\] (11)
+
+The above expression indicates that the spacing between the last few records (i.e \(j\sim{\cal{O}}(1)\)) is of order \(\sqrt{L}\), while most of the records are crowded at the beginning which is consistent with the behavior of the record occurrence probability (10).
+
+### Statistics of jumps
+
+The calculation of jump statistics Jain (2007a) is more involved than that of records because a jump event requires a minimization of the overtaking time. This constraint imposes a condition on the fitnesses of the squences that can possibly overtake the current leader in a time interval between \(t\) and \(t+dt\). The sequence at distance \(k^{\prime}\) can overtake the \(k\)th one (with fitness \(F\)) at time \(t\) if the fitness \(F(k^{\prime})=(E(k,t)+k^{\prime})/t\) and at time \(t+dt\), \(dt/t\to 0\) if
+
+\[F(k^{\prime})=\frac{E(k,t+dt)+k^{\prime}}{t+dt}=F+\frac{k^{\prime}-k}{t}-\frac{k^{\prime}-k}{t^{2}}dt+{\cal O}(dt^{2})~{}.\]
+
+Then the total collision rate \(W_{k^{\prime},k}(F,t)\) with which the \(k\)th sequence is overtaken by the \(k^{\prime}\)th one is given as
+
+\[W_{k^{\prime},k}(F,t)\approx\frac{k^{\prime}-k}{t^{2}}~{}p_{k^{\prime}}\left(F+\frac{k^{\prime}-k}{t}\right)~{},~{}k^{\prime}>k~{}\] (12)
+
+where \(p_{k}(F)\) is the distribution of the maximum of \(\alpha_{k}\) i.i.d. random variables distributed according to \(p(F)\) with support over the interval \(\left[F_{\rm{min}},F_{\rm{max}}\right]\). Using this collision rate, we can write the probability \({\cal P}_{k^{\prime},k}(t)\) that the sequence at distance \(k^{\prime}\) overtakes the \(k\)th one at time \(t\) as
+
+\[{\cal P}_{k^{\prime},k}(t)=\int_{F_{\rm{min}}}^{F_{\rm{max}}}dF~{}W_{k^{\prime},k}(F,t)~{}P_{k}(F,t)\] (13)
+
+where the probability \(P_{k}(F,t)\) that the \(k\)th sequence has the largest population at time \(t\) is given by
+
+\[P_{k}(F,t)=p_{k}(F)~{}\prod_{\begin{subarray}{c}j=0\\ j\neq k\end{subarray}}^{L}\int_{F_{\rm{min}}}^{F+\frac{j-k}{t}}dF^{\prime}~{}p_{j}(F^{\prime})~{}.\] (14)
+
+Note that unlike the records, the jump properties depend on the underlying distribution of the random variables. Below we present some results when the distribution \(p(F)=e^{-F}\).
+
+Integrating (13) over time, the probability distribution \(P_{k^{\prime},k}\) that \(k\)th sequence is overtaken by \(k^{\prime}\)th sequence is obtained,
+
+\[P_{k^{\prime},k}\approx\sqrt{\frac{L}{\pi k(L-k)}}~{}\left(\frac{k^{\prime}-k}{2k}\right)~{}e^{-\frac{L(k^{\prime}-k)^{2}}{4k(L-k)}}~{},~{}k<k^{\prime}<L/2~{}.\] (15)
+
+This form of the distribution implies that the overtaking sequence \(k^{\prime}\) is located within \({\cal O}(\sqrt{k})\) distance of the overtaken sequence \(k\). Thus the typical spacing between successive jumps for large \(k\) is roughly constant and goes as \(\sqrt{L}\) unlike in the case of records discussed in the last subsection. The jump distribution \(P_{k}\) for a jump to occur at distance \(k\) is obtained by integrating over \(k^{\prime}\) and we have Jain (2007a)
+
+\[P_{k}\approx\sqrt{\frac{L}{\pi k(L-k)}}~{}\theta_{H}\left(\frac{L}{2}-k\right)\] (16)
+
+where \(\theta_{H}\) is the Heaviside step function which takes care of the fact that the record distribution (10) vanishes at distance \(L/2\). Instead of integrating over time, by summing over the space variables \(k,k^{\prime}\) in (13), the probability \(P(t)\) that a jump occurs at time \(t\) can be obtained and is given by Jain (2007a)
+
+\[P(t)=\sqrt{\frac{L}{4\pi}}~{}\frac{1}{t^{2}}~{}\mathrm{sech}\left(\frac{1}{2t}\right)~{}.\] (17)
+
+The heavy tail distribution \(P(t)\sim t^{-2}\) can be understood using a simple argument Krug and Karl (2003) and implies that mean overtaking time is infinite. Finally, by either summing \(P_{k}\) over \(k\) or integrating \(P(t)\) over time, the total number of jumps \({\cal J}\) are found to be \(\sqrt{L\pi}/2\) which is much smaller than the number of records \({\cal R}\).
+
+## IV Summary
+
+In this article, we discussed the steady state and the dynamics of the quasispecies model which describes a self-replicating population evolving under mutation-selection dynamics. On rugged fitness landscapes, the population fitness increases in a punctuated fashion and we described several exact results concerning this mode of evolution. Our recent simulations indicate that the \(1/t^{2}\) law in (17) for the deterministic populations also holds for finite stochastically evolving populations Jain (2007a). At present, we do not have an analytical understanding of the latter result but it should be possible to test this law in long-term experiments such as those of Elena and Lenski (2003) on _E. Coli_.
+
+Acknowledgements: I am very grateful to Prof. J. Krug for introducing me to the area of theoretical evolutionary biology. I also thank the organisers of the Statphys conference at IIT, Guwahati for giving me an opportunity to present my work.
+
+## References
+
+* Darwin (1859) C. Darwin, _The origin of species by means of natural selection_ (John Murray, London, 1859).
+* Novella et al. (1995) I. Novella, E. Duarte, S. Elena, A. Moya, E. Domingo, and J. Holland, Proc. Natl. Acad. Sci. USA **92**, 5841 (1995).
+* Elena and Lenski (2003) S. F. Elena and R. E. Lenski, Nat. Rev. Genet. **4**, 457 (2003).
+* Jain and Krug (2007a) K. Jain and J. Krug, Genetics **175**, 1275 (2007a).
+* Jain and Krug (2005) K. Jain and J. Krug, J. Stat. Mech.: Theor. Exp. p. P04008 (2005).
+* Jain (2007a) K. Jain, Phys. Rev. E **76**, 031922 (2007a).
+* Gavrilets (2004) S. Gavrilets, _Fitness landscapes and the origin of species_ (Princeton University Press, 2004).
+* Jain and Krug (2007b) K. Jain and J. Krug, in _Structural approaches to sequence evolution: Molecules, networks and populations_, edited by U. Bastolla, M. Porto, H. Roman, and M. Vendruscolo (Springer, Berlin, 2007b), pp. 299–340, eprint arXiv:q-bio.PE/0508008.
+* Eigen (1971) M. Eigen, Naturwissenchaften **58**, 465 (1971).
+* Krug and Karl (2003) J. Krug and C. Karl, Physica A **318**, 137 (2003).
+* Nevzorov (2001) V.B. Nevzorov, _Records: Mathematical Theory_ (Providence, RI: American Mathematical Society, 2001).
+* Krug and Jain (2005) J. Krug and K. Jain, Physica A **358**, 1 (2005).
+

arxiv_ground_truth/0808.3215.md

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+# Quantitative Description of Strong-Coupling of Quantum Dots in Microcavities
+
+F.P. Laussy
+
+E. del Valle
+
+C. Tejedor
+
+###### Abstract
+
+We have recently developed a self-consistent theory of Strong-Coupling in the presence of an incoherent pumping [arXiv:0807.3194] and shown how it could reproduce quantitatively the experimental data [PRL **101**, 083601 (2008)]. Here, we summarize our main results, provide the detailed analysis of the fitting of the experiment and discuss how the field should now evolve beyond merely qualitative expectations, that could well be erroneous even when they seem to be firmly established.
+
+Keywords: : 42.50.Ct, 78.67.Hc, 42.55.Sa, 32.70.Jz 
+Figure 1: Theoretical fit (in semi-transparent red) of the data by Reithmaier _et al._ digitized from Ref. Reithmaier et al. (2004) (blue). The model F.P.Laussy et al. (2008a) provides \(S(\omega)=\frac{1}{2\pi}\mathrm{Re}\sum_{\sigma=-1,1}\frac{1+\sigma\mathcal{C}}{\Gamma_{+}+i[\sigma R+\omega-(\omega_{a}+\omega_{b})/2]}\) with \(\Gamma_{\pm}=({\gamma_{a}-P_{a}\pm(\gamma_{b}-P_{b})})/{4}\), \(R=\sqrt{g^{2}-(\Gamma_{-}+i\Delta/2)^{2}}\), \(D={\frac{g}{2}(\gamma_{a}P_{b}-\gamma_{b}P_{a})(i\Gamma_{+}-{\Delta}/2)}/{g^{2}\Gamma_{+}(P_{a}+P_{b})+P_{a}(\gamma_{b}-P_{b})(\Gamma_{+}^{2}+({\Delta}/{2})^{2})}\) and \(\mathcal{C}={\Gamma_{-}+i(\Delta/2+gD)}/R\). The data has been fitted on rescaled axes for numerical stability by a Levenberg–Marquardt method with \(\mathcal{N}S(\omega)-c\), with \(\mathcal{N}\) and \(c\) to account for the normalization and the background. Beside these two necessary parameters regardless of the model, each panel only has \(P_{a/b}\) and \(\omega_{a/b}\) (c) as fitting parameters. \(g\) and \(\gamma_{a/b}\) have been optimized globally, with best fits for \(g=61\mu\)eV, \(\gamma_{a}=220\mu\)eV and \(\gamma_{b}=140\mu\)eV. (a) shows the anticrossing from curves 1–15 put together. (b) keeps all fitting parameters the same but with \(P_{a}=0\) and vanishing \(P_{b}\). The dot emission now dominates and no anticrossing is observed, although the system is still in strong-coupling.
+
+_Strong-Coupling_ (SC) is the term consecrated by the quantum optics community to designate “quantum coupling”, where coherent interaction dominate over dissipation. A vivid picture represents this regime in terms of a cyclic _exchange_ of energy between the light and matter fields (the so-called “_Rabi oscillation_”). This oscillation is a particular case as can be seen straightforwardly by considering two conceptual cases where it does not occur: when the system is in an eigenstate of the SC hamiltonian, in which case it has no dynamics, and when it has reached a steady state (SS), where, regardless of which quantum state is realized, energy is balanced rather than exchanged. The SS is the relevant case for many experimental configurations with semiconductors Reithmaier et al. (2004); Yoshie et al. (2004); Peter et al. (2005), where a continuous incoherent pumping excites the system. An anticrossing is looked for in the luminescence spectra as a proof of SC, attributed to the emergence of “dressed” states.
+
+To compute a luminescence spectrum of a QD in a semiconductor, one must consider an innocent-looking but important consequence of pumping: it enforces a steady state that is a mixture of cavity photons and excited state of the QD (exciton). The lineshape of the cavity luminescence spectrum \(S(\omega)\) depends strongly on whether the system is _photon-like_ or _exciton-like_F.P.Laussy et al. (2008a), in the sense of which particle would be injected in the system in the limit of vanishing pumpings that keep a fixed ratio as they go to zero. The steady state should be computed self-consistently from the interplay of pumping and decay, rather than assuming that the system is, typically, in the excited state of the QD. The latter assumption would be reasonable if there was only one kind of pumping, namely in this case an electronic pumping. However, experimental evidence seems to imply that an incoherent cavity pumping accompanies the electronic pumping F.P.Laussy et al. (2008b). The most likely reason is that beside the QD that is strongly coupled to the cavity, there are many other dots in weak-coupling, that also get excited by the electronic pumping, and that populate the cavity when they de-excite. This, for the SC dot, is perceived as an incoherent cavity pumping.
+
+Even in the simplest description where the two modes (light, \(a\), and matter, \(b\)) are bosonic, the luminescence spectra enjoy a wealth of subtle characteristics when incoherent and continuous pumpings \(P_{a/b}\) for the cavity/exciton, are introduced. Those are discussed at length in Ref. F.P.Laussy et al. (2008a). Here we mention the most important ones: anticrossing is not systematically observed for a system in SC, depending on whether the system is more photon-like (\(P_{a}>P_{b}\)), or more exciton-like (\(P_{b}>P_{a}\)). More surprisingly, an “apparent” anticrossing can be observed in a system that is in weak-coupling, due to an interference that carves a hole in the spectra. These two facts together disconnect completely an anticrossing behaviour from the realization of SC. One should not, therefore, rely on this qualitative effect to ascertain SC. Instead, a fitting of the data should be attempted, so as to place a given experiment in a region of SC defining rigorously the dressed states, rather than from their ability to survive in the luminescence spectrum.
+
+We have considered one of the pioneering experiment Reithmaier et al. (2004), as a test of the model, and found an excellent agreement with a linear (bosonic) model, cf. Fig. 1, provided that both cavity and exciton pumping are taken into account. This is consistent with the experimental curves that feature a strong cavity emission. In panel (b) we show how the lineshapes transform when the cavity pumping is switched off: the exciton line dominates, and no anticrossing is observed. The system is nevertheless still in SC. The confrontation of the theory with the experimental data was done for illustration only, as the raw experimental data was not available by the time of our investigation. We have therefore digitized the data, what forbids an in-depth statistical analysis, since the experimental points are required rather than the interpolated curves published in Ref. Reithmaier et al. (2004), if only to know the numbers of degrees of freedom. Note that one expects better still results as our procedure added noise. We found the best agreements near resonance, which might be due to the exciton that, when it is less-strongly coupled at larger detunings, may go below the resolution of the detector, resulting in an apparently broader line. All these limitations can be circumvented with a careful statistical analysis (and treatment of the data to reconstruct linewidths below the experimental resolution). This is a standard procedure of a mature field to validate a theoretical model over another by statistical analysis of the experiment. Nullifying hypotheses such as: “_a Fermi (two-level) system accounts for the observed data better than a Boson model_” are important to assess the achievements made in terms of quantum emitters with these systems. This would also provide a meaningful and quantitative comparison between the various implementations (micropillars, microdisks and photonic crystals). Lacking the full experimental data, we have merely been unable to provide a confidence interval to our most-likelihoods estimators. Doing so, progress will be meaningfully quantified, and claims—rather than ranging between likely and convincing—will become unambiguously proven (within their interval of confidence).
+
+## References
+
+* Reithmaier et al. (2004) J. P. Reithmaier _et al._, _Nature_**432**, 197 (2004).
+* Yoshie et al. (2004) T. Yoshie _et al._, _Nature_**432**, 200 (2004).
+* Peter et al. (2005) E. Peter _et al._, _Phys. Rev. Lett._**95**, 067401 (2005).
+* F.P.Laussy et al. (2008a) F.P.Laussy _et al._, _arXiv:0807.3194_ (2008b).
+* F.P.Laussy et al. (2008b) F.P.Laussy _et al._, _Phys. Rev. Lett._**101**, 083601 (2008a).
+

arxiv_ground_truth/0811.0781.md

@@ -0,0 +1,248 @@
+# Crowding Effects on the Mechanical Stability and Unfolding Pathways of Ubiquitin.
+
+David L. Pincus¹ and D. Thirumalai_1,2,_¹
+
+Footnote 1: Corresponding author phone: 301-405-4803; fax: 301-314-9404; thirum@umd.edu
+
+###### Abstract
+
+The interior of cells is crowded thus making it important to assess the effects of macromolecules on the folding of proteins. Using the Self-Organized Polymer (SOP) model, which is a coarse-grained representation of polypeptide chains, we probe the mechanical stability of Ubiquitin (Ub) monomers and trimers ((Ub)₃) in the presence of monodisperse spherical crowding agents. Crowding increases the volume fraction (\(\Phi_{c}\))-dependent average force (\(\langle f_{u}(\Phi_{c})\rangle\)), relative to the value at \(\Phi_{c}=0\), needed to unfold Ub and the polyprotein. For a given \(\Phi_{c}\), the values of \(\langle f_{u}(\Phi_{c})\rangle\) increase as the diameter (\(\sigma_{c}\)) of the crowding particles decreases. The average unfolding force \(\langle f_{u}(\Phi_{c})\rangle\) depends on the ratio \(\frac{D}{R_{g}}\), where \(D\approx\sigma_{c}(\frac{\pi}{6\Phi_{c}})^{\frac{1}{3}}\) with \(R_{g}\) being the radius of gyration of Ub (or (Ub)₃) in the unfolded state. Examination of the unfolding pathways shows that, relative to \(\Phi_{c}=0\), crowding promotes reassociation of ruptured secondary structural elements. Both the nature of the unfolding pathways and \(\langle f_{u}(\Phi_{c})\rangle\) for (Ub)₃ are altered in the presence of crowding particles with the effect being most dramatic for the subunit that unfolds last. We predict, based on SOP simulations and theoretical arguments, that \(\langle f_{u}(\Phi_{c})\rangle\sim\Phi_{c}^{\frac{1}{3\nu}}\), where \(\nu\) is the Flory exponent that describes the unfolded (random coil) state of the protein.
+
+¹Biophysics Program, Institute for Physical Science and Technology
+²Department of Chemistry and Biochemistry
+University of Maryland, College Park, MD 20742
+
+Keywords: Depletion Effect, Entropic Stabilization, Crowders, AFM, Loading Rate, SOP model
+
+## Introduction.
+
+Cells exist in a crowded environment consisting of macromolecules (lipids, mRNA, ribosome, sugars, etc.), making it critical to investigate protein folding in the presence of crowding agents [1]. If the interactions between the crowding agents and the protein of interest are short-ranged and non-specific (as is often the case), then the volume excluded by the crowding agents prevents the polypeptide from sampling extended conformations. As a consequence, the entropy of the denatured state ensemble (DSE) decreases relative to the case when the crowding agents are absent. These arguments suggest that excluded volume of crowding agents should enhance the stability of the folded state provided that the crowding-induced changes in the native state are negligible [2, 3]. The entropic stabilization mechanism, described above, has been used in several theoretical models to quantitatively describe the extent of folded protein as a function of the volume fraction, \(\Phi_{c}\), of the crowding agents [3, 4]. More recently, a theory whose origins can be traced to the concept of intra-protein attraction due to depletion of crowding agents near the protein [5, 6, 7], predicts that the enhancement in stability, \(\Delta T(\Phi_{c})=T_{f}(\Phi_{c})-T_{f}(\Phi_{c}=0)\sim\Phi_{c}^{\alpha}\), where \(T_{f}(\Phi_{c})\) is the folding temperature at \(\Phi_{c}\) and \(\alpha\) is related to the Flory exponent that characterizes the size of the protein in the DSE [3]. From this prediction it follows that crowding affects the DSE to a greater extent than the folded state. Although the precise theoretical predictions of the power law change in \(\Delta T(\Phi_{c})\) as \(\Phi_{c}\) changes have not been verified, several experiments using a number of proteins have confirmed that indeed \(T_{f}(\Phi_{c})\) increases with \(\Phi_{c}\)[8, 9, 10]. It cannot be emphasized enough that the theory described here applies only to cases when the crowding interactions between crowding agents and proteins and between crowding particles themselves are purely repulsive.
+
+While much less is known about the effects of crowding on the folding kinetics, Cheung et al. [3] predicted that the entropic stabilization also suggests that the folding rates should increase at moderate values of \(\Phi_{c}\). They suggest that crowding can enhance folding rates by a factor \(e^{\Delta S(\Phi_{c})/k_{B}}\) where \(\Delta S(\Phi_{c})\)\((\sim\Phi_{c}^{\alpha})\) is the decrease in the entropy of the DSE relative to its value in the bulk. From the arguments of Cheung et al. [3] it follows that the equilibrium changes in the entropy (\(\Delta S(\Phi_{c})\)) of the DSE, with respect to the bulk, should also determine rate enhancement provided that neither the barriers to folding [11] nor the native state is perturbed significantly by crowding particles.
+
+Single molecule force spectroscopy, such as Atomic Force Microscopy (AFM) and Laser Optical Tweezers, have been used to monitor the behavior of biopolymers under tension are ideally suited to probe the enhancement in crowding-induced stability by a direct measurement of \(f_{u}(\Phi_{c})\). Indeed, Ping et al. [12] have recently investigated the effect of Dextran molecules on the mechanical stability of \((\text{Ub})_{8}\). The 8 Ub (Fig. 1**A**) modules were N-C linked (i.e., modules i and i+1 were chemically linked together in a head-to-tail manner). They found that the average force required to unfold a module, \(\langle f_{u}(\Phi_{c})\rangle\), increased by 21\(\%\) as the Dextran concentration, \(\rho\), was increased from 0 to 300 g/L at \(r_{f}=4.2\times 10^{3}\) pN/s. Similar results have been obtained recently by Yuan et al. at \(r_{f}=12.5\times 10^{3}\) pN/s [13].
+
+Motivated in part by experiments [12, 13], we used simulations to investigate the effects of crowding agents on the mechanical stability of a protein subject to external tension. We focused on ubiquitin (Ub), a 76-residue protein composed of 5 \(\beta\)-strands and 2 \(\alpha\)-helices (Fig. 1**A**), and confined our investigation to non-equilibrium ‘force-ramp’ experiments [14]. The primary data recorded during such an experiment is a trace of the force exerted on the tip as a function of the extension of the molecule; a force-extension curve (FEC). When the force exceeds some critical value, the FEC displays a sudden increase in length and is often accompanied by a concomitant sharp decrease in force. Presumably, the sharp change corresponds to the unfolding of the protein. Typical AFM experiments use tandem arrays of proteins which are chemically linked together (often through genetic engineering). We use the term module to denote a protein of the array. The FEC resulting from such an experiment reveals several equally spaced peaks punctuated by sharp increases in the extension of the molecule corresponding to the unfolding of individual modules. The height of these force peaks and their shape depend on the loading rate, \(r_{f}=k_{s}\times\mathit{v}\), where \(k_{s}\) is the cantilever’s spring constant and \(\mathit{v}\) is the (constant) speed at which the stage is retracted away from the cantilever [15].
+
+In order to compare to experiments our simulations are performed using coarse-grained models for which simulations can be done at \(r_{f}\) that are comparable to those in AFM experiments. Our work has led to a number of testable results: (1) At \(\Phi_{c}=0.3\) the average unfolding force for Ub increases by at most only 7% compared to \(\Phi_{c}=0\). We find that \(\langle f_{u}(\Phi_{c})\rangle\) in small crowding agents is greater than in larger particles. (2) In the presence of crowding agents, secondary structural elements reform multiple times even after initial rupture. (3) Although large crowding particles are predicted to have a smaller effect on \(\langle f_{u}(\Phi_{c})\rangle\) (for a given \(\Phi_{c}\)), they can profoundly affect the unfolding of poly Ub. We predict that \(\langle f_{u}(\Phi_{c})\rangle\) for a given subunit depends on the number of already unfolded portions of the poly protein. This result is important because many naturally occurring proteins that are subject to tensile stresses exist as tandem arrays of modules. It further suggests that the existence of such redundancy can more properly be understood in the context of a crowded cellular milieu.
+
+## Methods.
+
+### Self-Organized Polymer Model for Ub.
+
+We used a coarse-grained model for proteins to investigate crowding effects on the mechanical stability of Ub and \((\text{Ub})_{3}\) at loading rates that are comparable to those used in AFM experiments [12, 13]. We assumed that Ub could be described using the Self-Organized Polymer (SOP) model; a model that has been successfully used to make a number of predictions regarding the unfolding of proteins and RNA [16, 17], allosteric transitions in enzymes [18, 19], and movement of molecular motors on polar tracks [20]. Previous studies [21] have used more standard Go-models [22, 23] to probe various aspects of forced unfolding of Ub. The SOP energy function (\(E_{p}\)) for a protein with N amino-acids, specified in terms of the C_α_ coordinates \(\mathbf{r}_{i}\)\((i=1,2,...,N)\), is
+
+\[\begin{split}E_{p}&=E_{FENE}+E_{nb}^{att}+E_{nb}^{rep}\\ &=-\sum_{i=1}^{N-1}\frac{k}{2}R_{0}^{2}\ln\left[1-\frac{\left(r_{i,i+1}-r_{i,i+1}^{0}\right)^{2}}{R_{0}^{2}}\right]+\sum_{i=1}^{N-3}\sum_{j=i+3}^{N}\varepsilon_{h}\left[\left(\frac{r_{ij}^{0}}{r_{ij}}\right)^{12}-2\left(\frac{r_{ij}^{0}}{r_{ij}}\right)^{6}\right]\Delta_{ij}\\ &+\sum_{i=1}^{N-2}\sum_{j=i+2}^{N}\varepsilon_{l}\left(\frac{\sigma}{r_{ij}}\right)^{6}\left(1-\Delta_{ij}\right),\end{split}\] (1)
+
+where \(r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|\), \(r_{ij}^{0}=|\mathbf{r}_{i}^{0}-\mathbf{r}_{j}^{0}|\) is the value of \(r_{ij}\) in the native structure, \(k=2\times 10^{3}\) kcal/(mol\(\cdot\)\(\text{nm}^{2}\)), \(\varepsilon_{h}=1.4\) kcal/mol, \(\varepsilon_{l}=1.0\) kcal/mol, and \(\sigma=0.38\) nm. Note that \(k_{B}T\approx 0.6\) kcal/mol \(\approx 4.2\) pN\(\cdot\)nm. In Eq. (1) \(\Delta_{ij}=1\) if \(r_{ij}^{0}<0.8\) nm, and \(\Delta_{ij}=0\) otherwise. Native coordinates corresponded to those of the C_α_ atoms of the 1.8 Å resolution Protein Data Bank crystal structure 1UBQ [24]. For Ub \(N=76\) and \(N=228\) for \((\text{Ub})_{3}\). The first term in Eq. (1) is the FENE potential [25] that accounted for chain connectivity. The second (Lennard-Jones) term accounted for the non-bonded interactions that stabilize the native state, and the final (soft-sphere) term accounted for excluded-volume interactions (including those of an angular nature). The SOP model is different from the Go-model because there are no angular terms in SOP, and the connectivity is enforced differently as well. The SOP representation of the polypeptide chain is in the same spirit as other coarse-grained models used in polymers [26].
+
+### Crowding Particles and Interactions with Ub.
+
+We assumed that the crowding particles are spherical with diameter \(\sigma_{c}\). (\(\sigma_{c}=6.4\) nm in some simulations, while \(\sigma_{c}=1.0\) nm in others.) Crowders interacted amongst themselves and with the protein, respectively, via the following LJ potentials:
+
+\[E_{\text{cc}}=4\varepsilon_{l}\left(\left(\frac{\sigma_{\text{cc}}}{r}\right)^{12}-\left(\frac{\sigma_{\text{cc}}}{r}\right)^{6}+\frac{1}{4}\right)\Theta\left(r_{\text{min}}^{\text{cc}}-r\right)\] (2)
+
+\[E_{\text{cp}}=4\varepsilon_{l}\left(\left(\frac{\sigma_{\text{cp}}}{r}\right)^{12}-\left(\frac{\sigma_{\text{cp}}}{r}\right)^{6}+\frac{1}{4}\right)\Theta\left(r_{\text{min}}^{\text{cp}}-r\right),\] (3)
+
+where \(\sigma_{\alpha\beta}=(\sigma_{\alpha}+\sigma_{\beta})/2\), \(\sigma_{p}=\sigma=0.38\) nm, \(r_{\text{min}}^{\text{cc}}=2^{1/6}\sigma_{\text{cc}}\), and \(r_{\text{min}}^{\text{cp}}=2^{1/6}\sigma_{\text{cp}}\). The Heaviside functions truncate the potentials at their minima and thereby ensured that only the repulsive portions of Eqs. (2) and (3) contributed to interactions involving crowding agent.
+
+_Mimics of Crowding Using Asakura-Oosawa Theory:_ Even using a coarse-grained SOP representation of proteins, it is difficult to carry out converged simulations in the presence of crowding agents. The reason is that the number of crowding agents can be large. Moreover, the separation in the spatial and temporal scales of the protein and the crowding particles has to be carefully considered to obtain reliable results. In light of these difficulties, it is of interest to consider the effective attraction between the sites on the protein using the implicit pairwise potential computed by Asakura and Oosawa. The intramolecular attraction arises due to the depletion of crowding particles near the protein. To probe the efficacy of these models we employed in some simulations the Asakura-Oosawa model [5, 6, 7] of crowding effects. For these simulations, we added the following term to the bare SOP Hamiltonian (Eqn. (1)):
+
+\[E_{\text{AO}}\left(r_{ij}\right)=-\Phi_{c}k_{B}T\sum_{j\geq i+3}\left(\frac{\left(\sigma+\sigma_{c}\right)}{\sigma_{c}}\right)^{3}\left(1-\frac{3r_{ij}}{2\left(\sigma+\sigma_{c}\right)}+\frac{r_{ij}^{3}}{2\left(\sigma+\sigma_{c}\right)^{3}}\right)\text{ }\sigma<r<\sigma+\sigma_{c},\] (4)
+
+where \(\sigma=0.38\) nm, \(\sigma_{c}=6.4\) nm, \(\Phi_{c}=0.3\), \(k_{B}T\simeq 4.2\) pN\(\cdot\)nm, and \(r_{ij}\) is the distance separating protein beads \(i\) and \(j\).
+
+### \((\text{Ub})_{3}\) Intermodule Interactions.
+
+For simulations involving \((\text{Ub})_{3}\), residues in different modules interacted via:
+
+\[E_{\text{pp}}=\varepsilon_{l}\left(\frac{\sigma}{r}\right)^{6},\] (5)
+
+where r is the distance separating the two beads. Note that this potential is short-ranged and purely repulsive and that it is the same potential used for non-native intra-protein interactions [17].
+
+### Simulation Details.
+
+_\(\Phi_{c}=0\)_: Hundreds of simulations of \(5\times 10^{6}\) steps (\(\simeq 30\mu\)s) at \(T=300\) K were used to generate initial structures for use in the pulling simulations. The protein was completely free in solution (i.e., no forces were applied to either terminus), and no crowders were present during the equilibrations. The N-terminus of the protein was subsequently translated to the origin, and the protein was rotated such that its end-to-end vector, \(\mathbf{R}\), (i.e., the vector pointing from the N-terminal bead to the C-terminal bead) coincided with the pulling (+z) direction.
+
+An unfolding trajectory was initiated by selecting a random initial structure from amongst the set of thermally equilibrated structures, and tethering a harmonic spring to the C-terminal bead. The N-terminal bead was held fixed throughout the simulations. Tension was applied to the protein by displacing the spring along the +z axis and resulted in application of the following force to the C-terminal bead:
+
+\[f_{z}=-k_{s}\left([z(t)-z(0)]-\left[z_{s}(t)-z_{s}(0)\right]\right),\] (6)
+
+where \(k_{s}\) is the spring constant, \(z(t)=\mathbf{R}(t)\cdot\overset{\wedge}{\mathbf{z}}\), and \(z_{s}(t)\) corresponds to the displacement of the end of the spring. Note that \(k_{s}\) was also used to constrain the simulation to the z-axis; \(f_{x}=-k_{s}[x(t)-x(0)]\) and \(f_{y}=-k_{s}[y(t)-y(0)]\). The displacement of the spring was updated at every timestep.
+
+We simulated forced-unfolding of monomeric Ub at four different \(r_{f}\) (\(160\times 10^{3}\) pN/s, \(80\times 10^{3}\) pN/s, \(20\times 10^{3}\) pN/s, and \(4\times 10^{3}\) pN/s), while simulations on N-C-linked \((\text{Ub})_{3}\) were performed at \(r_{f}=640\times 10^{4}\) pN/s. All overdamped force-ramp simulations were performed at the same speed \(\mathit{v}=10312\) nm/s, and spring constants were varied over a range from 0.3879 pN/nm - 31.032 pN/nm to achieve the aforementioned \(r_{f}\) (via the relation \(r_{f}=k_{s}\mathit{v}\)). Our simulations were realistic because they maintained loading rates consistent with experiment and because \(r_{f}\) is the prime determinant of unfolding pathway [15].
+
+_\(\Phi_{c}\neq\) 0.0_: Simulations involving explicit crowders were carried out at a fixed volume fraction \(\Phi_{c}=0.3\) and with a fixed number \(N_{c}=100\) of crowding spheres. (\(N_{c}\) was fixed to render the problem computationally tractable). Using the relation \(\Phi_{c}=\frac{N_{c}\pi}{6}\left(\frac{\sigma_{c}}{L}\right)^{3}\), we adjusted the length of a side of the cubic simulation box ( \(L\) ) to maintain \(\Phi_{c}=0.3\). Thus, \(L=35.8\) nm when \(\sigma_{c}=6.4\) nm and \(L=5.6\) nm when \(\sigma_{c}=1.0\) nm. Explicit crowders were added to the simulation after loading an equilibrated structure but before the application of tension. Initial crowder positions were chosen randomly and in a serial manner from a uniform distribution. If the distance between an initial crowder position and that of another crowder or protein bead did not exceed the sum of their radii, then the prospective position was rejected and another random position chosen to avoid highly unfavorable steric overlaps.
+
+Periodic boundary conditions (PBC) and the minimum image convention [27] were employed in the simulations. Two sets of coordinates were stored for protein beads at every timestep; PBC were applied to one set and the other was propagated without PBC. Distances between protein beads were calculated from the uncorrected set of coordinates without minimum imaging, while protein-crowder distances were calculated from the PBC coordinates with minimum imaging.
+
+To improve simulation efficiency, a cell list [27] was used to calculate crowder-crowder and protein-crowder interactions. The entire simulation volume was partitioned into 64 subvolumes, and it was only necessary to calculate interactions within a subvolume and between beads of the subvolume and those of 13 of its 26 neighbors. The cell-list was updated at every timestep to ensure the accuracy of the simulations.
+
+The equations of motion in our overdamped simulations (used in all force-ramp simulations) were integrated with a timestep \(h=0.01\tau_{L}\) (\(\tau_{L}=2.78\) ps) using the method of Ermak and McCammon [28]. The friction coefficient of the crowders, \(\zeta_{c}\), was determined via the relation \(\frac{\zeta_{c}}{\zeta}=\frac{\sigma_{c}}{\sigma}\), where \(\sigma_{c}\) is the crowder diameter, \(\sigma=0.38\) nm is the diameter of a protein bead, and \(\zeta=83.3\times(\frac{m}{\tau_{L}})=9\times 10^{-9}\) g/s is the friction coefficient associated with a protein bead of mass \(m=3\times 10^{-22}\) g. Simulated-times were translated into real-times using \(\tau_{H}=(\frac{\zeta\epsilon_{h}}{k_{B}T})\times\tau_{L}\times(\frac{\tau_{L}}{m})\)[29]. At \(T=300\) K, \(\tau_{H}=543.06\) ps, and since \(h=0.01\times\tau_{L}\) the real-time per step is 5.4306 ps.
+
+## Results and Discussion.
+
+### Monomeric Ub at \(\Phi_{c}=0.0\).
+
+At \(\Phi_{c}=0.0\), forced unfolding of Ub was simulated at four different \(r_{f}\) (\(4\times 10^{3}\) pN/s, \(20\times 10^{3}\) pN/s, \(80\times 10^{3}\) pN/s, \(160\times 10^{3}\) pN/s), where the lowest value corresponds approximately to the value used in the pulling experiments of Ping et al. [12], and all \(r_{f}\) are experimentally accessible.
+
+_Force Profiles_: Fig. 2(**A** and **B**) provides examples of FEC’s collected at the highest and lowest \(r_{f}\). We used a nominal contour length of (\(N-1\)) \(\sigma=75\times 0.38\) nm and unfolding forces, \(f_{u}\), to determine contour-length increments, \(\Delta\mathcal{L}\), for each trajectory at \(r_{f}=160\times 10^{3}\) pN/s (Fig. 2**A**). We identified \(f_{u}\) with the peak of the FEC before the stick-slip transition [30, 31]. The average extension \(\langle\Delta\mathcal{L}\rangle=23.991\pm 0.010\) nm is in excellent agreement with the experimental result of 24 \(\pm\) 5 nm found by Carrion-Vazquez et al. [30]. The projection (\(z_{u}\)) of the end-to-end vector at \(f_{u}\) in the z-direction varied between 4.1 nm and 4.7 nm, depending on \(r_{f}\). Since the native end-to-end distance, \(z_{0}=3.7\) nm, \(z_{u}-z_{0}\equiv\Delta z_{u}\) ranges from 0.4-1.0 nm. The lower end of this range is slightly larger than the 0.25 nm transition-state distance for the mechanical unfolding of the structurally similar titin immunoglobulin domains [32]. Indeed, we expect \(\Delta z_{u}>0.25\) nm, because of the non-equilibrium nature of the simulations. Larger \(r_{f}\) typically lead to larger \(\langle\Delta z_{u}\rangle\) (\(\sim k_{B}T/\langle f_{u}\rangle\ln(r_{f})\)).
+
+Average unfolding forces, \(\langle f_{u}(\Phi_{c})\rangle\), depended approximately logarithmically on \(r_{f}\)[15] (Fig. 2**C**), and \(\langle f_{u}(r_{f}=4\times 10^{3}\text{pN/s})\rangle=136\) pN is in fair agreement with the experimental value of \(166\pm 33\) pN observed by Ping et al. [12] at \(r_{f}=4.2\times 10^{3}\) pN/s. The \(\langle z_{u}\rangle\) also showed a logarithmic dependence on \(r_{f}\), but the difference between the value calculated at \(r_{f}=160\times 10^{3}\) pN/s and that calculated at \(r_{f}=4\times 10^{3}\) pN/s is small (\(\simeq\) 2 Å). Although the underlying free-energy landscape is time-dependent in a non-equilibrium force-ramp pulling experiment, these results suggest that the distance from the native state to the transition state is small. It is likely that at loading rates that are achieved in laser optical tweezer experiments (\(\sim\) 10 pN/s), the location to the transition state would increase because the response of biopolymers to loading rate changes from being plastic (low \(r_{f}\)) to brittle (high \(r_{f}\)) [33].
+
+_Unfolding Pathways_: In the dominant pathway, unfolding proceeded in a fairly Markovian fashion with the primary order of events following the sequence \(\beta\)1/\(\beta\)5 \(\rightarrow\)\(\beta\)3/\(\beta\)5 \(\rightarrow\)\(\beta\)3/\(\beta\)4 \(\rightarrow\)\(\beta\)1/\(\beta\)2 (Figs. 1**B** and 3). This is precisely the same sequence seen in the simulations by Li et al.[23]. Alternative pathways, reminiscent of kinetic partitioning [34] observed in forced-unfolding of GFP [35] and lysozyme [36], were also infrequently sampled. For example, at \(r_{f}=4\times 10^{3}\) pN/s, \(\sim 6\%\) of the trajectories unfolded as follows: \(\beta\)1/\(\beta\)5 \(\rightarrow\)\(\beta\)1/\(\beta\)2 \(\rightarrow\)\(\beta\)3/\(\beta\)5 \(\rightarrow\)\(\beta\)3/\(\beta\)4 (Fig. 1**B**), while the remaining \(\sim 94\%\) followed the dominant pathway. At the highest loading rate, one frequently observed the following sequence of events \(\beta\)1/\(\beta\)5 \(\rightarrow\)\(\beta\)3/\(\beta\)5 \(\rightarrow\)\(\beta\)1/\(\beta\)2 \(\rightarrow\)\(\beta\)1/\(\beta\)2 (reform) \(\rightarrow\)\(\beta\)3/\(\beta\)4 \(\rightarrow\)\(\beta\)1/\(\beta\)2, where \(\beta\)1/\(\beta\)2 ruptured but then reformed prior to the rupture of \(\beta\)3/\(\beta\)4. As illustrated in Fig. 2(**A** and **B**), unfolding events at smaller \(r_{f}\) tended to result in larger molecular extensions.
+
+The non-equilibrium character of a pulling experiment decreased with decreasing \(r_{f}\), and smaller \(r_{f}\) resulted in smaller force-drops after the unfolding force, \(f_{u}\), is reached. Since the force applied by the spring to the end of the protein did not fall off as sharply at lower \(r_{f}\), more of the protein was extended during an unfolding event. The smaller extensions following unfolding events at higher \(r_{f}\) (relative to those observed at lower \(r_{f}\)) were responsible for the \(\beta\)1/\(\beta\)2 unfolding/refolding events mentioned above because the applied tension was very low after the initial rupture event (Fig. 2(**A** and **B**)). At lower \(r_{f}\), this situation no longer held because the initial unfolding event resulted in a chain extension that was a significant fraction of the chain’s contour length (Fig. 2**B**).
+
+### Crowding Effects on Ub (\(\Phi_{c}=0.3\)).
+
+Depletion forces stabilize proteins and shift the folding equilibrium towards more compact states [37]. These forces result from an increase in the entropy of the crowding agents that more than compensates for an increase in the free-energy of a protein molecule upon compaction. Simulations of forced-unfolding of Ub in the presence of explicit crowders of diameters \(\sigma_{c}=6.4\) nm and \(\sigma_{c}=1.0\) nm were used to assess the contribution of the depletion forces to mechanical stability. Sixteen trajectories were collected for each \(r_{f}\) investigated. Three \(r_{f}\) (\(20\times 10^{3}\) pN/s, \(80\times 10^{3}\) pN/s, \(160\times 10^{3}\) pN/s), were explored for the \(\sigma_{c}=6.4\) nm sized depletants. Only the two highest \(r_{f}\) (\(80\times 10^{3}\) pN/s, \(160\times 10^{3}\) pN/s), were explored for the \(\sigma_{c}=1.0\) nm sized crowders.
+
+_Small crowding particles increase unfolding forces_: Example FEC’s collected in the presence of crowders of diameter \(\sigma_{c}=6.4\) nm and \(\sigma_{c}=1.0\) nm are illustrated in Fig. 4(**A** and **B**). The \(\sigma_{c}=6.4\) nm curves in Fig. 4 look qualitatively very similar to those seen at \(\Phi_{c}=0.0\), while the FEC’s collected at \(\sigma_{c}=1.0\) nm look qualitatively different. For example, larger \(\langle f_{u}\rangle\) than those observed at either \(\Phi_{c}=0.0\) or in the presence of the \(\sigma_{c}=6.4\) nm crowders are apparent in the FEC’s collected at \(\sigma_{c}=1.0\) nm (Fig. 4(**A** and **B**)). Indeed, Fig. 4**C** reveals that this observation is quantitatively accurate. Although the \(\langle f_{u}\rangle\) in the presence of the \(\sigma_{c}=6.4\) nm crowders were statistically indistinguishable from the \(\langle f_{u}\rangle\) at \(\Phi_{c}=0\) (compare Fig. 4**C** with Fig. 2**C**), the average unfolding forces in the presence of the \(\sigma_{c}=1.0\) nm crowders were statistically greater than those measured at \(\Phi_{c}=0.0\). At \(r_{f}=160\times 10^{3}\) pN/s, the \(\langle f_{u}(\Phi_{c}=0.3)\rangle\) in the presence of the \(\sigma_{c}=1.0\) nm crowders exceeded that at \(\Phi_{c}=0.0\) by 3\(\%\), while at \(r_{f}=80\times 10^{3}\) pN/s the increase was 4\(\%\) (compare Fig. 4**C** with Fig. 2**C**). In one respect these results are not surprising; it follows from the AO theory and Eq. (8) that smaller crowders stabilize Ub more than larger ones. On the other hand, the extent of stabilization (as measured by increases in \(\langle f_{u}\rangle\)) was small. We should emphasize that although the increase in the unfolding force is small, the stability change upon crowding is significant. The enhancement in stability is \(\Delta G\sim\langle f_{u}(\Phi_{c})\rangle\langle z_{DSE}\rangle\sim 5\)\(k_{B}T\) using a 3% increase in the unfolding force, and \(\langle z_{DSE}\rangle\) the location of the unfolded basing \(\approx 5\) nm.
+
+_Crowding leads to transient local refolding_: The unfolding pathways in the presence of crowding agents of both sizes were very similar to those seen at \(\Phi_{c}=0.0\). Nevertheless, at \(\Phi_{c}=0.3\) there tended to be more unfolding/refolding events as the molecule extended past \(z_{u}\) (Fig. 3) than at \(\Phi_{c}=0.0\). As illustrated, strand-pairing between \(\beta\)1 and \(\beta\)2 and between \(\beta\)3 and \(\beta\)4 persisted to a greater extent after the initial unfolding event at \(\Phi_{c}=0.3\) than at \(\Phi_{c}=0.0\). As Ub passed through the point (\(z_{u}\),\(f_{u}\)), its termini were extended to distances greater than the diameter of even the larger crowders. Depletion forces resulting from the presence of 6.4 nm crowders act on these larger length scales. Assuming that the tension applied to the C-terminus is small enough (e.g., after rupture events at higher loading rates), then such depletion forces can promote reformation of contacts between secondary structural elements several times during the course of a trajectory. If this is indeed the case, then it suggests that unfolding polyUb may be different from the unfolding of monomeric Ub, because depletion effects should increase with number of modules in the tandem (see below).
+
+_AO Model For Forced-Unfolding in the Presence of Crowders_: We used the Asakura-Oosawa AO model [5, 6, 7] (Eq. (4)) of the depletion interaction to model the effects of a crowded environment on Ub. The AO theory has been successfully used to model the effects of a crowded environment on polymers and colloids [6, 7, 38, 39], and we found that it gives qualitatively accurate results for Ub. We used Eq. (4) to approximate the effective interaction between spherical protein beads immersed in a crowded solution of volume fraction \(\Phi_{c}=0.3\). From the form of the AO potential between protein beads, it follows that the range of the potential is proportional to \(\sigma_{c}\), but the strength is greater for smaller crowders. Indeed, as revealed in the previous section, simulations in the presence of explicit crowders showed that 1.0 nm crowders resulted in larger average unfolding forces than 6.0 nm crowders (Fig. 4**C**).
+
+Although the AO-potential yields qualitatively accurate results, use of the AO-potential of Eq. (4) to implicitly model non-bonded interactions in Ub did not yield quantitatively accurate results. At \(r_{f}=160\times 10^{3}\) pN/s \(\langle f_{u}(\Phi_{c}=0.0)\rangle=175.78\pm 1.52\) pN, while \(\langle f_{u}(\Phi_{c}=0.3)\rangle=266.26\pm 2.70\) pN. Thus, simulations with the AO potential led to a mean unfolding force that is roughly 50\(\%\) greater than in its absence. This disagrees sharply with the \(\langle f_{u}(\Phi_{c}=0.3)\rangle=173.64\pm 3.49\) pN resulting from our own simulations in the presence of explicit crowding agent at \(\Phi_{c}=0.3\) at \(r_{f}=160\times 10^{3}\) pN/s ( Fig. 4**C** ). Indeed, the result also stands in marked contrast to the experimental results of Ping et al. [12] on octameric Ub, which saw a maximum increase in \(\langle f_{u}\rangle\) of 21\(\%\) (at \(\Phi_{c}>0.3\), \(r_{f}=4200\) pN/s, and with \(\sigma_{c}\simeq 7.0\) nm) over the \(\langle f_{u}\rangle=166\) pN at \(\Phi_{c}=0\)[12].
+
+There are a couple of origins to the discrepancy. First, the AO-potential was derived to understand the equilibrium of colloidal spheres and plates in the presence of smaller-sized spherical crowding agents. Our experiments were of a non-equilibrium nature, so it is somewhat unreasonable to expect such simulations to yield quantitatively accurate unfolding forces or dynamics. Second, as pointed out by Shaw and Thirumalai [37], three-body terms are required to properly model depletion effects even in good solvents let alone in the poor-solvent conditions of our simulations. To elaborate, let us consider the volume excluded to crowders by a Ub molecule, V_ex_(Ub), to be the volume enclosed by a union of spheres of radii \(S_{i}\) = \(\frac{\sigma+\sigma_{c}}{2}\). With an AO potential, the volume excluded to the spherical crowding agents is:
+
+\[V_{E}\text{(Ub)}\simeq\sum_{i=1}^{N}V(S_{i})-\sum_{j>i}V(S_{i}\cap S_{j}),\] (7)
+
+where \(N\) is the number of residues in monomeric Ub (76), \(V(S_{i})\) is the volume of \(S_{i}\) and \(V(S_{i}\cap S_{j})\) is the volume associated with the overlap of \(S_{i}\) and \(S_{j}\). Eq. (7) neglects the overlap of three or more spheres. The importance of such overlaps, for soft-spheres, increases as the crowders become much larger than the protein beads, and the neglect of such overlaps is the reason for the quantitative inaccuracy. As the size of the crowders increases (i.e., as the thickness of the depletion layer surrounding the protein beads increases), the surface enclosing Ub becomes more spherical, loses detail, and undoubtedly changes less in response to changes in the conformation of the molecule. Since depletion forces are proportional to the change in \(V_{E}\)(Ub) with respect to changes in Ub conformation, Eq. (7) overestimates the size of depletion forces in the presence of large crowders. Despite these limitations the AO model, which is simple, can be used to provide qualitative predictions. Finally, we note that experiments typically use polyproteins to study force induced unfolding (e.g., Ping et al. [12] used (Ub)₈ in their experiments). For polyproteins the quantitative accuracy of the AO theory for unfolding in the presence of crowders of diameter \(\sigma_{c}=6.4\) nm is likely to increase, because the volume excluded to the crowders will change significantly with changes in the conformation of the polyprotein.
+
+### \((\text{Ub})_{3}\) at \(\Phi_{c}=0.0\) and \(\Phi_{c}=0.3\).
+
+From arguments based on volume exclusion that lead to crowding-induced entropic stabilization of the folded structures, it follows that crowding effects should be more dramatic on poly Ub than the monomer. In order to illustrate the effect of crowding on stretching of (Ub)₃ we chose \(\sigma_{c}=6.4\) nm, which had negligible effect on \(\langle f_{u}(\Phi_{c})\rangle\) for the monomer. However, we found significant influence of the large crowding particles when (Ub)₃ was forced to unfold in their presence. The FEC (Fig. 5) shows three peaks that corresponded to unfolding of the three domains, when simulations were performed at crowder volume fractions \(\Phi_{c}=0.0\) and \(\Phi_{c}=0.3\) at \(r_{f}=640\times 10^{4}\) pN/s. Fig. 6 presents average unfolding forces as a function of the unfolding event number. Although the first two unfolding events were statistically indistinguishable at \(\Phi_{c}=0.0\) and \(\Phi_{c}=0.3\), the final event occurred at much larger \(\langle f_{u}\rangle\) in the presence of crowders than in their absence.
+
+_Order of Unfolding was Stochastic:_\((\text{Ub})_{3}\) has 3 chemically identical modules. In the pulling simulations the N-terminus of module \(A\) was held fixed, while force was applied to the C-terminus of module \(C\). Figure 7 illustrates the time-dependence of contacts between secondary structure elements of modules \(A\), \(B\), \(C\). It is clear from Fig. 7**A** that module \(C\) unfolded first, followed by module \(B\), and finally by module \(A\). On the other hand, the order of events in Fig. 7**B** was \(A\to C\to B\). The frequency with which the \(3!=6\) possible permutations of these orders at \(\Phi_{c}=0\) and at \(\Phi_{c}=0.3\) were observed is presented in Table 1. It is clear that at both \(\Phi_{c}=0\) and at \(\Phi_{c}=0.3\), the most probably order of events was \(C\to B\to A\). At \(\Phi_{c}=0\) this order was only marginally more probable than the order \(C\to A\to B\), while at \(\Phi_{c}=0.3\)\(C\to B\to A\) became overwhelmingly more probable than any other unfolding order. In none of the simulations at \(\Phi_{c}=0\) or \(\Phi_{c}=0.3\), did the rupture of module \(C\) occur as the final event.
+
+_Unfolding Within a Module Depended on Proximity to the Point of Force Application:_ At \(r_{f}=640\times 10^{4}pN/s\), \((\text{Ub})_{3}\) is fairly brittle, and the rupture of contacts within a module occured nearly simultaneously. Nevertheless, by carefully examining time-dependent contact maps such as those illustrated in Fig. 7, we were able to determine (1) at both \(\Phi_{c}=0\) and \(\Phi_{c}=0.3\), \(\beta 1/\beta 5\) contacts were the first to rupture, and (2) only for module \(C\) was this rupture event _invariably_ followed by the loss of \(\beta 3/\beta 5\) contacts. When other modules ruptured, loss of \(\beta 1/\beta 5\) contacts was occasionally followed by loss of the \(\beta 1/\beta 2\) strand-pair contacts.
+
+\((\text{Ub})_{3}\)_Must Achieve a Larger_\(R_{g}\)_to Rupture at_\(\Phi_{c}=0.3\): Figure 8**A** illustrates the time dependence of \(\langle R_{g}\rangle\) at \(\Phi_{c}=0\) and at \(\Phi_{c}=0.3\). Interestingly, the plot reveals that after the second rupture event, the \(\langle R_{g}(\Phi_{c},t)\rangle\) increased more rapidly in the presence of crowders than in their absence. This is likely a reflection of the fact that at \(\Phi_{c}=0\) modules \(A\) and \(C\) were the first two modules to unfold in 44% of the trajectories, while at \(\Phi_{c}=0.3\) these two modules were the first to unfold in only 19% of trajectories. Thus, \(\langle R_{g}(\Phi_{c},t)\rangle\) increased more rapidly in the presence of crowders, because the \(R_{g}\) of \((\text{Ub})_{3}\) with two adjacent modules unfolded is larger than that with two unfolded but non-adjacent modules. Interestingly, these differences are masked in the time-dependent increase of the end-to-end distance (Fig. 8**B**). Figure 8**A** also reveals that the horizontal inflection points marking the third unfolding event occur at different times at \(\Phi_{c}=0.3\) and in the absence of crowders. The difference between these two times was responsible for the difference in average unfolding forces of \(\approx 14\) pN illustrated in Fig. 6. Thus, it is clear that despite the highly non-equilibrium nature of this pulling experiment, depletion effects were substantial and it required a much greater force to reach \(f_{u}\) in the presence of these crowders than in their absence.
+
+We predict that systematic experiments will reveal that polyUb molecules composed of larger numbers of modules will show greater increases in \(\langle f_{u}(\Phi_{c})\rangle\) relative to \(\langle f_{u}(\Phi_{c}=0)\rangle\) than polyUb molecules composed of fewer repeats. The size of these differences should increase with decreasing loading rate. Finally, it may even be possible to observe differences in the \(\langle f_{u}\rangle\) as a function of unfolding event number (as in Fig. 6). The increase in \(\langle f_{u}(\Phi_{c})\rangle\), at a fixed \(\Phi_{c}\), for poly Ub is likely to be even more significant for small crowding agents as shown in Eq. (8) (see below for further discussion).
+
+## Conclusions.
+
+General theory, based on the concept of depletion effects (see Eqs. (4) and (8)), shows that crowding should enhance the stability of proteins, and hence should result in higher forces to unfold proteins. However, predicting the precise values of \(\langle f_{u}(\Phi_{c})\rangle\) is difficult because of the interplay of a number of factors such as the size of the crowding agents and the number of amino acid residues in the protein. Despite the complexity a few qualitative conclusions can be obtained based on the observation that, when only excluded volume interactions are relevant, then the protein or polyprotein would prefer to be localized in a region devoid of crowding particles [40]. The size of such a region \(D\approx\sigma_{c}({\frac{\pi}{6\Phi_{c}}})^{\frac{1}{3}}\). If \(D\gg R_{g}\) then the crowding would have negligible effect on the unfolding forces. The condition \(D\gg R_{g}\) can be realized by using large crowding particles at a fixed \(\Phi_{c}\). In the unfolded state, \(R_{g}\approx 0.2N^{0.6}\) nm [41] which for Ub leads to \(R_{g}\approx 2.7\) nm. Thus, \(\frac{D}{R_{g}}=0.4\sigma_{c}\). These considerations suggest that the crowder with \(\sigma_{c}=6.4\) nm would have negligible effect on the unfolding force, which is in accord with the simulations. On the other hand, \(\frac{D}{R_{g}}\approx 0.4\) when \(\sigma_{c}=1\) nm, and hence we expect that the smaller crowders would have measurable effect on the unfolding forces. Our simulations are in harmony with this prediction. We expect that for the smaller crowding agent \(\langle f_{u}(\Phi_{c})\rangle\) would scale with \(\Phi_{c}\) in a manner given by Eq. (8). In general, appreciable effect of crowding on the unfolding forces can be observed only for large proteins or for polyproteins using relatively small crowding agents.
+
+Although we have only carried out simulations for Ub and (Ub)₃ at one non-zero \(\Phi_{c}\), theoretical arguments can be used to predict the changes in \(\langle f_{u}(\Phi_{c})\rangle\) as \(\Phi_{c}\) increases. The expected changes in the force required to unfold a protein can be obtained using a generalization of the arguments of Cheung et al.[3] In the presence of crowding agents the protein is localized in a region that is largely devoid of the crowding particles [40]. The most probable size of the region is \(D\sim\sigma_{c}\Phi_{c}^{-1/3}\) where \(\sigma_{c}\) is the size of the crowding agent. If the structures in the DSE are treated as a polymer with no residual structure then the increase in entropy of the DSE upon confinement is \(\Delta S/k_{B}\sim(R_{g}/D)^{1/\nu}\) where \(R_{g}\) is the dimension of the unfolded state of the protein. If native state stabilization is solely due to the entropic stabilization mechanism we expect:
+
+\[\langle f_{u}(\Phi_{c})\rangle\sim T\Delta S/L_{c}\sim(\frac{R_{g}}{\sigma_{c}})^{2}\Phi_{c}^{1/3\nu}(\frac{k_{B}T}{L_{c}})\] (8)
+
+where \(f_{u}(\Phi_{c})\) is the critical force for unfolding the protein, \(L_{c}\) is the gain in contour length at the unfolding transition, and \(\nu(\approx 0.588)\) relates \(R_{g}\) to the number of amino acids through the relation \(R_{g}\sim a_{D}N^{\nu}\) (\(a_{D}\) varies between 2-4 Å). A few comments regarding Eq. (8) are in order. (1) The \(\Phi_{c}\) dependence in Eq. (8) does not depend on the nature of the most probable region that is free of crowding particles. As long as the confining region, which approximately mimics the excluded volume effects of the macromolecule, is characterized by a single length D, we expect Eq. (8) to be valid. (2) The additional assumption used in Eq. (8) is that \(N\gg 1\), and hence there may be deviations due to finite size effects. (3) The equivalence between crowding and confinement breaks down at large \(\Phi_{c}\) values. Consequently, we do not expect Eq. (8) to fit the experimental data at all values of \(\Phi_{c}\). (4) It follows from Eq. (8) that, for a given \(R_{g}\), small crowding agents are more effective in stabilizing proteins than large ones. Thus, the prediction based on Eq. 8 is supported by our simulations. (5) From the variation of \(\langle f_{u}(\Phi_{c})\rangle\) with \(\Phi_{c}\) Ping et al. [12] suggest that \(\langle f_{u}\rangle\sim\Phi_{c}\). However, the large errors in the measurements cannot rule out the theoretical prediction in Eq. (8). We have successfully fit their experimental results using Eq. (8) (Fig. 8**C**). Additional quantitative experiments are required to validate the theoretical prediction.
+
+It is difficult to map the concentrations in g/L used in the study of Ping et al. [12] to an effective volume fraction because of uncertainties in the molecular weight of Dextran used in the study. Hence, a quantitative comparison between theory and experiments is challenging. A naive estimate may be obtained using the values reported by Weiss et al. [42]. Given that the Dextran used in the study is thought to have an average molecular weight of 40 kDa and an estimated average hydrodynamic radius of 3.5 nm [42], we find that \(\rho=300\) g/L corresponds to a volume fraction \(\Phi_{c}=0.8\), which is very large. Nevertheless, \(\Phi_{c}\) must be large when \(\rho=300\) g/L. Alternatively, we estimated \(\sigma_{c}/2\) for Dextran using \(\sigma_{c}/2\approx a_{D}N^{1/3}\) where \(N\) is the number of monomers in a 40 kDa Dextran is \(40/0.162\approx 147\). If the monomer size \(a_{D}\approx 0.4-0.45\) nm, then we find \(\Phi_{c}\approx 0.3-0.4\). If we assume that \(\rho=300\) g/L corresponds to \(\Phi_{c}\simeq 0.4\) and that \(\langle f_{u}(\Phi_{c})\rangle\sim\Phi_{c}^{5/9}\) (Eq. (8)), then at \(\Phi_{c}=0.3\) we would expect a nearly 18\(\%\) increase in \(\langle f_{u}\rangle\). Similarly, if \(\rho=300\) g/L corresponds to \(\Phi_{c}\simeq 0.3\) then we would expect an increase in \(\langle f_{u}\rangle\) of approximately 21\(\%\). In any case, we can say that at physiologically relevant volume fractions (\(\Phi_{c}\)\(\in\) [0.1,0.3]), the percent increase in \(\langle f_{u}\rangle\) is likely to be \(\leq 20\%\). Our simulations for \(\sigma_{c}=1.0\) nm predict an increase of 3-4%, which shows that a more detailed analysis is required to obtain an accurate value of \(\sigma_{c}\) for Dextran before a quantitative comparison with experiments can be made. The larger increase seen in experiments may also be a reflection of the use of (Ub)₈ rather than a monomer.
+
+Regardless of the crowder size we find that the unfolding pathways are altered in the presence of crowding agents. It is normally assumed that the rupture of secondary structure elements is irreversible if the applied force exceeds a threshold value. However, when unfolding experiments are carried out in the presence of crowding particles, that effectively localize the protein in a smaller region than when \(\Phi_{c}=0\), reassociation between already ruptured secondary structures is facilitated as shown here. Thus, forced-unfolding cannot be described using one dimensional free energy profiles with \(z_{u}\) as the reaction coordinate [33].
+
+We find that the average unfolding force for the final rupture event of the unfolding of \((\text{Ub})_{3}\) occurred at much larger values in the presence of crowders than in their absence. With \(\sigma_{c}=6.4\) nm, which has practically no effect on the unfolding force of the monomer, and \(\Phi_{c}=0.3\) even with unfolding of two modules the interactions between the stretched modules and protein are small (\(D\approx 1.2\sigma_{c}\)). Only upon unfolding of the third Ub do crowding effects become relevant, which leads to an increase in \(\langle f_{u}(\Phi_{c})\rangle\). Our results suggest that \(\langle f_{u}(\Phi_{c})\rangle/\langle f_{u}(0)\rangle\) should increase with the number of modules in the array and that it may be possible to detect differences in \(\langle f_{u}\rangle\) which are conditional on the unfolding event number. We speculate that naturally occurring polyproteins that are subject to mechanical stress have evolved to take advantage of precisely such enhanced depletion effects.
+
+## Acknowledgments
+
+This work was supported by a grant from the National Science Foundation (CHE 05-14056). DLP is grateful for a Ruth L. Kirschstein Postdoctoral Fellowship from the National Institute of General Medical Sciences (F32GM077940). Computational time and resources for this work were kindly provided by the National Energy Research Scientific Computing (NERSC) Center.
+
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+* 26. Pincus, D. L.; Cho, S. S.; Hyeon, C.; Thirumalai, D. Minimal models for proteins and RNA: From folding to function. In _Progress in Nucleic Acid Research and Molecular Biology_; Conn, P. M., Ed.; (in press).
+* 27. Allen, M. P.; Tildesley, D. J. _Computer Simulation of Liquids;_ Oxford University Press: 1987.
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+* 30. Carrion-Vazquez, M.; Li, H.; Lu, H.; Marszalek, P.; Oberhauser, A.; Fernandez, J. _Nature Structural Biology_**2003,**10, 738-743.
+* 31. Bockelmann, U.; Essevaz-Roulet, B.; Heslot, F. _Phys. Rev. E_**1998,**58, 2386-2394.
+* 32. Carrion-Vazquez, M.; Oberhauser, A.; Fowler, S.; Marszalek, P.; Broedel, S.; Clarke, J.; Fernandez, J. _Proc. Natl. Acad. Sci. USA_**1999,**96, 3694-3699.
+* 33. Hyeon, C.; Thirumalai, D. _J. Phys. Cond. Matt._**2007,**19, 113101.
+* 34. Guo, Z.; Thirumalai, D. _Biopolymers_**1995,**36, 83-102.
+* 35. Mickler, M.; Dima, R.; Dietz, H.; Hyeon, C.; Thirumalai, D.; Rief, M. _Proc. Natl. Acad. Sci. USA_**2007,**104, 20268-20273.
+* 36. Peng, Q.; Li, H. _Proc. Natl. Acad. Sci. USA_**2008,**105, 1885-1890.
+* 37. Shaw, M. R.; Thirumalai, D. _Phys. Rev. A_**1991,**44, R4797-R4800.
+* 38. Verma, R.; Crocker, J.; Lubensky, T.; Yodh, A. _Phys. Rev. Lett._**1998,**81, 4004-4007.
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+* 41. Kohn, J. E.; Millet, I. S.; Jacob, J.; Zagrovic, B.; Dillon, T. M.; Cingel, N.; Dothager, R. S.; Seifert, S.; Thiyagarajan, P.; Sosnick, T. R.; Hasan, M. Z.; Pande, V. S.; Ruczinski, I.; Doniach, S.; Plaxco, K. W. _Proc. Natl. Acad. Sci. USA_**2005,**101, 14475-14475.
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+
+## Tables
+
+\begin{table}
+\begin{tabular}{|c|c|c|}
+\hline
+Unfolding Order & Frequency Observed at \(\Phi_{c}=0\) & Frequency Observed at \(\Phi_{c}=0.3\) \\
+\hline \hline
+\(C\to B\to A\) & 0.44 & 0.56 \\
+\hline
+\(C\to A\to B\) & 0.38 & 0.13 \\
+\hline
+\(B\to A\to C\) & 0.00 & 0.00 \\
+\hline
+\(B\to C\to A\) & 0.13 & 0.25 \\
+\hline
+\(A\to B\to C\) & 0.00 & 0.00 \\
+\hline
+\(A\to C\to B\) & 0.06 & 0.06 \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Module Unfolding Order Frequencies at \(\Phi_{c}=0\) and \(\Phi_{c}=0.3\).
+
+## Figure Captions.
+
+Fig. 1: (**A**) Cartoon representation of the native structure of ubiquitin (PDB accession id 1UBQ) in the presence of spherical crowding agents. The five beta-strands, labeled \(\beta\)1 through \(\beta\)5, are colored in yellow. The two alpha-helices (\(\alpha\)1 and \(\alpha\)2) are shown in purple. The N- and C-terminal beads are represented as spheres. In our simulations the N-terminal bead was held fixed while the C-terminal bead was pulled via a tethered spring. (**B**) Snapshots from an unfolding trajectory illustrating the main ubiquitin (Ub) unfolding pathway (brown-dashed arrows) and an alternate unfolding pathway (green-dotted arrow). In both pathways the initial unfolding event corresponds to separation of the C-terminal strand \(\beta\)5 from the N-terminal strand \(\beta\)1. Along the main pathway, this is quickly followed by separation of \(\beta\)5 from \(\beta\)3. The penultimate rupture event along the main pathway corresponds to disruption of the \(\beta\)3/\(\beta\)4 strand-pair, while the N-terminal \(\beta\)1/\(\beta\)2 strand pair is the last to break. The trajectory illustrated here was generated at \(\Phi_{c}=0.0\) and \(r_{f}=160\times 10^{3}\) pN/s. An alternate pathway was observed at \(\Phi_{c}=0.0\) and \(r_{f}=4\times 10^{3}\) pN/s. Along this pathway separation of \(\beta\)5 from \(\beta\)1 is followed by separation of \(\beta\)1/\(\beta\)2. The final two rupture events correspond to those of the \(\beta\)5/\(\beta\)3 contacts and \(\beta\)3/\(\beta\)4 strand-pair respectively. (Figures generated with VMD [43])
+
+Fig. 2: Force-extension curves (FEC’s) at two different loading rates, (**A**) \(r_{f}=160\times 10^{3}\) pN/s, and (**B**) \(r_{f}=4\times 10^{3}\) pN/s. Data from the simulation is presented as a red trace. For each trajectory a black arrow points to the unfolding force, \(f_{u}\). \(z_{u}\) corresponds to the extension of the molecule along the pulling (i.e., z-) direction evaluated at \(f_{u}\). \(\Delta\mathcal{L}\) (dotted blue line) is the contour length increment, and is a measure of the amount of chain released in an unfolding event. We measured \(\Delta\mathcal{L}\) as \(\mathcal{L}-z_{u}\), where \(\mathcal{L}=(N-1)\times\sigma=75\times 0.38\) nm is a nominal contour length of 28.5 nm. Stars in each subfigure mark the minimum force observed after an unfolding event, and chain conformations corresponding to the starred points in figures **A** and **B** are illustrated at the center of the figure. Unfolding events at smaller \(r_{f}\) resulted in larger molecular extensions before significant resistance was encountered. (Yellow arrows correspond to beta-strands and purple cylinders correspond to \(\alpha\)-helices). (Figures generated with VMD [43]). (**C**) \(\langle f_{u}\rangle\) vs. \(r_{f}\) evaluated at \(\Phi_{c}=0\). The red curve corresponds to a linear-least squares fit to the set of basis functions \(\{1,\ln(r_{f})\}\) and demonstrates that \(\langle f_{u}\rangle\sim\ln(r_{f})\). (Note that the abscissa is a log-scale). Each point is labeled Mean \(\pm\) standard error. Statistics at \(r_{f}=160\times 10^{3}\) pN/s, \(80\times 10^{3}\) pN/s, \(20\times 10^{3}\) pN/s, and \(4\times 10^{3}\) pN/s were calculated from 50, 49, 50, and 16 trajectories respectively.
+
+Fig. 3: Rupture events at \(\Phi_{c}=0.0\) (**A**) and at \(\Phi_{c}=0.3\) and \(\sigma_{c}=6.4\) nm (**B**). The figure illustrates that after the initial rupture event at step 0, subsequent unfolding was affected by the crowding agent. \(\beta\)1/\(\beta\)2 contacts and \(\beta\)3/\(\beta\)4 persisted longer and there were many more local refolding events in the presence of the crowders (**B**) than in their absence (**A**). (**C**) Snapshots from an unfolding trajectory at \(\Phi_{c}=0.3\) and \(r_{f}=4\times 10^{3}\) pN/s illustrate the primary difference between unfolding at \(\Phi_{c}=0.3\) and at \(\Phi_{c}=0.0\). The brown-dotted arrow shows unfolding without any local-refolding events, while the sequence of black arrows illustrates local-rupture and -refolding events. The hallmark of forced-unfolding in a crowded environment is repeated breaking and reforming of contacts after an initial rupture event. (Figures generated with VMD [43])
+
+Fig. 4: (**A**) Examples of force-extension traces resulting from simulation in the presence of spherical crowding agents of diameter \(\sigma_{c}=6.4\) nm and obtained at \(r_{f}=80\times 10^{3}\) pN/s. (**B**) Examples of force-extension traces resulting from simulation in the presence of spherical crowding agents of diameter \(\sigma_{c}=1.0\) nm and obtained at \(r_{f}=80\times 10^{3}\) pN/s. Both subfigures are labeled as in Fig. 2(**A** and **B**). (**C**) \(\langle f_{u}\rangle\) vs. \(r_{f}\) evaluated at \(\Phi_{c}=0.3\). Black triangles and red circles correspond to spherical crowders of diameter \(\sigma_{c}=1.0\) nm and \(\sigma_{c}=6.4\) nm respectively. Each point is labeled Mean \(\pm\) standard error. Statistics for each point were calculated from 16 independent trajectories. Only the \(\sigma_{c}=1.0\) nm had an appreciable effect on \(\langle f_{u}\rangle\) when compared to those obtained at identical \(r_{f}\) and at \(\Phi_{c}=0.0\) (see Fig. 2**C**)
+
+Fig. 5: FEC’s for \((\text{Ub})_{3}\) forced unfolding at \(\Phi_{c}=0.0\) (**A**) and at \(\Phi_{c}=0.3\) (**B**). Trajectories were generated at \(r_{f}=640\times 10^{4}\) pN/s and with crowders of diameter \(\sigma_{c}=6.4\) nm. Black arrows mark each trajectory’s three unfolding events.
+
+Fig. 6: \(\langle f_{u}\rangle\) vs. unfolding event number for the unfolding of \((\text{Ub})_{3}\) in the presence of spherical crowders of diameter \(\sigma_{c}=6.4\) nm (\(\Phi_{c}=0.3\), black triangles) and in their absence (\(\Phi_{c}=0.0\), red circles). The individual modules of the polyUb tandem were N-C-linked and the loading rate was \(640\times 10^{4}\) pN/s. Each point is labeled Mean \(\pm\) standard error. Statistics for each point were calculated from 16 independent unfolding trajectories. An unfolding event corresponded to the unfolding of an individual module. Note that although the crowders had little effect on \(\langle f_{u}\rangle\) for the first and second unfolding events, they had a substantial effect on the last unfolding event. \(\langle f_{u}\rangle\) increased by \(\approx 14\) pN for the last unfolding event.
+
+Fig. 7: Illustration of the stochastic nature of module unfolding. In (**A**) module \(C\) (the most proximal to the applied force) unfolds first, followed by module \(B\), and finally by rupture of module \(A\). In (**B**) the order of module unfolding events is \(A\to C\to B\). The two trajectories illustrated here were both collected at \(\Phi_{c}=0\) and \(r_{f}=640\times 10^{4}\) pN/s. Table 1 provides the frequencies at which the different possible orders were observed at \(\Phi_{c}=0\) and at \(\Phi_{c}=0.3\). The diameter of the crowders is \(\sigma_{c}=6.4\) nm.
+
+Fig. 8: (**A**) \(\langle R_{g}(t)\rangle\) versus time at \(\Phi_{c}=0\) (red) and at \(\Phi_{c}=0.3\) (black). Although the \(\langle R_{g}\rangle\) increases more rapidly with time after the second rupture event at \(\Phi_{c}=0.3\) than at \(\Phi_{c}=0\), the inset reveals that a larger \(R_{g}\) must be achieved to initiate the final rupture event in the presence of crowding particles with \(\sigma_{c}=6.4\) nm. (See text for additional discussion). (**B**) \(\langle z(t)\rangle\) versus time at \(\Phi_{c}=0\) (red) and at \(\Phi_{c}=0.3\) (black). \(z(t)\) cannot discriminate between unfolding at the different volume fractions and hence is less suitable (than \(R_{g}\)) as a potential reaction coordinate. (**C**) The experimental results of Ping et al. [12] (red circles) for unfolding-force, \(\langle f_{u}\rangle\) as a function of Dextran concentration, \(\rho\). The black line is a fit assuming \(\langle f_{u}\rangle\sim\rho\) and the blue assuming \(\langle f_{u}\rangle\sim\rho^{5/9}\). Although both fits are consistent with the data, based on theoretical considerations (see Eq. (8)) we prefer the blue fit (see text). (Standard deviations taken from Ping et al. [12])
+
+Figure 1:
+
+Figure 2:
+
+Figure 3:
+
+Figure 4:
+
+Figure 5:
+
+Figure 6:
+
+Figure 7:
+
+Figure 8:
+

arxiv_ground_truth/0811.0851.md

@@ -0,0 +1,220 @@
+Solitaire: Recent Developments
+**John D. Beasley**
+September 2003¹
+
+Footnote 1: Original version at http://gpj.connectfree.co.uk/gpjj.htm
+ Converted to LaTeX by George Bell with minor modifications to the text, November 2008.
+
+johnbeasley@mail.com
+
+**Abstract**
+
+This special issue on Peg Solitaire has been put together by John Beasley as guest editor, and reports work by John Harris, Alain Maye, Jean-Charles Meyrignac, George Bell, and others. Topics include: short solutions on the \(6\times 6\) board and the 37-hole “French” board, solving generalized cross boards and long-arm boards. Five new problems are given for readers to solve, with solutions provided.
+
+## 1 Introduction and historical update
+
+There has recently been a flurry of activity on the game of Peg Solitaire, and I have suggested to George Jelliss that The Games and Puzzles Journal[6] might be a convenient place for people to report new discoveries. His reaction was that he would like to introduce the game to readers by dedicating a special number to it, after which he will consider contributions as they arise, and he has asked me to provide the material for this special edition. It updates the material given in my book _The Ins and Outs of Peg Solitaire_[1] and what was given there will not normally be repeated here, but enough background will be given to put any reader not previously familiar with the game’s development in the picture. _The Ins and Outs_ is now out of print and will probably remain so, but it can be found in most academic and many UK public service libraries, and there appears to be a steady trickle of copies on the secondhand market². The 1992 edition differs from the 1985 only in the addition of a page summarizing intervening developments and discoveries, and I can supply photocopies of this on request.
+
+Footnote 2: Try ABEbooks.com.
+
+The game’s historical background is now well known. It originated in France in the late seventeenth century (there are references in French sources back to 1697), and it appears to have been the “Rubik’s Cube” of the court of Louis XIV. I summarized its early history [1, p. 3–7] and little appears to have been discovered since, but one statement in the book now needs modification. I took a very cautious view of a passing reference in a letter written by Horace Walpole in 1746, fearing that it might have referred to a card game, but David Parlett, who has looked into the games of the period much more deeply and extensively than I, tells me that my fears were groundless: “Patience dates from the late eighteenth century, did not reach England until the nineteenth, and was not called Solitaire when it did” [4, p. 157]. So the spread of our Solitaire to England by the middle of the eighteenth century can be taken as established.
+
+There is one matter in which discovery remains conspicuous by its absence. It has frequently been written that the game was invented by a prisoner in the Bastille, but I reported in 1985 that the earliest reference to this appeared to be in an English book of 1801, and nobody has yet drawn my attention to anything earlier. An uncorroborated English source of 1801 is of course quite valueless as evidence for an alleged occurrence in France over a century before, and anyone who repeats this tale without citing a French source earlier than 1801 should regard himself as perpetuating myth rather than history. Sadly, the more picturesque a legend surrounding the origin of a game or puzzle, the greater the likelihood that somebody has invented it along the way.
+
+## 2 The 6 x 6 board: the work of John Harris
+
+I gave solutions to various problems on the \(6\times 6\) board in the 1985 edition of _The Ins and Outs_, and on page 252 of the 1992 edition I added a note that John Harris had found all possible 15-move solutions, one by hand and the rest by computer. I refrained from giving details on the grounds that he might still wish to publish them himself, but to the best of my knowledge he has not done so, and others are beginning to reproduce his results. I therefore think I should summarize what he sent me in 1985–86, if only to establish his priority.
+
+Figure 1: The \(6\times 6\) square board divided into 16 “Merson regions” A–P.
+
+Robin Merson observed back in 1962 that the holes of the \(6\times 6\) board could be divided into 16 regions such that only the first jump of a multi-jump move could open up a new region (Figure 1); any later jump had to be between regions already opened. It follows that it takes at least 15 moves to clear the board if the initial vacancy is in a non-corner square, and 16 moves if it is in a corner (because the first move refills this corner and we are still left with 15 regions to be opened). Harris found a 16-move solution to the problem “vacate a1 and play to finish there” back in 1962, and Harry O. Davis subsequently found 15-move solutions to the problems “vacate c1, finish at f4”; and “vacate c3, finish at f6”. All these are in _The Ins and Outs_. That to “start and finish at a1” ends with an elegant eight-sweep loop.
+
+Subsequently (letter to me dated 26 August 1985) Harris found a 15-move solution to the problem “vacate c3, finish at c6”: c1-c3, a1-c1, d1-b1, f1-d1, a3-a1-c1-e1 (5), a5-a3, c4-a4-a2-c2-c4, d4-b4, c6-c4-a4, e3-c3-a3-a5-c5 (10), f3-f1-d1-d3, f5-f3, d6-d4-d2-f2-f4-d4, f6-d6, a6-c6-e6-e4-c4-c6. “Don’t know how to find these,” he wrote, “just copied a Davis beginning and got lucky.” Harris then attacked the problem by computer, and by August 1986 he had found 15-move solutions to all the problems with non-corner starts apart from “start and finish at c1”. His computer proved this to require 16 moves. The remaining corner-start problems, “vacate a1, finish at a4 or d4” had been solved in 16 moves by Davis, and his solutions had appeared in the instructions to Wade Philpott’s 1974 game SWEEP. Harris’s solution to “start and finish at c3” used 5 single jumps, then 6 double jumps, then 4 jumps from corners: c1-c3, a2-c2, d2-b2, d4-d2, d6-d4 (5), f3-d3-d5, b6-d6-d4, f5-d5-d3, e1-e3-e5, c4-c2-e2, a4-c4-c6 (11), a6-a4-a2-c2, f6-d6-b6-b4-b2-d2-d4, f1-f3-f5-d5-d3, a1-c1-e1-e3-c3. “This has to be my favorite solution,” he wrote.
+
+Readers who revel in the power of modern computers may care to note that all this was done on a TRS-80, which if memory serves me right offered a mere 64Kb of RAM for operating system, program, and data together, backed up by a single 52Kb disc drive. Harris’s results have recently been confirmed by Jean-Charles Meyrignac.
+
+## 3 Solutions on the classical 33-hole and 37-hole boards
+
+The classical 33-hole (3-3-7-7-7-3-3) and 37-hole (3-5-7-7-7-5-3) boards offer no simple test for minimality such as is provided by the need to open up each of Merson’s regions on a \(6\times 6\) board, and there is usually a gap of two or three between the length of the shortest solution actually discovered and the number of moves that can be proved necessary by simple means. This gap can be filled only by an exhaustive analysis by computer.
+
+### The 33-hole board
+
+In the original 1985 edition of _The Ins and Outs_, I listed the shortest solutions found by Ernest Bergholt and Harry O. Davis to the single-vacancy single-survivor problems on the standard 33-hole board, and I reported some last-minute computer calculations by myself which demonstrated them indeed to be the shortest possible. However, this was “proof of non-existence by failure to find despite a search believed exhaustive”, and to achieve it on the machine at my disposal I had to resort to some fairly complicated testing to identify and reject blind alleys. I therefore took the view that the proof should be regarded as provisional pending independent confirmation.
+
+No such confirmation had been reported to me when the 1992 edition went to press, but on 24 October 2002 Jean-Charles Meyrignac reported [7] that he had programmed the calculation independently and had verified that the solutions of Bergholt and Davis were indeed optimal³. My 1984 machine offered only 32Kb of RAM for program and data together, even less than that provided by Harris’s TRS-80, though I did have two 100Kb disc drives. Meyrignac, with a more powerful present-day machine at his disposal, had no need for complicated restriction testing and could perform a complete enumeration, reproducing all known solutions as well as demonstrating that there were none shorter.
+
+Footnote 3: The report on his web site merely said “All solutions”, but he has clarified the matter in an e-mail to me.
+
+### The 37-hole “French” board (see Figure 5a)
+
+Although this was historically the first board to be used, minimal solutions on it appear to have received less attention than those on the 33-hole board, and in _The Ins and Outs_ I could only report some relatively recent findings by Leonard Gordon and Harry O. Davis. Four of Gordon’s solutions were subsequently beaten by Alain Maye (work dating from 1985-86 but only recently brought to my notice), and I would have reported this in the 1992 edition of _The Ins and Outs_ had I been aware of it. Meyrignac has now performed an exhaustive enumeration by computer, which shows that the problem “vacate c1, play to a single survivor” can be solved in 20 moves irrespective of which of the holes b4/e1/e4/e7 is chosen to receive the survivor (Gordon and Maye had got each case down to 21), and proves the remaining solutions of Gordon, Davis, and Maye to be optimal. Table 1 below has been proved by Meyrignac to be definitive.
+
+\begin{table}
+\begin{tabular}{|c|c|c|l|}
+\hline
+Vacate & Finish & Length & Investigator \\
+\hline \hline
+\multirow{4}{*}{c1} & e1 & 20 & \multirow{4}{*}{Meyrignac (by computer)} \\
+ & b4 & 20 & \\
+ & e4 & 20 & \\
+ & e7 & 20 & \\
+\hline
+\multirow{3}{*}{d3} & d2 & 21 & Gordon \\
+ & a5 & 21 & Davis \\
+ & d5 & 21 & Maye \\
+\hline
+\multirow{3}{*}{d6} & d2 & 20 & Gordon \\
+ & a5 & 20 & Gordon \\ \cline{1-1}
+ & d5 & 20 & Maye \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Summary of shortest solutions on the 37-hole French board.
+
+Maye’s solutions:
+Vacate d3, finish at d5: d1-d3, b2-d2, d3-d1, f2-d2, e4-e2 (5), c4-c2, a3-c3, d1-d3-b3, g3-e3, a5-a3-c3 (10), b5-b3-d3-f3, g5-g3-e3, d5-b5, b6-b4, c7-c5 (15), c1-c3, f5-f3-d3-b3-b5-d5-f5, e1-e3, f6-f4, e7-e5 (20), d7-d5-d3-f3-f5-d5.
+Vacate d6, finish at d5: b6-d6, c4-c6, c7-c5, a4-c4-c6, e7-c7-c5 (5), e6-c6-c4, b2-b4, d3-b3, c1-c3-c5, a3-c3 (10), e4-e6, f6-d6, g5-e5, e2-c2-c4-c6-e6-e4-e2, d2-f2 (15), g3-e3, g4-e4-e2-c2, a5-c5, e1-c1-c3, d5-b5-b3-d3-d5.
+The most interesting of Meyrignac’s is “vacate c1, finish at e4”, which ends with an eight-sweep: e1-c1, d3-d1, b3-d3, c5-c3, c7-c5 (5), e4-c4-c6, f2-d2-d4, b2-d2, g4-e4-e2, g3-e3 (10), a5-c5, f6-f4, d5-f5-f3-d3-b3, c1-e1-e3, a3-c3 (15), e7-e5, d7-d5-f5, g5-e5, b6-d6, a4-c4-e4-e2-c2-c4-c6-e6-e4!
+All Maye’s and Meyrignac’s solutions can be found on Meyrignac’s web site [7].
+
+## 4 Generalized cross boards and long-arm boards
+
+### Generalized cross boards
+
+George Bell has been studying a class of boards he calls “generalized cross boards”. These have a similar cross shape to the standard 33-hole board, but the four \(3\times n\) “arms” are allowed to have different lengths \(n_{1},n_{2},n_{3},n_{4}\) (including zero). The standard 33-hole board is of course such a board (\(n_{1}=n_{2}=n_{3}=n_{4}=2\)), as is Wiegleb’s 45-hole board (\(n_{1}=n_{2}=n_{3}=n_{4}=3\)). Shown in Figure 2 is a 48-hole example with \(n_{1}=5\), \(n_{2}=3\), \(n_{3}=2\), \(n_{4}=3\).
+
+Figure 2: The 48-hole generalized cross board \(n_{1}=5\), \(n_{2}=3\), \(n_{3}=2\), \(n_{4}=3\).
+
+All these generalized cross boards are built up from rows of three, so they are automatically null-class boards. We can therefore hope that “single-vacancy complement problems”, where we play to leave a single peg in the hole initially vacated, will be solvable, and we shall describe a board as“solvable at X” if the problem “vacate X, play to finish at X” is solvable on it. Making extensive use of the computer for investigation, George has shown that there are exactly 12 generalized cross boards which are solvable at every location. Table 2 lists all such boards—they range in size from 24 to 42 holes, and of course they include the standard 33-hole board (but not Wiegleb’s board, which is not solvable at the middle square at the end of an arm).
+
+\begin{table}
+\begin{tabular}{|c|c|c|c|c|l|l|}
+\hline
+\(n_{1}\) & \(n_{2}\) & \(n_{3}\) & \(n_{4}\) & Holes & Symmetry & Comment \\
+\hline \hline
+2 & 1 & 2 & 0 & 24 & Lateral & \\
+\hline
+2 & 1 & 2 & 1 & 27 & Rectangular & \\
+2 & 2 & 1 & 1 & 27 & Diagonal & \\
+3 & 2 & 0 & 1 & 27 & & \\
+\hline
+3 & 2 & 1 & 1 & 30 & & \\
+\hline
+2 & 2 & 2 & 2 & 33 & Square & The standard 33-hole board \\
+3 & 2 & 2 & 1 & 33 & & \\
+\hline
+3 & 3 & 2 & 1 & 36 & & \\
+3 & 2 & 3 & 1 & 36 & Lateral & \\
+\hline
+3 & 2 & 3 & 2 & 39 & Rectangular & “semi-Wiegleb” \\
+3 & 3 & 2 & 2 & 39 & Diagonal & \\
+\hline
+3 & 3 & 3 & 2 & 42 & Lateral & \\ \hline
+\end{tabular}
+\end{table}
+Table 2: The 12 generalized cross board solvable at every location.
+
+Most of these problems are easy, but some are not. Perhaps the hardest is given by the middle square at the end of a long arm on the 39-hole board “3,2,3,2”, which has two “standard” arms and two “Wiegleb” arms. This is presented as a _problem to solve_ in the last section, and its solution is unique to within symmetry and order of jumps [2].
+
+No generalized cross board other than these twelve is solvable everywhere. George demonstrates this by applying conventional analysis to show that no such board with an arm of length 5 or more can be solvable everywhere (via the same technique as he uses for the general 6-arm case below), and then performing a relatively simple and quick computer analysis of the 45 remaining cases. However, the computer analysis must be laboriously run over each case individually, and he stresses that the results await independent verification. His analysis of Wiegleb’s board confirms my own [1, p. 199–201].
+
+### Boards with longer arms
+
+The investigation above showed that no generalized cross board with an arm longer than three was solvable everywhere, but George wondered what would happen if a longer arm were attached to a board of some other shape. He came up with the 36-hole “mushroom board” (Figure 3), which proved that a board with a 4-arm could be solvable at all locations, in particular at the middle of the end of the arm (always likely to be the most difficult square). For convenience, we invert the mushroom so that this key square is at the top, and we continue to call it “d1”, adding a z-file to the left of the a-file.
+
+Figure 3: The 36-hole “mushroom board” with a solvable d1-complement.
+
+The d1-complement on this board can be solved by d3-d1, d5-d3, b5-d5, d6-d4-d2, f5-d5, d8-d6-d4, e3-e5, e1-e3, e6-e4-e2, h5-f5, g7-g5-e5, b7-d7, e7-c7-c5, c4-c6, b6-d6, z5-b5, a7-a5-c5, c2-c4-c6-e6-e4, c1-e1-e3-e5, f7-f5-d5-d3-d1. The a6-complement is another tricky one (it fails if the arm is only of length 2, or is absent altogether), but the other single-vacancy complement problems are not difficult.
+
+This board has only lateral symmetry, but it is not difficult to construct 4-arm boards solvable everywhere that have square symmetry. One example is the 129-hole board obtained by taking a \(9\times 9\) square and attaching a 4-arm to the middle of each side.
+
+Initial experimentation suggested that any board with a 5-arm would be unsolvable at the mid-end of the arm, but a proof covering all cases was elusive and eventually we found a 90-hole board which was solvable there. Subsequent exploration brought the number of holes down to 75, and further reduction may be possible. The 75-hole board is shown in Figure 4.
+
+Figure 4: A 75-hole board with a solvable d1-complement.
+
+and the d1-complement problem solves by d3-d1, d5-d3, d7-d5, f7-d7, e5-e7, e3-e5, e1-e3, e8-e6-e4-e2, g6-e6, c8-e8, e9-e7-e5, i6-g6, i8-i6, j6-h6-f6, g8-g6-e6-e4, k8-i8-g8-e8, e11-e9-e7, g9-e9, g10-e10-e8-e6, c1-e1-e3-e5-e7, k7-i7-g7, h10-h8, d10-d8-d6-d4-d2, b6-d6, j9-h9-h7-f7-d7-d5, c4-c6, c7-c5, c2-c4-c6, z6-b6-d6-d4, x6-z6, y8-y6-a6, z8-z6-b6, a8-a6-c6, b8-b6-d6, d12-d10, b11-d11-d9, b10-d10-d8, b9-d9-d7-d5-d3-d1.
+
+This board has no symmetry whatever, and we have not investigated the solvability of problems other than the d1-complement. It appears to us that the square-symmetrical 141-hole board obtained by attaching 5-arms to the sides of a \(9\times 9\) square is not solvable at the mid-end of the arm, but the 285-hole board obtained by doing the same to a \(15\times 15\) square is solvable everywhere.
+
+A 5-arm is the limit. A board with a 6-arm is unsolvable at the mid-end of the arm whatever the size and shape of the rest of the board. The proof is in two stages: (a) identifying every combination of moves which refills d1 and clears the rest of the arm, and (b) showing that each leaves a deficit when measured by the “golden ratio” resource count developed by Conway to resolve the problem of the Solitaire Army (see [1, chapter 12]).
+
+## 5 Five new problems for solution
+
+Table 3 shows the symbols used to describe which holes are required to be full (a peg is present) at the start and finish of each problem. The same symbols are used in _The Ins and Outs_[1]. A “marked peg” is one specifically identified, and generally not allowed to jump until near the end, when it sweeps all remaining pegs off the board.
+
+\begin{table}
+\begin{tabular}{|c|l|l|}
+\hline
+Symbol & Start & Finish \\
+\hline \hline
+(none) & Empty & Empty \\
+ & Full & Empty \\
+ & Marked & Empty \\
+\hline
+ & Empty & Full \\
+ & Full & Full \\
+ & Marked & Full \\ \hline
+\end{tabular}
+\end{table}
+Table 3: Symbols used to describe peg solitaire problems.
+
+**Problem 1**
+On the 37-hole board, possibly by myself [John Beasley]⁴: Vacate d4, mark the pegs at a4 and g4, and play to interchange these pegs and clear the rest of the board (Figure 5a).
+
+Footnote 4: I am reluctant to make an unqualified claim to this, because “vacate d4, finish at a4 and g4” is a natural problem to try on the 37-hole board and it must have occurred to somebody to see if it could be done interchanging the pegs originally in these holes, but I haven’t seen it anywhere else.
+
+Figure 5: (a) Problem 1 on the “French” board. (b) Problem 2 on the “semi-Wiegleb” board. (c) Problem 3 on the \(8\times 8\) square board.
+
+**Problem 2**
+On the 39-hole “semi-Wiegleb” 3-3-3-7-7-7-3-3-3 board, by George Bell: Vacate d1, and play to finish there (Figure 5b).
+
+This was discovered in the course of the investigation described in Section 4.1. George’s computer originally threw out a solution in 24 moves, my solution by hand took 23; a subsequent analysis by computer to find the shortest possible solution got the number down to 21.
+
+**Problem 3**
+On the \(8\times 8\) board, by John Harris, 1986: Vacate d6, play to finish at h6 (Figure 5c).
+
+“Here is something I found with 63 poker chips and a chessboard.” John does it in 25 moves, only one more than the number immediately established as necessary by the \(8\times 8\) version of Merson’s “region” analysis.
+
+Figure 6: (a) Problem 4 on the 41-hole diamond board. (b) Problem 5 on the \(6\times 6\) square board.
+
+**Problem 4**
+On the 41-hole diamond board, by John Harris, 1985: Allowing diagonal jumps, vacate c7, mark f2, and play to finish at b4 with a 23-sweep (Figure 6a).
+
+“Can the 41 cell board be cleared in less than 12 moves? Probably. Is a longer sweep possible on this board? Don’t know, it is possible to set up a 26 peg sweep, but not if you start with a single vacancy.”
+
+**Problem 5**
+On the \(6\times 6\) board, by John Harris, 1985: Allowing diagonal jumps, start and finish at b2, solving the problem in 13 moves and ending with a symmetrical 16-sweep (Figure 6b).
+
+John’s proof that 13 moves are required: each of the 12 Merson regions around the edge requires a first escape, and the first jump has to be by a centre peg. “It is so simple, maybe even a computer could do it! There could be a 16 peg sweep, 12 move game by starting with the vacancy somewhere else, but it is unlikely to be symmetrical.”
+
+Readers are requested to try to solve the problems for themselves. This is the best way to gain a full understanding of any problem. Problems 1, 4 and 5 are best solved indirectly—first try to determine the board position before the final sweep(s). Then, start from the complement of this board position and attempt to reduce the board to one peg at the location of the stating vacancy (see the “time-reversal trick” [3, p. 817–8]).
+
+George Bell has created an interactive JavaScript puzzle[5] where you can try all five problems.
+
+## References
+
+* [1] J. Beasley, _The Ins and Outs of Peg Solitaire_, Oxford Univ. Press, 1992.
+* [2] G. Bell and J. Beasley, New problems on old solitaire boards, Board Game Studies, **8** (2005), http://www.boardgamesstudies.org, arXiv:math/0611091 [math.CO]
+* [3] E. Berlekamp, J. Conway and R. Guy, Purging pegs properly, in _Winning Ways for Your Mathematical Plays_, 2nd ed., Vol. 4, Chap. 23: 803–841, A K Peters, 2004.
+* [4] D. Parlett, _The Oxford History of Board Games_, Oxford Univ. Press, 1999.
+* [5] G. Bell, http://www.geocities.com/gibell.geo/pegsolitaire/index.html#games
+* [6] G. Jelliss, The Games and Puzzles Journal, http://www.gpj.connectfree.co.uk/index.htm, published online 2001–2006.
+* [7] J. C. Meyrignac, http://euler.free.fr/PegInfos.htm
+
+## Solutions
+
+**Solution to Problem 1 (Figure 5a) d2-d4, b2-d2, d1-d3, f2-d2, c4-c2, e3-c3, c2-c4, a3-c3, c4-c2, c1-c3, g3-e3, e4-e2, e1-e3, c6-c4, a5-c5, c4-c6, c7-c5, d5-b5, e6-e4, g5-e5, e4-e6, e7-e5, d7-d5-f5, after which the board position of Figure 7a is reached, and the rest is easy.**
+
+Figure 7: (a,b,c) The final sweep positions for Problems 1, 4 and 5.
+
+**Solution to Problem 2** (Figure 5b)
+I originally played d3-d1, d5-d3, b5-d5, c3-c5, c1-c3, c6-c4-c2, e5-c5, a6-c6, d6-b6, a4-a6-c6-c4, e1-c1-c3-c5, c8-c6-c4, b4-d4-d2, e7-e5, e9-e7, e4-e6-e8, g6-e6, d8-d6-f6, g4-g6-e6, c9-e9-e7-e5, e2-e4, f4-d4, f5-d5-d3-d1. This was the result of a detailed analysis of debts and surpluses using pencil and paper, and had George not told me that the problem was solvable I would have assumed it wasn’t; indeed, at one point I was sure I had proved it. George’s computer subsequently reduced the number of moves to 21 by playing d3-d1, d5-d3, f4-d4-d2, b5-d5, e6-e4, e3-e5 (6), c7-c5, c9-c7, b4-d4, e1-e3, c2-c4-c6-c8 (11), a6-c6, g6-e6, d6-f6, d8-d6-b6 (15), a4-a6-c6, e8-e6-e4-e2, e9-c9-c7-c5-e5 (18), g4-g6-e6-e4, c1-e1-e3-e5, f5-d5-d3-d1.
+
+**Solution to Problem 3** (Figure 5c)
+f6-d6, c6-e6, f8-f6-d6, c8-c6-e6, a8-c8 (5), d8-b8, h8-f8-d8-d6-f6, g6-e6-e8, g4-g6-g8, a6-a8-c8 (10), e4-g4, h4-f4, c4-e4-g4, d2-d4, a4-a6-c6-c4-e4-e6 (15), b3-d3, c1-c3, a2-a4-c4-c2, a1-c1, d1-b1-b3 (20), f2-f4-f6-d6-d4-d2, f1-d1-d3-f3, h2-h4-f4-f2, h1-f1-f3-h3, h6-h8-f8-d8-b8-b6-b4-b2-d2-f2-h2-h4-h6. Move 8 (g6-e6-e8) is the one that is not an initial exit from one of the Merson regions.
+
+**Solution to Problem 4** (Figure 6a)
+e5-c7, c3-e5, f4-d6, g7-e5, h4-f6-d4 (5), c7-e5-g5, e9-c7, b6-d8, i5-g7-e9-c7, a5-c3-e5 (10), e1-c3, and we are set up for the sweep in Figure 7b, f2-h4-f4-f2-d2-f4-d4-d2-b4-b6-d4-d6-b6-d8-d6-f6-d8-f8-f6-h6-h4-f6-d4-b4.
+
+**Solution to Problem 5** (Figure 6b)
+d4-b2, a1-c3 (2), b6-d4-b2, a3-a1-c3, a5-a3-c5, d6-b6, a6-c6 (7), f2-d4-b2, c1-a1-c3, e1-c1-e3, f4-f2, f1-f3 (12) and we are set up for the sweep in Figure 7c, f6-d6-b6-b4-d6-d4-b4-b2-d4-f6-f4-f2-d2-f4-d4-d2-b2. John uses a binumeric notation in order to bring out the symmetry.
+

arxiv_ground_truth/0811.4001.md

@@ -0,0 +1,688 @@
+# Separation of Relatively Quasiconvex Subgroups
+
+Jason Fox Manning
+
+Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900
+j399m@buffalo.edu
+
+Eduardo Martínez-Pedroza
+
+Department of Mathematics & Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada.
+emartinez@math.mcmaster.ca
+
+###### Abstract.
+
+Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, relatively quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic \(3\)–manifolds would follow from LERF for closed hyperbolic \(3\)–manifolds.
+
+The method is to reduce, via combination and filling theorems, the separability of a relatively quasiconvex subgroup of a relatively hyperbolic group \(G\) to the separability of a quasiconvex subgroup of a hyperbolic quotient \(\bar{G}\). A result of Agol, Groves, and Manning is then applied.
+
+## 1. Main Results
+
+A subgroup \(H\) of a group \(G\) is called _separable_ if for any \(g\in G\setminus H\) there is a homomorphism \(\pi\) onto a finite group such that \(\pi(g)\not\in\pi(H)\). A group is called _residually finite_ if the trivial subgroup is separable, and a group is called _subgroup separable_ or _LERF_ if every finitely generated subgroup is separable. For example, Hall showed that free groups are LERF in [15]. It follows from a theorem of Mal_′_cev [21] that polycyclic (and in particular finitely generated nilpotent) groups are LERF. A group is called _slender_ if every subgroup is finitely generated. Polycyclic groups are also slender, by a result of Hirsch [16].
+
+Given a relatively hyperbolic group with peripheral structure consisting of LERF and slender subgroups, we study separability of relatively quasiconvex subgroups. This is connected, via filling constructions, to residual finiteness of hyperbolic groups. It is not known whether all word-hyperbolic groups are residually finite. Consequences of a positive or negative answer to this question have been explored by several authors; see for example [2, 19, 20, 24, 27, 29]. In particular, the main result of [2] is the following.
+
+**Theorem 1.1****.**: _[_2_]_ _If all hyperbolic groups are residually finite, then every quasiconvex subgroup of a hyperbolic group is separable._
+
+We extend this result, answering a question in [2], as follows:
+
+**Theorem 1.2****.**: _Suppose that all hyperbolic groups are residually finite. If \(G\) is a torsion free relatively hyperbolic group with peripheral structure consisting of subgroups which are LERF and slender, then any relatively quasiconvex subgroup of \(G\) is separable._
+
+This extension has some interesting corollaries. A _pinched Hadamard manifold_ is a simply connected Riemannian manifold with pinched negative curvature. In [5], Bowditch gave several equivalent definitions of geometrical finiteness for discrete subgroups of the isometry group of a pinched Hadamard manifold, generalizing the notion of geometrical finiteness in Kleinian groups. The next theorem summarizes some useful facts about these groups. (Statement (1) can be found in [13] or [6]; statement (2) follows from the Margulis lemma; and statement (3) is [18, Corollary 1.3].)
+
+**Theorem 1.3****.**: _[_13, 18, 6_]_ _Let \(X\) be a pinched Hadamard manifold and let \(G\) be a geometrically finite subgroup of \(\text{Isom}(X)\)._
+
+1. (1)\(G\) _is relatively hyperbolic, relative to a collection of representatives of conjugacy classes of maximal parabolic subgroups._
+2. (2)_Maximal parabolic subgroups of_ \(G\) _are virtually nilpotent._
+3. (3)_A subgroup_ \(H\) _of_ \(G\) _is relatively quasiconvex if and only if_ \(H\) _is geometrically finite._
+
+Rank one symmetric spaces are pinched Hadamard manifolds. We therefore have the following corollary of Theorem 1.2:
+
+**Corollary 1.4****.**: _Suppose that all hyperbolic groups are residually finite. Let \(G\) be a discrete, geometrically finite subgroup of the isometry group of a rank one symmetric space. (For example, \(G\) could be a lattice.) All the geometrically finite subgroups of \(G\) are separable._
+
+In case the symmetric space is \(\mathbb{H}^{3}\), more can be said (see Section 5 for the proof).
+
+**Corollary 1.5****.**: _If all hyperbolic groups are residually finite, then all finitely generated Kleinian groups are LERF._
+
+Briefly, Theorem 1.2 is proved by combining one of Martínez-Pedroza’s combination theorems in [22] with Theorem 1.1 and the Dehn filling technique of [14, 27]. We next give a more detailed discussion.
+
+**Definition 1.6****.**: A relatively quasiconvex subgroup \(H\) of \(G\) is called _fully quasiconvex_ if for any subgroup \(P\in\mathcal{P}\) and any \(f\in G\), either \(H\cap P^{f}\) is finite or \(H\cap P^{f}\) is a finite index subgroup of \(P^{f}\). (Here \(P^{f}=fPf^{-1}\).)
+
+Using the work in [22], we show the following.
+
+**Theorem 1.7****.**: _Let \(G\) be a group hyperbolic relative to a collection of slender and LERF subgroups. Suppose that \(Q\) is a relatively quasiconvex subgroup of \(G\) and \(g\) is an element of \(G\) not in \(Q\). Then there exists a fully quasiconvex subgroup \(H\) which contains \(Q\) and does not contain \(g\)._
+
+**Remark 1.8****.**: In case \(G\) is a finite volume hyperbolic \(3\)–manifold group and \(Q\) is the fundamental group of a quasi-fuchsian surface, Theorem 1.7 can be proved using geometric arguments like those in [10, 11]. Such geometric arguments were applied (in a different way) to separability questions in [3] (cf. [30]).
+
+Using the work in [2] and Theorem 1.7, we prove the following separation result of relatively quasiconvex subgroups by maps onto word hyperbolic groups. (A part of this theorem can be interpreted as saying that \(G\) is “quasiconvex extended residually hyperbolic”.)
+
+**Theorem 1.9****.**: _Let \(G\) be a torsion free group hyperbolic relative to a collection of slender and LERF subgroups. For any relatively quasiconvex subgroup \(Q\) of \(G\) and any element \(g\in G\) such that \(g\not\in Q\), there is a fully quasiconvex subgroup \(H\) of \(G\), and a surjective homomorphism \(\pi\colon\, G\longrightarrow\bar{G}\) such that_
+
+1. (1)\(Q<H\)_,_
+2. (2)\(\bar{G}\) _is a word-hyperbolic group,_
+3. (3)\(\pi(H)\) _is a quasiconvex subgroup of_ \(\bar{G}\)_,_
+4. (4)\(\pi(g)\not\in\pi(H)\)_._
+
+We can prove Theorem 1.2 from Theorem 1.9 as follows:
+
+Proof of Theorem 1.2.: Let \(Q<G\) be relatively quasiconvex, and let \(g\in G\setminus Q\). By Theorem 1.9 there is a fully quasiconvex \(H<G\) containing \(Q\) but not \(g\), and a quotient \(\pi\colon\, G\to K\) so that \(\pi(g)\notin\pi(H)\), \(K\) is hyperbolic, and \(\pi(K)\) is quasiconvex.
+
+Assuming all hyperbolic groups are residually finite, Theorem 1.1 implies that there is a finite group \(F\) and a quotient \(\phi\colon\, K\to F\) so that \(\phi(\pi(g))\notin\phi(\pi(H))\). Since \(\phi(\pi(H))\) contains \(\phi(\pi(Q))\), the map \(\phi\circ\pi\) serves to separate \(g\) from \(Q\). ∎
+
+**Remark 1.10****.**: The torsion-free hypotheses in Theorems 1.9 and 1.2 are not really necessary. We sketch the necessary changes to our argument in Appendix B. If one is primarily interested in Theorem 1.2 in the special case of virtually polycyclic peripheral subgroups, we have the following simple argument, pointed out to us by the referee:
+
+Let \(G\) be a relatively hyperbolic group, relative to a collection \(\mathcal{P}=\{P_{1},\ldots,P_{m}\}\) of virtually polycyclic subgroups. An easy argument shows that each \(P_{i}\) contains a finite index normal subgroup which is torsion free. Moreover, \(G\) contains only finitely many finite order non-parabolic elements, up to conjugacy [26, Theorem 4.2]. It then follows from Osin’s version of the relatively hyperbolic Dehn filling theorem that there is a filling \(G\stackrel{{\pi}}{{\to}}G(N_{1}^{\prime},\ldots,N_{m}^{\prime})\), so that \(G(N_{1}^{\prime},\ldots,N_{m}^{\prime})\) is hyperbolic, and no non-trivial torsion element of \(G\) is in the kernel of \(\pi\). Assuming hyperbolic groups are residually finite, the group \(G(N_{1}^{\prime},\ldots,N_{m}^{\prime})\) has a torsion-free subgroup \(S\) of finite index. The preimage \(G_{0}=\pi^{-1}(S)\) is a torsion-free finite index subgroup of \(G\). Again under the assumption that hyperbolic groups are residually finite, Theorem 1.2 implies that \(G_{0}\) is QCERF, and so (applying Corollary 2.2 below) \(G\) must also be QCERF.
+
+The paper is organized as follows. In Section 2.5 we give a definition of relatively hyperbolic group which suits the purposes of this paper and is equivalent to the more standard definitions in the literature. In Section 3 we recall a combination theorem for relatively quasiconvex subgroups from [22] and prove Theorem 1.7. In Section 4 we recall some definitions and results on fillings of relatively hyperbolic groups and prove Theorem 1.9. In Section 5, we give two applications to separability questions on hyperbolic \(3\)–manifolds: Corollary 1.5 and Proposition 5.3. In Appendix A, we prove a result we need on the equivalence of various definitions of relative quasiconvexity, and in Appendix B, we sketch how to prove our results in the presence of torsion.
+
+## 2. Preliminaries
+
+### Separability
+
+Let \(G\) be a group. Recall that the _profinite topology_ on \(G\) is the smallest topology on \(G\) in which all finite index subgroups and their cosets are closed. The group \(G\) is residually finite if and only if this topology is Hausdorff. A subgroup \(H\) is separable if and only if it is a closed subset of \(G\), with this topology.
+
+Given a subgroup \(G_{0}<G\), one can ask whether the profinite topology on \(G_{0}\) coincides with the topology induced by the profinite topology on \(G\). In general, the topologies are quite different, but in case \(G_{0}\) is finite index, the topologies coincide. In particular, we have:
+
+**Lemma 2.2****.**: _Let \(G_{0}<G\) be a finite index subgroup, and let \(C_{0}\subseteq G_{0}\). The following conditions are equivalent:_
+
+1. (1)\(C_{0}\) _is closed in the profinite topology on_ \(G_{0}\)_._
+2. (2)\(C_{0}\) _is closed in the profinite topology on_ \(G\)_._
+3. (3)\(C_{0}=C\cap G_{0}\) _for some set_ \(C\) _which is closed in the profinite topology on_ \(G\)_._
+
+Proof.: Suppose \(C_{0}\) is closed in \(G_{0}\). Since the finite index subgroups of \(G_{0}\) and their cosets generate the topology on \(G_{0}\), we can write \(C_{0}\) as a finite union of arbitrary intersections of cosets
+
+(1) \[C_{0}=\bigcup_{i=1}^{n}\bigcap_{i\in I_{j}}g_{i}K_{i}\]
+
+where every \(K_{i}\) is a finite index subgroup of \(G_{0}\). But since \(G_{0}\) is finite index in \(G\), each of the \(K_{i}\) appearing in equation (1) is also finite index in \(G\). This \(C_{0}\) is a finite union of intersections of closed sets in the profinite topology on \(G\), so \(C_{0}\) is closed in \(G\). Thus condition (1) implies condition (2).
+
+Trivially, condition (2) implies condition (3). To see that condition (3) implies condition (1), we first establish the following.
+
+**Claim 2.3****.**: _If \(g\in G\), and \(K<G\) is finite index, then \(gK\cap G_{0}\) is closed in the profinite topology on \(G_{0}\)._
+
+Proof.: Let \(K_{0}=K\cap G_{0}\). Since \(K_{0}\) is finite index in \(K\), the subgroup \(K\) is a finite union of cosets \(K=\cup_{i=1}^{p}g_{i}K_{0}\), and so \(gK=\cup_{i=1}^{p}gg_{i}K_{0}\) is as well. Since a coset of \(K_{0}\) in \(G\) either lies inside \(G_{0}\) or in its complement, it follows that \(gK\cap G_{0}\) is a finite union of cosets of \(K_{0}\) in \(G_{0}\). Since \(K_{0}\) is finite index in \(G_{0}\), these cosets are closed in the profinite topology on \(G_{0}\) and therefore \(gK\cap G_{0}\) is closed as well. ∎
+
+Suppose then that \(C_{0}=C\cap G_{0}\) for \(C\) closed in \(G\). We have \(C=\bigcup_{i=1}^{n}\bigcap_{i\in I_{j}}g_{i}K_{i}\) where now the \(K_{i}\) are arbitrary finite index subgroups of \(G\), and the \(g_{i}\) are arbitrary elements of \(G\). We thus have
+
+(2) \[C_{0}=C\cap G_{0}=\bigcup_{i=1}^{n}\bigcap_{i\in I_{j}}(g_{i}K_{i})\cap G_{0}.\]
+
+By the claim, every \((g_{i}K_{i})\cap G_{0}\) appearing in equation (2) is closed in \(G_{0}\), and so \(C_{0}\) is also closed in the profinite topology on \(G_{0}\). ∎
+
+**Corollary 2.4****.**: _Let \(H<G\) be a pair of groups, and let \(G_{0}\) be a finite index subgroup of \(G\). Let \(H_{0}=H\cap G_{0}\). The subgroup \(H\) is separable in \(G\) if and only if \(H_{0}\) is separable in \(G_{0}\)._
+
+Another immediate corollary is that LERFness and QCERFness are commensurability invariants.
+
+### Relative hyperbolicity
+
+The notion of relative hyperbolicity has been studied by several authors with different equivalent definitions. The definition in this subsection is based on the work by D. Osin in [26]. Let \(G\) be a group, \(\mathcal{P}\) denote a collection of subgroups \(\{P_{1},\dots,P_{m}\}\), and \(S\) be a finite generating set which is assumed to be symmetric, i.e, \(S=S^{-1}\). Denote by \(\Gamma(G,\mathcal{P},S)\) the Cayley graph of \(G\) with respect to the generating set \(S\cup\bigcup\mathcal{P}\). If \(p\) is a path between vertices in \(\Gamma(G,\mathcal{P},S)\), we will refer to its initial vertex as \(p_{-}\), and its terminal vertex as \(p_{+}\). The path \(p\) determines a word \(\mathrm{Label}(p)\) in the alphabet \(S\cup\bigcup\mathcal{P}\) which represents an element \(g\) so that \(p_{+}=p_{-}g\).
+
+**Definition 2.6******(Weak Relative Hyperbolicity)**.**: The pair \((G,\mathcal{P})\) is _weakly relatively hyperbolic_ if there is an integer \(\delta\geq 0\) such that \(\Gamma(G,\mathcal{P},S)\) is \(\delta\)–hyperbolic. We may also say that \(G\) is _weakly relatively hyperbolic, relative to_\(\mathcal{P}\).
+
+**Definition 2.7******( [26])**.**: Let \(q\) be a combinatorial path in the Cayley graph \(\Gamma(G,\mathcal{P},S)\). Sub-paths of \(q\) with at least one edge are called _non-trivial_. For \(P_{i}\in\mathcal{P}\), a _\(P_{i}\)–component_ of \(q\) is a maximal non-trivial sub-path \(s\) of \(q\) with \(Label(s)\) a word in the alphabet \(P_{i}\). When we don’t need to specify the index \(i\), we will refer to \(P_{i}\)–components as _\(\mathcal{P}\)–components_.
+
+Two \(\mathcal{P}\)–components \(s_{1}\), \(s_{2}\) are _connected_ if the vertices of \(s_{1}\) and \(s_{2}\) belong to the same left coset of \(P_{i}\) for some \(i\). A \(\mathcal{P}\)–component \(s\) of \(q\) is _isolated_ if it is not connected to a different \(\mathcal{P}\)–component of \(q\). The path \(q\) is _without backtracking_ if every \(\mathcal{P}\)–component of \(q\) is isolated.
+
+A vertex \(v\) of \(q\) is called _phase_ if it is not an interior vertex of a \(\mathcal{P}\)–component \(s\) of \(q\). Let \(p\) and \(q\) be paths between vertices in \(\Gamma(G,\mathcal{P},S)\). The paths \(p\) and \(q\) are _\(k\)–similar_ if
+
+\[\max\{dist_{S}(p_{-},q_{-}),dist_{S}(p_{+},q_{+})\}\leq k,\]
+
+where \(dist_{S}\) is the metric induced by the finite generating set \(S\) (as opposed to the metric in \(\Gamma(G,\mathcal{P},S)\)).
+
+**Remark 2.8****.**: A geodesic path \(q\) in \(\Gamma(G,\mathcal{P},S)\) is without backtracking, all \(\mathcal{P}\)–components of \(q\) consist of a single edge, and all vertices of \(q\) are phase.
+
+**Definition 2.9******(Bounded Coset Penetration (BCP))**.**: The pair \((G,\mathcal{P})\) satisfies the _BCP property_ if for any \(\lambda\geq 1\), \(c\geq 0\), \(k\geq 0\), there exists an integer \(\epsilon(\lambda,c,k)>0\) such that for \(p\) and \(q\) any two \(k\)–similar \((\lambda,c)\)–quasi-geodesics in \(\Gamma(G,\mathcal{P},S)\) without backtracking, the following conditions hold:
+
+1. (i.)The sets of phase vertices of \(p\) and \(q\) are contained in the closed \(\epsilon(\lambda,c,k)\)–neighborhoods of each other, with respect to the metric \(dist_{S}\).
+2. (ii.)If \(s\) is any \(\mathcal{P}\)–component of \(p\) such that \(dist_{S}(s_{-},s_{+})>\epsilon(\lambda,c,k)\), then there exists a \(\mathcal{P}\)–component \(t\) of \(q\) which is connected to \(s\).
+3. (iii.)If \(s\) and \(t\) are connected \(\mathcal{P}\)–components of \(p\) and \(q\) respectively, then
+
+\[\max\{dist_{S}(s_{-},t_{-}),dist_{S}(s_{+},t_{+})\}\leq\epsilon(\lambda,c,k).\]
+
+**Remark 2.10****.**: Our definition of the BCP property corresponds to the conclusion of Theorem 3.23 in [26].
+
+**Definition 2.11******(Relative Hyperbolicity)**.**: The pair \((G,\mathcal{P})\) is _relatively hyperbolic_ if the group \(G\) is weakly relatively hyperbolic relative to \(\mathcal{P}\) and the pair \((G,\mathcal{P})\) satisfies the Bounded Coset Penetration property. If \((G,\mathcal{P})\) is relatively hyperbolic then we say \(G\) is _relatively hyperbolic, relative to_\(\mathcal{P}\); if there is no ambiguity, we just say that \(G\) is relatively hyperbolic.
+
+**Remark 2.12****.**: Definition 2.11 given here is equivalent to Osin’s [26, Definition 2.35] for finitely generated groups: To see that Osin’s definition implies 2.11, apply [26, Theorems 3.23]; to see that 2.11 implies Osin’s definition, apply [26, Lemma 7.9 and Theorem 7.10]. For the equivalence of Osin’s definition and the various other definitions of relative hyperbolicity see [18] and the references therein.
+
+The definition of relative hyperbolicity is independent of finite generating set \(S\).
+
+## 3. Combination of Parabolic and quasiconvex Subgroups
+
+In this section, \(G\) will be relatively hyperbolic, relative to a finite collection of subgroups \(\mathcal{P}\), and \(S\) will be a finite generating set for \(G\). Denote by \(\Gamma(G,\mathcal{P},S)\) the Cayley graph of \(G\) with respect to the generating set \(S\cup\bigcup\mathcal{P}\).
+
+### Parabolic and Quasiconvex Subgroups
+
+**Definition 3.2****.**: The _peripheral_ subgroups of \(G\) are the elements of \(\mathcal{P}\). A subgroup of \(G\) is called _parabolic_ if it can be conjugated into a peripheral subgroup.
+
+**Proposition 3.3****.**: _[_26_, Proposition 2.36]_ _The following conditions hold._
+
+1. (1)_For any_ \(g_{1}\)_,_ \(g_{2}\in G\)_, the intersection_ \(P_{i}^{g_{1}}\cap P_{j}^{g_{2}}\) _is finite unless_ \(i=j\)_._
+2. (2)_The intersection_ \(P_{i}^{g}\cap P_{i}\) _is finite for any_ \(g\not\in P_{i}\)_._
+
+_In particular, if \(Q\) is a subgroup of \(G\), then any infinite maximal parabolic subgroup of \(Q\) is of the form \(Q\cap P_{i}^{f}\) for some \(f\in Q\) and \(P_{i}\in\mathcal{P}\)._
+
+**Definition 3.4****.**: [26, Definition 4.9] A subgroup \(Q\) of \(G\) is called _quasiconvex relative to \(\mathcal{P}\)_ (or simply _relatively quasiconvex_ when the collection \(\mathcal{P}\) is fixed) if there exists a constant \(\sigma\geq 0\) such that the following holds: Let \(f\), \(g\) be two elements of \(Q\), and \(p\) an arbitrary geodesic path from \(f\) to \(g\) in the Cayley graph \(\Gamma(G,\mathcal{P},S)\). For any vertex \(v\in p\), there exists a vertex \(w\in Q\) such that \(dist_{S}(v,w)\leq\sigma,\) where \(dist_{S}\) is the word metric induced by \(S\).
+
+**Remark 3.5****.**: For more on different definitions of relative quasiconvexity in the literature, see Appendix A.
+
+**Theorem 3.6****.**: _[_18_, Theorem 9.1]_ _Let \(Q\) be a finitely generated relatively quasiconvex subgroup of \(G\). The number of infinite maximal parabolic subgroups of \(Q\) up to conjugacy in \(Q\) is finite. Furthermore, if \(\mathcal{O}\) is a set of representatives of these conjugacy classes, then \(Q\) is relatively hyperbolic, relative to \(\mathcal{O}\)._
+
+**Remark 3.7****.**: In [18], an extended definition of relative hyperbolicity is used which includes some countable but non-finitely generated groups. Using this extended definition, the assumption of finite generation in Theorem 3.6 is superfluous.
+
+We note that in case all the peripheral subgroups are slender, relatively quasiconvex subgroups are necessarily finitely generated (see [18, Corollary 9.2]).
+
+### Combination of Quasiconvex Subgroups
+
+For \(g\in G\), \(|g|_{S}\) denotes the distance from \(g\) to the identity element in the word metric induced by \(S\).
+
+**Theorem 3.9****.**: _[_22_, Theorem 1.1]_ _Let \(Q\) be a relatively quasiconvex subgroup of \(G\), and let \(P\) be a maximal parabolic subgroup of \(G\). Suppose that \(P^{f}=P_{i}\) for some \(P_{i}\in\mathcal{P}\) and \(f\in G\)._
+
+_There are constants \(C=C(Q,P)\geq 0\) and \(c=c(Q,P)\geq 0\) with the following property. Suppose \(D\geq C\) and \(R\) is a subgroup of \(P\) such that_
+
+* •\(P\cap Q<R\)_, and_
+* •\(|g|_{S}>D\) _for any element_ \(g\in R\setminus Q\)_._
+
+_It follows that:_
+
+1. (1)_The subgroup_ \(H=\langle Q\cup R\rangle\) _is relatively quasiconvex and the natural map_ \(Q\ast_{Q\cap R}R\longrightarrow H\) _is an isomorphism._
+2. (2)_Every parabolic subgroup of_ \(H\) _is conjugate in_ \(H\) _to a parabolic subgroup of_ \(Q\) _or_ \(R\)_._
+3. (3)_For any_ \(g\in H\)_, either_ \(g\in Q\)_, or any geodesic from_ \(1\) _to_ \(g\) _in the relative Cayley graph_ \(\Gamma(G,\mathcal{P},S)\) _has at least one_ \(P_{i}\)_-component_ \(t\) _such that_ \(|t|_{S}>D-c\)_._
+
+Proof.: Conclusions (1) and (2) rephrase [22, Theorem 1.1]. The proof of conclusion (3) is divided into two cases: \(P\in\mathcal{P}\) and \(P\notin\mathcal{P}\).
+
+**Case 1****.**: \(P\in\mathcal{P}\).
+
+We summarize part of the argument for [22, Theorem 1.1] for conclusions (1) and (2); we then explain how conclusion (3) follows in this case.
+
+Let \(g\in Q\ast_{Q\cap R}R\setminus Q\). The element \(g\) has a _normal form_
+
+(3) \[g=g_{1}h_{1}\dots g_{k}h_{k}\]
+
+where \(g_{j}\in Q\setminus Q\cap R\) for \(1<j\leq k\), \(h_{j}\in R\setminus Q\cap R\) for \(1\leq j<k\), either \(g_{1}=1\) or \(g_{1}\in Q\setminus Q\cap R\), and either \(h_{k}=1\) or \(h_{k}\in R\setminus Q\cap R\). We use the normal form to produce a path \(o\) in \(\Gamma(G,\mathcal{P},S)\) from \(1\) to the image of \(g\) by the natural map \(Q\ast_{Q\cap R}R\longrightarrow H\) as follows. For each \(j\) between \(1\) and \(k\), let \(u_{j}\) be a geodesic path in \(\Gamma(G,\mathcal{P},S)\) from \(g_{1}h_{1}\cdots h_{j}\) to \(g_{1}h_{1}\cdots h_{j}g_{j}\) (so that \(\mathrm{Label}(u_{j})\) represents \(g_{j}\)). Similarly, let \(v_{j}\) be a geodesic path from \(g_{1}h_{1}\cdots g_{j-1}\) to \(g_{1}h_{1}\cdots g_{j-1}h_{j}\) (so that \(\mathrm{Label}(v_{j})\) represents \(h_{j}\)). A path \(o\) from \(1\) to \(g\) in \(\Gamma(G,\mathcal{P},S)\) is given by
+
+\[o=u_{1}v_{1}\dots u_{k}v_{k}.\]
+
+(See Figure 1.)
+
+Figure 1. Part of the polygonal path \(o\) in \(\Gamma(G,\mathcal{P},S)\). For each \(i\), the \(P_{i}\)–component of \(o\) containing the subsegment \(v_{j}\) is long with respect to the \(S\)–metric. This implies that the path \(o\) is a quasi-geodesic with different end-points.
+
+Each subsegment \(v_{j}\) is part of a \(\mathcal{P}\)-component \(t_{j}\) of the path \(o\). Let \(D\) be as in the hypothesis of the theorem. The penultimate inequality in the proof of Claim 2 in the proof of Lemma 5.1 of [22] is
+
+(4) \[|t_{j}|_{S}=dist_{S}((t_{j})_{-},(t_{j})_{+})>D-2M(P,Q,\sigma),\]
+
+where \(\sigma\) is the quasiconvexity constant for \(Q\), and \(M(P,Q,\sigma)\) is the constant provided by [22, Lemma 4.2] for the subgroups \(Q\), \(P\), and the constant \(\sigma\). Let \(\eta\) be the constant from Proposition 3.1 of [22]. If \(D-2M(P,Q,\sigma)>\eta\), then \(o\) is a \((\lambda,0)\)–quasi-geodesic with distinct endpoints. Let \(C=\eta+2M(P,Q,\sigma)\).
+
+It follows from the argument just sketched that if \(D>C\), then the natural map \(Q\ast_{Q\cap R}R\longrightarrow H\) is an isomorphism. It can further be shown that \(H\) is a relatively quasiconvex subgroup and that the parabolic subgroups of \(H\) are conjugate into \(Q\) or \(R\) by elements of \(H\) (See [22, Lemmas 5.2 and 5.3]for details.). In other words, parts (1) and (2) hold for \(P\in\mathcal{P}\) and \(C\) as above.
+
+If \(g\in H\setminus Q\) and \(p\) is a geodesic from \(1\) to \(g\) in \(\Gamma(G,\mathcal{P},S)\), then the \((\lambda,0)\)–quasi-geodesic \(o\) and the geodesic \(p\) are \(0\)–similar. Since \(o\) has a \(\mathcal{P}\)–component of \(S\)–length at least \(D-2M(P,Q,\sigma)\), the Bounded Coset Penetration property (Definition 2.9) implies that \(p\) has a \(\mathcal{P}\)–component of \(S\)–length at least \(D-2M(P,Q,\sigma)-2\epsilon(0,\lambda,0)\). We have verified (3) of the Theorem for \(c\) equal to
+
+\[c(Q,P)=2M(P,Q,\sigma)+2\epsilon(0,\lambda,0)\]
+
+in the special case that \(f=1\) and \(P\in\mathcal{P}\).
+
+**Case 2****.**: \(P\notin\mathcal{P}\), but \(P^{f}\in\mathcal{P}\) for some \(f\in G\setminus P\).
+
+Since \(P^{f}\in\mathcal{P}\) and \(Q^{f}\) is relatively quasiconvex, by [22, Theorem 1.1] and Case 1, all three conclusions of Theorem 3.9 hold for \(Q^{f}\) and \(P^{f}\) and some constants \(C^{\prime}=C(Q^{f},P^{f})>0\) and \(c^{\prime}=c(Q^{f},P^{f})>0\). Define
+
+\[C=C^{\prime}+2|f|_{S}+3\epsilon(1,0,|f|_{S}),\]
+
+and
+
+\[c=c^{\prime}+2|f|_{S}+2\epsilon(1,0,|f|_{S}),\]
+
+where \(\epsilon(1,0,|f|_{S})\) is the constant of Definition 2.9 on the Bounded Coset Penetration property. Now we show that the theorem holds for the subgroups \(P\) and \(Q\), and the constants \(C\) and \(c\). Let \(R\) be a subgroup of \(Q\) satisfying the hypothesis of the theorem for a constant \(D>C\).
+
+If \(r\in R\setminus Q\), then \(|r|_{S}\geq D\), by hypothesis. It follows that
+
+\[|r^{f}|_{S}\geq D-2|f|_{S}\geq C^{\prime}.\]
+
+We therefore have:
+
+1. (1)The subgroup \(H^{f}=\langle Q^{f}\cup R^{f}\rangle\) is relatively quasiconvex and the natural map \(Q^{f}\ast_{(Q\cap R)^{f}}R^{f}\longrightarrow H^{f}\) is an isomorphism. Since relative quasiconvexity is preserved by conjugation, \(H=\langle Q\cup R\rangle\) is relatively quasiconvex. Obviously the map \(Q\ast_{Q\cap R}R\longrightarrow H\) is also an isomorphism. In other words, conclusion (1) holds for \(Q\) and \(P\) and the constant \(C\).
+2. (2)Every parabolic subgroup of \(H^{f}\) is conjugate in \(H^{f}\) to a parabolic subgroup of \(Q^{f}\) or \(R^{f}\). Parabolicity is preserved under conjugation, so the same property (conclusion (2) of the theorem) holds for the subgroups \(Q\), \(R\), and \(H\), and the constant \(C\).
+3. (3)For any \(h\in H^{f}\), either \(h\in Q^{f}\), or any geodesic from \(1\) to \(h\) in the relative Cayley graph \(\Gamma(G,\mathcal{P},S)\) has at least one \(P_{i}\)–component \(t\) such that \(|t|_{S}>D-2|f|_{S}-c^{\prime}\).
+
+It remains to see why conclusion (3) of the Theorem holds with the chosen constant \(c\). Let \(g\in H\setminus Q\) and let \(p\) be a geodesic from \(1\) to \(g\) in \(\Gamma(G,\mathcal{P},S)\). We must show that \(p\) has a \(P_{i}\)–component of \(S\)–length at least \(D-c\). Let \(q\) be a geodesic from \(1\) to \(fgf^{-1}\). Since \(fgf^{-1}\) belongs to \(H^{f}\setminus Q^{f}\), the geodesic \(q\) has a \(P_{i}\)–component \(u\) of \(S\)–length at least \(D-2|f|_{S}-c^{\prime}\). Let \(r\) be the geodesic starting at \(f\), with the same label as \(p\). Thus \(r\) joins \(f\) to \(fg\), and the geodesics \(q\) and \(r\) are \(|f|_{S}\)–similar (see Figure 2).
+
+Figure 2. The geodesics \(q\) and \(r\) are \(|f|_{S}\)–similar and \(q\) contains a large \(\mathcal{P}\)–component \(u\). By the BCP-property, \(r\) has a large \(\mathcal{P}\)–component \(v\). Since \(p\) and \(r\) have the same word-label, \(p\) has a large \(\mathcal{P}\)–component \(w\).
+
+By the Bounded Coset Penetration property, \(r\) has a \(P_{i}\)–component \(v\) of \(S\)–length at least
+
+\[D-2|f|_{S}-c^{\prime}-2\epsilon(1,0,|f|_{S})=D-c.\]
+
+Since \(p=f^{-1}r\) and \(r\) have the same labels, it follows that \(p\) has a \(P_{i}\)–component \(w\) of \(S\)–length at least \(D-c\).
+
+∎
+
+**Corollary 3.10****.**: _[_22_, Lemma 5.4]_ _Suppose that \(G\), \(Q\), \(P\), and \(R\) are as in the hypothesis of Theorem 3.9 and that \(Q\cap P\) is a proper subgroup of \(R\)._
+
+_If \(\{K_{1},\dots,K_{n}\}\) is a collection of representatives of the maximal infinite parabolic subgroups of \(Q\) up to conjugacy in \(Q\) so that \(K_{1}=P\cap Q\), then \(\{R,K_{2},\dots,K_{n}\}\) is a collection of representatives of the maximal parabolic subgroups of \(H\) up to conjugacy in \(H\)._
+
+Proof.: By Theorem 3.9 (2), a maximal parabolic subgroup of \(H\) is conjugate to \(R\) or \(K_{i}\) for some \(i=2,\dots n\). Hence \(\{R,K_{2},\dots,K_{n}\}\) is a collection of representatives of maximal parabolic subgroups. That all these subgroups are different up to conjugacy follows from the algebraic structure of \(H\) as an amalgamated product. In particular, since \(K_{i}\) and \(K_{j}\) are not conjugate in \(Q\), they are not conjugate in \(Q\ast_{Q\cap R}R\). The subgroup \(R<Q\ast_{Q\cap R}R\) is not conjugate to a subgroup of \(Q\) since \(Q\cap R\) is a proper subgroup of \(R\). ∎
+
+Proof of Theorem 1.7.: Suppose that every subgroup in \(\mathcal{P}\) is LERF and slender. Let \(Q\) be a relatively quasiconvex subgroup of \(G\) and let \(g\) be an element of \(G\) not in \(Q\). Let \(\{K_{1},\dots,K_{n}\}\) be a collection of representatives of maximal infinite parabolic subgroups of \(Q\) up to conjugacy in \(Q\); such a collection exists by Theorem 3.6. By Proposition 3.3, for each \(K_{i}\) there is a peripheral subgroup \(P_{i}\in\mathcal{P}\) and \(f_{i}\in G\) such that \(K_{i}<P_{i}^{f_{i}}\).
+
+We will construct an ascending sequence of relatively quasiconvex subgroups
+
+\[Q=Q_{0}<Q_{1}<\dots<Q_{n}=H\]
+
+such that for each \(k\in\{1,\dots,n\}\) the following properties hold.
+
+1. (1)For \(0\leq j\leq k\), the subgroup \(Q_{k}\cap P_{j}^{f_{j}}\) is finite index in \(P_{j}^{f_{j}}\).
+2. (2)\(\{Q_{k}\cap P_{1}^{f_{1}},\dots,Q_{k}\cap P_{k}^{f_{k}},K_{k+1},\dots,K_{n}\}\) is a collection of representatives of the maximal parabolic subgroups of \(Q_{k}\) up to conjugation in \(Q_{k}\).
+3. (3)\(g\not\in Q_{k}\).
+
+It will follow that \(H=Q_{n}\) is a fully quasiconvex subgroup which contains \(Q\) and does not contain the element \(g\).
+
+Choose a geodesic \(p\) from \(1\) to \(g\) in the relative Cayley graph \(\Gamma(G,\mathcal{P},S)\). Let \(L\) be an upper bound for the \(S\)–length of the \(P_{k}\)–components of the path \(p\).
+
+We now show how to construct \(Q_{k}\), assuming that \(Q_{k-1}\) has already been constructed. Let \(C\) and \(c\) be the constants provided by Theorem 3.9 for the subgroups \(Q_{k-1}\) and \(P_{k}^{f_{k}}\). Since \(P_{k}^{f_{k}}\) is slender, \(K_{k}=Q_{k-1}\cap P_{k}^{f_{k}}\) is finitely generated. Define a finite set \(F\subset P_{k}^{f_{k}}\setminus K_{k}\) by
+
+\[F=\left\{\begin{array}[]{ll}\{p\in P_{k}^{f_{k}}\setminus K_{k}\mid|p|_{S}\leq L+C+c\}\cup\{g\}&\mbox{if }g\in P_{k}^{f_{k}}\\ \{p\in P_{k}^{f_{k}}\setminus K_{k}\mid|p|_{S}\leq L+C+c\}&\mbox{otherwise.}\end{array}\right.\]
+
+Because \(P_{k}^{f_{k}}\cong P_{k}\) is LERF, we may find a finite index subgroup \(R_{k}\) of \(P_{k}^{f_{k}}\) satisfying
+
+* •\(K_{k}<R_{k}\), and
+* •\(f\notin R_{k}\) for all \(f\in F\).
+
+In particular, \(|h|_{S}>L+C+c\) for any \(h\in R_{k}\setminus Q_{k-1}\). Let \(Q_{k}=\langle Q_{k-1}\cup R_{k}\rangle\). Note that the hypotheses of the combination Theorem 3.9 (and hence those of Corollary 3.10) are satisfied for the relatively quasiconvex subgroup \(Q_{k-1}\) and the parabolic subgroup \(R_{k}\).
+
+We now verify properties (1)–(3) for the subgroup \(Q_{k}\) just constructed. Property (1) follows from the fact that \(Q_{k}\) contains \(Q_{j}\cap P_{j}^{f_{j}}\) for each \(j\) between \(1\) and \(k-1\), and also contains \(R_{k}\). Corollary 3.10 implies that property (2) is satisfied. By Theorem 3.9(3) any geodesic in the Cayley graph \(\Gamma(G,S\cup\bigcup\mathcal{P})\) from \(1\) to an element of element of \(Q_{k}\) which is not in \(Q_{k-1}\cup R_{k}\) has a \(P_{k}\)–component of \(S\)–length greater than \(L+C\); it follows that the element \(g\) does not belong to \(Q_{k}\), and so property (3) is also satisfied. This concludes the construction of the group \(Q_{k}\), and the theorem follows by taking \(H=Q_{n}\). ∎
+
+## 4. Fillings of Relatively Hyperbolic Groups
+
+Let \(G\) be torsion free and relatively hyperbolic, relative to a collection of subgroups \(\mathcal{P}=\{P_{1},\dots,P_{m}\}\), and let \(S\subset G\) be a finite generating set of \(G\). Suppose that \(S\cap P_{i}\) is a generating set of \(P_{i}\) for each \(i\).
+
+**Definition 4.1****.**: A _filling_ of \(G\) is determined by a collection of subgroups \(\{N_{i}\}_{i=1}^{m}\) such that for each \(i\), \(N_{i}\) is a normal subgroup of \(P_{i}\); these subgroups are called _filling kernels_. The quotient of \(G\) by the normal subgroup generated by \(\bigcup_{i=1}^{m}N_{i}\) is denoted by \(G(N_{1},\dots,N_{m})\).
+
+The following result is due (in the present setting) independently to Groves and Manning [14, Theorem 7.2 and Corollary 9.7] and to Osin [27, Theorem 1.1]. (Osin actually proves a more general result, in which \(G\) may have torsion.)
+
+**Theorem 4.2****.**: _[_14, 27_]_ _Let \(F\) be a finite subset of \(G\). There exists a constant \(B\) depending on \(G\), \(\mathcal{P}\), \(S\), and \(F\) with the following property. If a collection of filling kernels \(\{N_{i}\}_{i=1}^{m}\) satisfies \(|f|_{S}>B\) for every nontrivial \(f\in\cup_{i}N_{i}\), then_
+
+1. (1)_the natural map_ \(\imath_{i}:P_{i}/N_{i}\longrightarrow G(N_{1},\dots,N_{m})\) _is injective,_
+2. (2)\(G(N_{1},\dots,N_{m})\) _is relatively hyperbolic relative to_ \(\{\imath_{i}(P_{i}/N_{i})\}_{i=1}^{m}\)_, and_
+3. (3)_the projection_ \(G\longrightarrow G(N_{1},\dots,N_{m})\) _is injective on_ \(F\)_._
+
+### Fillings and Quasiconvex Subgroups
+
+Let \(H\) be a relatively quasiconvex subgroup of \(G\). A filling \(G\longrightarrow G(N_{1},\dots,N_{m})\) is an _\(H\)–filling_ if whenever \(H\cap P_{i}^{g}\) is non-trivial, \(N_{i}^{g}\subset P_{i}^{g}\cap H\).
+
+**Theorem 4.4****.**: _[_2_, Propositions 4.3 and 4.5]_ _Let \(H<G\) be a finitely generated relatively quasiconvex subgroup and \(g\in G\setminus H\). There is a finite subset \(F\subset G\) depending on \(H\) and \(g\) with the following property._
+
+_If \(\pi\colon\, G\longrightarrow G(N_{1},\dots,N_{m})\) is an \(H\)–filling which is injective on \(F\), then \(\pi(H)\) is a relatively quasiconvex subgroup of \(G(N_{1},\dots,N_{m})\), and \(\pi(g)\not\in\pi(H)\)._
+
+**Remark 4.5****.**: Propositions 4.3 and 4.5 in [2] use a different definition of relative quasiconvexity than the one we use here. In particular, their definition requires the subgroup to be finitely generated. We show in Appendix A (specifically Corollary A.11), that the definition used in [2] is equivalent to the one we use here under the assumption that the subgroup is finitely generated.
+
+Proof of Theorem 1.9.: Suppose that every subgroup in \(\mathcal{P}\) is LERF and slender. Let \(Q\) be a relatively quasiconvex subgroup of \(G\) and let \(g\) be an element of \(G\) not in \(Q\). By Theorem 1.7, there is a fully quasiconvex subgroup \(H\) which contains \(Q\) and does not contain \(g\). We must choose filling kernels \(\{N_{i}\}_{i=1}^{m}\) such that \(\pi\colon\, G\longrightarrow G(N_{1},\dots,N_{m})\) is an \(H\)–filling, \(G(N_{1},\dots,N_{m})\) is a word hyperbolic group, \(\pi(H)\) is a quasiconvex subgroup, and \(\pi(g)\not\in\pi(H)\).
+
+By Theorem 3.6, there is a collection \(\{K_{1},\dots,K_{n}\}\) of representatives of the infinite maximal parabolic subgroups of \(H\) up to conjugacy in \(H\). For each \(r\in\{1,\dots,n\}\) there is an integer \(i_{r}\in\{1,\dots,m\}\) and an element \(f_{r}\in G\) such that \(K_{r}^{f_{r}}\) is a finite index subgroup of \(P_{i_{r}}\). The index \(i_{r}\) is determined by \(r\), but there may be many distinct \(f_{r}\) with this property. On the other hand, there will be only finitely many conjugates \({K_{i}}^{g}\) in \(P_{i_{r}}\). Let \(I_{r}\) be the intersection of these conjugates; the group \(I_{r}\) is a finite index normal subgroup of \(P_{i_{r}}\). For \(i\in\{1,\dots,m\}\) define the subgroup \(M_{i}\) of \(P_{i}\) as
+
+\[M_{i}=\left\{\begin{array}[]{ll}P_{i}&\textrm{if $\{r\mid i_{r}=i\}=\emptyset$ }\\ &\\ \bigcap\{I_{r}\mid i_{r}=i\}&\textrm{if $\{r\mid i_{r}=i\}\not=\emptyset$ }.\end{array}\right.\]
+
+Put another way, if some conjugate of \(H\) intersects \(P_{i}\) nontrivially,
+
+\[M_{i}=\bigcap\{{K_{r}}^{g}\mid{K_{r}}^{g}\cap{P_{i}}\neq\{1\}\mbox{, }r\in\{1,\ldots,n\}\mbox{, and }g\in G\}.\]
+
+In other words, \(M_{i}\) is the intersection of \(P_{i}\) with all the \({K_{r}}^{g}\) which lie in \(P_{i}\). Because \(M_{i}\) is a finite intersection of finite index normal subgroups of \(P_{i}\), \(M_{i}\) is a finite index normal subgroup of \(P_{i}\). From the definition, \(M_{i}<{K_{r}}^{g}\) whenever \(r\in\{r\mid i_{r}=i\}\) and \({K_{r}}^{g}<P_{i}\).
+
+By Theorem 4.4, there is a finite subset \(F\subset G\) such that if \(\pi\colon\, G\longrightarrow\bar{G}\) is any \(H\)–filling which is injective on \(F\), then \(\pi(g)\not\in\pi(H)\) and \(\pi(H)\) is relatively quasiconvex. Let \(B>0\) be the constant from Theorem 4.2, applied to this finite subset \(F\).
+
+Since each \(P_{i}\) is residually finite, there is some finite index \(\hat{N_{i}}\lhd P_{i}\) so that \(|p|_{S}>B\) for all \(p\in\hat{N}_{i}\setminus\{1\}\). Let \(N_{i}=\hat{N}_{i}\cap M_{i}\). Since \(N_{i}\) is an intersection of two finite index normal subgroups of \(P_{i}\), \(N_{i}\) is a finite index normal subgroup of \(P_{i}\). Moreover, \(|p|_{S}>B\) for every \(p\in N_{i}\setminus\{1\}\).
+
+**Lemma 4.6****.**: \(G\to G(N_{1},\ldots,N_{m})\) _is an \(H\)–filling._
+
+Proof.: Let \(P_{i}\in\mathcal{P}\), \(f\in G\), and suppose that \(H\cap{P_{i}}^{f}\neq\{1\}\). We must show that \({N_{i}}^{f}<H\). Obviously it suffices to show that \({M_{i}}^{f}<H\). Since \(G\) is torsion free, \(H\cap{P_{i}}^{f}\) is infinite and \(H\cap{P_{i}}^{f}={K_{r}}^{h}\) for some \(h\in H\). The group
+
+\[{M_{i}}^{f}=\bigcap\{{K_{r}}^{g}\mid{K_{r}}^{g}\cap{P_{i}}^{f}\neq\{1\}\mbox{, }r\in\{1,\ldots,n\}\mbox{, and }g\in G\}\]
+
+must therefore be contained in \(H\cap{P_{i}}^{f}\), and thus in \(H\). ∎
+
+We now claim the subgroup \(H<G\) and the filling \(G\to\bar{G}:=G(N_{1},\ldots,N_{m})\) satisfy the conclusions of Theorem 1.9. Indeed, conclusion (1) of Theorem 1.9 is satisfied by construction. By Theorem 4.2.(2), the quotient \(\bar{G}\) is hyperbolic relative to a collection of finite groups, hence \(\bar{G}\) is hyperbolic. Conclusion (2) is thus established. According to Lemma 4.6, \(G\to\bar{G}\) is an \(H\)–filling. By Theorem 4.2.(3), \(G\to\bar{G}\) is injective on \(F\). Since the peripheral subgroups are slender, \(H\) is finitely generated [18, Corollary 9.2]. We therefore may apply Theorem 4.4 to obtain conclusions (3) and (4) of Theorem 1.9. Having established all the conclusions, we have proved the theorem. ∎
+
+## 5. Applications to \(3\)–manifolds
+
+### Kleinian groups
+
+Recall that a _Kleinian group_ is a discrete subgroup of \(\mathrm{Isom}(\mathbb{H}^{3})\), the group of isometries of hyperbolic \(3\)–space. In this subsection we show that if all hyperbolic groups are residually finite, then all finitely generated Kleinian groups are LERF.
+
+Proof of Corollary 1.5.: By Selberg’s lemma [4], every finitely generated Kleinian group contains a torsion-free subgroup of finite-index. Applying Corollary 2.4 it suffices to consider torsion-free Kleinian groups. Let \(G\) be a torsion-free Kleinian group and let \(H<G\) be a finitely generated subgroup. There are two cases:
+
+**Case 1****.**: \(\mathbb{H}^{3}/G\) has finite volume.
+
+Either \(H\) is geometrically finite or not. Suppose first that \(H\) is not geometrically finite. By a result of Canary [9, Corollary 8.3] together with the positive solution to the Tameness Conjecture [1, 8], \(H\) must be a _virtual fiber subgroup_ of \(G\). This implies that there is a finite index subgroup of \(G_{0}<G\) whose intersection \(H_{0}\) with \(H\) is normal in \(G_{0}\), and so that \(G_{0}/H_{0}\cong\mathbb{Z}\). The group \(H_{0}\) is obviously separable in \(G_{0}\), so \(H\) is separable in \(G\) by Corollary 2.4.
+
+If \(H\) is geometrically finite, then it is relatively quasiconvex, by Theorem 1.3, and we may apply Theorem 1.2.
+
+**Case 2****.**: \(\mathbb{H}^{3}/G\) has infinite volume.
+
+In this case it follows from the Scott core theorem and Thurston’s geometrization theorem for Haken manifolds that \(G\) is isomorphic to a geometrically finite Kleinian group \(G^{\prime}\) (see, for example, [23, Theorem 4.10]). If \(H^{\prime}\) is the image of \(H\) in \(G^{\prime}\) then obviously \(H\) is separable in \(G\) if and only if \(H^{\prime}\) is separable in \(G^{\prime}\). We therefore may as well assume that \(G\) is geometrically finite to begin with.
+
+Since \(G\) is geometrically finite and infinite covolume, every finitely generated subgroup of \(G\) is geometrically finite by an argument of Thurston (see [25, Proposition 7.1] or [23, Theorem 3.11] for a proof). In particular, \(H\) is geometrically finite, so Theorem 1.3 implies that \(H\) is a relatively quasiconvex subgroup of \(G\). To finish, we apply Theorem 1.2 again. ∎
+
+### Subgroup separability in finite volume \(3\)–manifolds
+
+Here we prove that if compact hyperbolic \(3\)–manifold groups are QCERF then finite volume hyperbolic \(3\)–manifold groups are LERF.
+
+**Proposition 5.3****.**: _If all fundamental groups of compact hyperbolic \(3\)–manifolds are QCERF, then all fundamental groups of finite volume hyperbolic \(3\)–orbifolds are LERF._
+
+Proof.: Let \(G=\pi_{1}(M)\), where \(M\) is some finite volume hyperbolic \(3\)–orbifold. Applying Corollary 2.4, we may pass to a finite cover and assume that \(M\) is an orientable manifold. It follows that \(G\) is relatively hyperbolic, relative to some finite collection \(\mathcal{P}=\{P_{i},\ldots,P_{m}\}\) of subgroups, each of which is isomorphic to \(\mathbb{Z}\oplus\mathbb{Z}\).
+
+Now suppose \(Q<G\) is some finitely generated subgroup. We must show that \(Q\) is separable. If \(Q\) is geometrically infinite, then we argue as we did in the proof of 1.5 that \(Q\) is the fundamental group of a virtual fiber, and thus separable. We may thus suppose that \(Q\) is geometrically finite, and therefore a relatively quasiconvex subgroup of \(G\).
+
+Let \(g\in G\setminus Q\). We then apply Theorem 1.7 to enlarge \(Q\) to a fully quasiconvex subgroup \(H\) not containing \(g\). Let \(F\subset G\) be the finite set obtained by applying Theorem 4.4 to \(G\), \(H\), and \(g\). Let \(B_{1}\) be the constant from Theorem 4.2 applied to \(G\) and \(F\), with respect to some generating set \(S\) for \(G\). We will choose a cyclic filling kernel \(N_{i}<P_{i}\) for each \(P_{i}\in\mathcal{P}\). The Hyperbolic Dehn Surgery Theorem of Thurston [28, 17] implies there is some constant \(B_{2}\) so that if the generators of the \(N_{i}\) are chosen to have length greater that \(B_{2}\), then the filling \(G(N_{1},\ldots,N_{m})\) will be the (orbifold) fundamental group of a hyperbolic orbifold obtained by attaching orbifold solid tori to the boundary components of a compact core of \(M\). Let \(B=\max\{B_{1},B_{2}\}\).
+
+For each \(P_{i}\in\mathcal{P}\) we choose some cyclic \(N_{i}<P_{i}\). For each \(i\) let \(n_{i}\in P_{i}\) satisfy \(|n_{i}|_{S}>B\). There are at most finitely many conjugates \({P_{i}}^{t_{1}},\ldots{P_{i}}^{t_{k}}\) so that \({P_{i}}^{t_{j}}\cap H\) is nonempty; for each such \(j\), the group \({P_{i}}^{t_{j}}\cap H\) is finite index in \({P_{i}}^{t_{j}}\). If there are no such conjugates, we choose \(N_{i}=\langle n_{i}\rangle\). Otherwise, we let \(N_{i}=\langle n_{i}^{\alpha}\rangle\), where the power \(\alpha\in\mathbb{N}\) is chosen so that \(n_{i}^{\alpha}\in H^{{t_{j}}^{-1}}\) for each \(j\), and so \(|n_{i}^{\alpha}|_{S}>B\).
+
+With the \(N_{i}<P_{i}\) chosen as above, the \(H\)–filling \(G(N_{1},\ldots,N_{m})\) is a compact hyperbolic \(3\)–orbifold group. Let \(\pi\colon\, G\to G(N_{1},\ldots,N_{m})\) be the quotient map. By Theorem 4.4, \(\pi(H)\) is a quasiconvex subgroup of \(G(N_{1},\ldots,N_{m})\), not containing \(\pi(g)\). By assumption, compact hyperbolic \(3\)–manifold groups (and thus compact hyperbolic \(3\)–orbifold groups) are QCERF. There is therefore some finite group \(F\), and some \(\phi\colon\, G(N_{1},\ldots,N_{m})\to F\) with \(\phi(\pi(g))\notin\phi(\pi(H))\). Since \(\phi(\pi(H))\) contains \(\phi(\pi(Q))\), we have separated \(g\) from \(Q\) in a finite quotient. ∎
+
+**Remark 5.4****.**: No part of the proof of Proposition 5.3 rests in an essential way on the results in this paper or in [2], but only on facts about hyperbolic \(3\)–manifolds and \(3\)–orbifolds which could be deduced by geometric arguments, based on the Gromov-Thurston \(2\pi\) Theorem. On the other hand, Proposition 5.3 is a nice illustration of the general principle that a relatively hyperbolic group which can be approximated by QCERF Dehn fillings must itself be QCERF.
+
+## 6. Acknowledgments
+
+Thanks very much to Ian Agol and Daniel Groves, both for useful conversations, and for helpful comments on an earlier draft of this note. We also thank the referee for insightful comments and corrections.
+
+Manning was partially supported by NSF grant DMS-0804369. Martínez-Pedroza was funded as a Britton Postdoctoral Fellow at McMaster University, grant A4000.
+
+## Appendix A On the equivalence of definitions of quasiconvexity
+
+The current paper relies heavily on results about relatively quasiconvex subgroups of relatively hyperbolic groups proved in the papers [22] and [2]. These papers unfortunately use different definitions of relative quasiconvexity, but we show in this appendix that the two definitions agree, at least for finitely generated subgroups.
+
+First, a few words about the literature. Dahmani [12] and Osin [26] studied classes of subgroups of relatively hyperbolic groups which they called _relatively quasiconvex_, intending to generalize the notion of quasiconvexity in hyperbolic groups. Dahmani’s definition was a dynamical one, whereas Osin’s was Definition 3.4; Osin’s definition was used in [22]. Hruska in [18] gave several definitions of relative quasiconvexity in the setting of countable (not necessarily finitely generated) relatively hyperbolic groups, including definitions based on Osin’s and Dahmani’s, and showed that they are equivalent. The authors of [2] were mainly interested in relatively hyperbolic structures on groups which were already hyperbolic, and used a definition of relative quasiconvexity (based more closely on the usual metric notion of quasiconvexity) different from any of those in [18]. The definition in [2] applies only to a finitely generated subgroup of a finitely generated relatively hyperbolic group.
+
+Throughout this section \(G\) will be relatively hyperbolic, relative to a finite collection of subgroups \(\mathcal{P}=\{P_{1},\dots,P_{n}\}\), and \(S\) will be a finite generating set for \(G\). The _cusped space_ (recalled below) for \(G\) with respect to \(\mathcal{P}\) and \(S\) will be denoted by \(X(G,\mathcal{P},S)\), and \(d(\cdot,\cdot)\) will denote the path metric on the cusped space. In particular, we will not need the word metric on \(G\), but only the metric induced by this path metric. For more detailed definitions and background on cusped spaces for relatively hyperbolic groups we refer the reader to [14] and [18]; we sketch the construction and recall some terminology here for the reader’s convenience.
+
+Let \(A\) be a discrete metric space with metric \(\rho\). The _combinatorial horoball based on \(A\)_ is a graph \(\mathcal{H}(A)\) with vertex set \(A\times\mathbb{Z}_{\geq 0}\), so that
+
+* •\((a,n)\) is connected by an edge to \((a,n+1)\) for any \(a\in A\), and
+* •for \(n\geq 1\), \((a,n)\) is connected to \((a^{\prime},n)\) whenever \(\rho(a,a^{\prime})\leq 2^{n}\).
+
+Edges of the first type are called _vertical_; edges of the second type are called _horizontal_. We say that a vertex \((a,n)\) of \(\mathcal{H}(A)\) has _depth \(n\)_. If \(A\) is a subset of a path metric space \(Y\), we may _attach a horoball to \(Y\) along \(A\)_ by gluing \(A\subseteq Y\) to \(A\times\{0\}\subset\mathcal{H}(A)\), and taking the obvious path metric on the union. If \(G\) is finitely generated by \(S\), we take \(Y\) to be the Cayley graph of \(G\) with respect to \(S\); any subset of \(G\) inherits a discrete metric from the path metric on the Cayley graph. The _cusped space_\(X(G,\mathcal{P},S)\) is the space obtained by simultaneously attaching horoballs to \(Y\) along all left cosets \(tP\) for \(P\in\mathcal{P}\). A vertex \(v\) of \(X(G,\mathcal{P},S)\) corresponds either to a group element, if it lies in the Cayley graph of \(G\), or otherwise to a triple \((tP,g,n)\) where \(tP\) is a left coset of an element of \(\mathcal{P}\), the element \(g\) lies in \(tP\), and \(n>0\) is the depth of \(v\) in the attached horoball \(\mathcal{H}(tP)\). In what follows we do not distinguish between \(v\) and the corresponding group element or triple.
+
+The group \(G\) is relatively hyperbolic, relative to \(\mathcal{P}\), if and only if the space \(X(G,\mathcal{P},S)\) is Gromov hyperbolic (see for example [14, Theorem 3.25]). If so, then \(G\) acts on \(X(G,\mathcal{P},S)\)_geometrically finitely_,¹ meaning in particular:
+
+Footnote 1: Hruska uses the term “cofinitely.”
+
+1. (1)Given any \(n\geq 0\), let \(X_{n}\) be the subset of \(X(G,\mathcal{P},S)\) obtained by deleting all vertices of height greater than \(n\). Then \(G\) acts cocompactly on \(X_{n}\) (which is an example of what Hruska calls a _truncated space_ for the action of \(G\) on \(X(G,\mathcal{P},S)\)).
+2. (2)For fixed \(n\), there are only finitely many components of of \(X(G,\mathcal{P},S)\setminus X_{n}\), up to the action of \(G\). (The components of \(X(G,\mathcal{P},S)\setminus X_{n-1}\) are called _\(n\)–horoballs_, for \(n\geq 1\). If \(n\) is understood, we call them _horoballs_. A \(0\)–horoball is a \(1\)–neighborhood of a \(1\)–horoball, and is equal to \(\mathcal{H}(tP)\) for some coset \(tP\) of some \(P\in\mathcal{P}\).)
+
+### Relatively Quasiconvex Subgroups according to Agol–Groves–Manning.
+
+Suppose that \(H\) is a relatively hyperbolic group, and let \(\mathcal{D}=\{D_{1},\dots,D_{m}\}\) be the peripheral subgroups of \(H\) and \(T\) a finite generating set for \(H\). Let \(\phi\colon\, H\to G\) be a homomorphism. If every \(\phi(D_{i})\in\mathcal{D}\) is conjugate in \(G\) into some \(P_{j}\in\mathcal{P}\), we say that the map \(\phi\)_respects the peripheral structure on_\(H\).
+
+Given such a map \(\phi\), one can extend it to a map \(\check{\phi}\) between zero-skeletons of cusped spaces in the following way: For each \(D_{i}\in\mathcal{D}\), choose an element \(c_{i}\in G\) (of minimal length) and some \(P_{j_{i}}\) such that \(\phi(D_{i})\subseteq cP_{j_{i}}c^{-1}\). For \(h\in H\), \(\check{\phi}(h)=\phi(h)\). For a vertex \((sD_{i},h,n)\) in a horoball of \(X(H,\mathcal{D},T)\), define
+
+\[\check{\phi}(sD_{i},h,n)=(\phi(s)c_{i}P_{j_{i}},\phi(h)c_{i},n).\]
+
+**Lemma A.1****.**: _[_2_, Lemma 3.1]_ _Let \(\phi\colon\, H\to G\) be a homomorphism which respects the peripheral structure on \(H\). The extension \(\check{\phi}\) defined above is \(H\)–equivariant and lipschitz. If \(\phi\) is injective, then \(\check{\phi}\) is proper._
+
+Recall that, for \(C\geq 0\), a subset \(A\) of a geodesic metric space \(X\) is _\(C\)–quasiconvex_ if every geodesic with endpoints in \(A\) lies in a \(C\)–neighborhood of \(A\). The subset is _quasiconvex_ if it is \(C\)–quasiconvex for some \(C\). The following is a slight paraphrase of the definition from [2]:
+
+**Definition A.2****.**: (QC-AGM) [2, Definition 3.11] Let \(G\) be as above, and let \(H<G\) be finitely generated by a set \(T\). We say that \(H\) is _(QC-AGM) relatively quasiconvex_ in \((G,\mathcal{P})\) if, for some finite collection of subgroups \(\mathcal{D}\) of \(H\),
+
+1. (1)\(H\) is relatively hyperbolic, relative to \(\mathcal{D}\), and
+2. (2)if \(\iota\colon\, H\to G\) is the inclusion, then the map \(\check{\iota}\colon\, X(H,\mathcal{D},T)^{0}\to X(G,\mathcal{P},S)^{0}\) described in Lemma A.1 has quasiconvex image.
+
+### Relatively Quasiconvex Subgroups according to Hruska.
+
+The following definition is direct from Hruska [18], where it is called _QC-3_. Hruska shows in [18] that this definition is equivalent to several others, including our Definition 3.4.
+
+**Definition A.3****.**: (QC-H)[18, Definition 6.6] A subgroup \(H\leq G\) is _(QC-H) relatively quasiconvex_ if the following holds. Let \((X,\rho)\) be some (any) proper Gromov hyperbolic space on which \((G,\mathcal{P})\) acts geometrically finitely. Let \(X-U\) be some (any) truncated space for \(G\) acting on \(X\). For some (any) basepoint \(x\in X-U\) there is a constant \(\mu\geq 0\) such that whenever \(c\) is a geodesic in \(X\) with endpoints in the orbit \(Hx\), we have
+
+\[c\cap(X-U)\subseteq\mathcal{N}_{\mu}({Hx}),\]
+
+where the neighborhood is taken with respect to the metric \(\rho\) on \(X\).
+
+**Remark A.4****.**: The meaning of “some (any)” in Definition A.3 just means that the word “some” can be replaced by “any” without affecting which subgroups of \(G\) are (QC-H) relatively quasiconvex. Thus “Definition” A.3 has some non-definitional content, established in [18, Proposition 7.5 and 7.6].)
+
+**Definition A.5****.**: Let \(A\subset X=X(G,\mathcal{P},S)\) be a horoball, and let \(R>0\). We say that a geodesic _\(\gamma\) penetrates the horoball \(A\) to depth \(R\)_ if there is a point \(p\in\gamma\cap A\) at distance at least \(R\) from \(X\setminus A\). We say that _\(A\) is \(R\)–penetrated by the subgroup \(H\)_ if there is a geodesic \(\gamma\) with endpoints in \(H\) penetrates the horoball \(A\) to depth \(R\).
+
+The goal of this subsection is to prove the following proposition.
+
+**Proposition A.6****.**: _Let \(H<G\) be (QC-H) relatively quasiconvex. Then there is a constant \(R=R(G,\mathcal{P},S,H)\), so that whenever a \(0\)–horoball is \(R\)–penetrated by \(H\), the intersection of \(H\) with the stabilizer of that horoball is infinite._
+
+Before the proof, we quote a proposition from [18] and prove two lemmas.
+
+**Proposition A.7****.**: _[_18_, Proposition 9.4]_ _Let \(G\) have a proper, left invariant metric \(d\), and suppose \(xH\) and \(yK\) are arbitrary left cosets of subgroups of \(G\). For each constant \(L\) there is a constant \(L^{\prime}=L^{\prime}(G,d,xH,yK)\) so that in the metric space \((G,d)\) we have_
+
+\[\mathcal{N}_{L}({xH})\cap\mathcal{N}_{L}({yK})\subseteq\mathcal{N}_{L^{\prime}}({xHx^{-1}\cap yKy^{-1}}).\]
+
+**Lemma A.8****.**: _Let \(H\) be a (QC-H) relatively quasiconvex subgroup of \(G\). Let \(A\) be a \(0\)–horoball of \(X(G,\mathcal{P},S)\), whose stabilizer is \(P^{t}\) for \(P\in\mathcal{P}\). If \(A\) is \(R\)–penetrated by \(H\) for all \(R>0\), then \(H\cap P^{t}\) is infinite._
+
+Proof.: It suffices to show that, for every \(M>0\), there is some \(h\) in \(H\cap P^{t}\) with \(d(1,h)>M\).
+
+Let \(\mu\) be the quasiconvexity constant of Definition A.3 for \(H\) and the space \(X^{\prime}\) which consists of all vertices in \(X(G,\mathcal{P},S)\) at depth \(0\). Let \(C\) be the constant given by Proposition A.7 such that
+
+\[\mathcal{N}_{\mu}({H})\cap tP\subseteq\mathcal{N}_{C}({H\cap tPt^{-1}}),\]
+
+where the neighborhoods are taken in the cusped space.
+
+Suppose that \(\gamma\) is a geodesic with endpoints in \(H\) which penetrates the horoball \(A\) to depth \(M+C\). The first and last points of \(\gamma\cap A\) are group elements, \(a\) and \(b\), both in the coset \(tP\). Since \(H\) is (QC-H) relatively quasiconvex, \(a\) and \(b\) are elements of \(\mathcal{N}_{\mu}({H})\cap tP\) and therefore (using Proposition A.7) there are elements \(h_{1}\) and \(h_{2}\) in \(H\cap P^{t}\) such that \(d(h_{1},a)\leq C\) and \(d(h_{2},b)\leq C\). Since \(d(a,b)\geq 2(M+C)\),
+
+\[d(1,h_{1}^{-1}h_{2})=d(h_{1},h_{2})\geq 2(M+C)-2C\geq 2M>M.\qed\]
+
+**Lemma A.9****.**: _Let \(H\) be a (QC-H) relatively quasiconvex subgroup of \(G\). Let \(\mu\) be the quasiconvexity constant of Definition A.3 for \(H\) and the space \(X^{\prime}=X_{0}\) which is obtained from \(X(G,\mathcal{P},S)\) by deleting all vertices of positive depth._
+
+_Let \(R>0\), and let \(A\) be a \(0\)–horoball, stabilized by \(P^{t}\) for \(P\in\mathcal{P}\). If \(A\) is \(R\)–penetrated by \(H\), then there is a horoball \(A^{\prime}\) so that_
+
+1. (1)\(A^{\prime}=hA\) _for some_ \(h\in H\)_,_
+2. (2)\(d(A^{\prime},1)\leq\mu\)_, and_
+3. (3)\(A^{\prime}\) _is_ \(R\)_–penetrated by_ \(H\)_._
+
+Proof.: Suppose that \(\gamma\) is a geodesic with endpoints \(h_{1}\) and \(h_{2}\) in \(H\) which penetrates the horoball \(A\) to depth \(R\). Let \(a\) and \(b\) be the first and last vertices of \(\gamma\cap A\). By (QC-H) relative quasiconvexity, there is some \(h\in H\) so that \(d(a,h)\leq\mu\). The geodesic \(h^{-1}\gamma\) goes between \(h^{-1}h_{2}\) and \(h^{-1}h_{2}\), and penetrates the horoball \(A^{\prime}=h^{-1}A\) to depth \(R\). Moreover,
+
+\[d(1,A^{\prime})\leq d(1,h^{-1}a)=d(a,h)\leq\mu.\qed\]
+
+Proof of Proposition A.6.: Suppose there is no such number \(R\). There must be a sequence of integers \(R_{i}\to\infty\) and a sequence of \(0\)–horoballs \(\{A_{i}\}\), so that, for each \(i\), the horoball \(A_{i}\) is \(R_{i}\)–penetrated by \(H\), but the intersection of the stabilizer of \(A_{i}\) with \(H\) is finite.
+
+For \(h\in H\), the stabilizer of \(hA_{i}\) is conjugate (by \(h\)) to the stabilizer of \(A_{i}\). Using Lemma A.9, we can therefore assume that \(d(1,A_{i})\leq\mu\) for each \(i\). By passing to a subsequence, we can therefore assume that the sequence \(\{A_{i}\}\) is constant. It follows that \(A_{0}\) is \(R_{i}\)–penetrated by \(H\) for all \(i\). Lemma A.8 then implies that the intersection of \(H\) with the stabilizer of \(A_{0}\) is infinite, which is a contradiction. ∎
+
+### Equivalence of the two definitions.
+
+In this section, \(G\) will be a relatively hyperbolic group, relative to a finite collection of subgroups \(\mathcal{P}\), and \(S\) will be a finite generating set for G. Let \(X(G,\mathcal{P},S)\) be the cusped space for \(G\) with respect to \(\mathcal{P}\) and \(S\), and let \(\delta\) be its hyperbolicity constant.
+
+**Theorem A.10****.**: _Let \(H\) be a finitely generated subgroup of \(G\). Then \(H\) is (QC-H) relatively quasiconvex if and only if \(H\) is (QC-AGM) relatively quasiconvex._
+
+Proof.: One direction is easy. Suppose that \(H<G\) is (QC-AGM) relatively quasiconvex, generated by the finite set \(T\), and with peripheral subgroups \(\mathcal{D}\). Recall that to define \(\check{\iota}\colon\, X(H,\mathcal{D},T)^{0}\longrightarrow X(G,\mathcal{P},S)^{0}\), an element \(c_{i}\in G\) was chosen for each \(D_{i}\in\mathcal{D}\) so that \(D_{i}\subset c_{i}P_{j_{i}}c_{i}^{-1}\). Let \(C=\max\{d(1,c_{i})\mid D_{i}\in\mathcal{D}\}\), and let \(C_{q}\) be the constant of quasiconvexity in the definition of (QC-AGM) quasiconvexity. As remarked at the beginning of the Appendix, the cusped space \(X=X(G,\mathcal{P},S)\) is acted on geometrically finitely by \(G\), and the subspace \(X-U=X_{0}\subset X(G,\mathcal{P},S)\) obtained by deleting \(1\)–horoballs is a truncated space for the action. Moreover, as explained in Remark A.4, it suffices to find a \(\mu\) which works for this choice of \(X\) and \(X-U\), and for the \(H\)–orbit of \(1\) in \(X\). Let \(x\), \(y\in H\), and let \(\gamma\) be any geodesic joining them in \(X\). Let \(z\) be a vertex of \(\gamma\) contained in \(X_{0}\). By (QC-AGM), there is some point \(w\in\check{\iota}(X(H,\mathcal{D},T)^{0})\) so that \(d(z,w)\leq C_{q}\). It follows that \(w\in X_{C_{q}}\), but any point in \(\check{\iota}(H,\mathcal{D},T)\cap X_{C_{q}}\) is at most \(C+C_{q}\) away from some point in \(H\). It follows that \(z\) is no further than \(\mu:=C+2C_{q}\) from \(H\), and so \(H\) is (QC-H) relatively quasiconvex.
+
+We now establish the other direction. Let \(H\) be a subgroup of \(G\), and suppose that \(H\) is (QC-H) relatively quasiconvex. Let \(\mathcal{D}\) be a collection of representatives of the \(H\)–conjugacy classes of infinite maximal parabolic subgroups of \(H\). By [18, Theorem 9.1], \(H\) is relatively hyperbolic, relative to \(\mathcal{D}\). By Lemma A.1, the inclusion \(\iota\colon\, H\longrightarrow G\) extends to a lipschitz map of (\(0\)–skeletons of) cusped spaces
+
+\[\check{\iota}\colon\, X(H,\mathcal{D},T)^{0}\longrightarrow X(G,\mathcal{P},S)^{0}.\]
+
+We need to prove that the image \(Y=\check{\iota}(X(H,\mathcal{D},T)^{0})\) of \(\check{\iota}\) is quasiconvex.
+
+Let \(R\) be the constant provided by Proposition A.6 for the subgroup \(H\). Let \(X^{\prime}=X_{100\delta+R}\) be the subspace of \(X(G,\mathcal{P},S)\) consisting of all vertices at depth at most \(100\delta+R\). Since \(H\) is (QC-H) relatively quasiconvex, there is a constant \(\mu\) such that for any geodesic \(\zeta\) in \(X(G,\mathcal{P},S)\) with endpoints in \(H\),
+
+\[\zeta\cap X^{\prime}\subset\mathcal{N}_{\mu}({H})\subset\mathcal{N}_{\mu}({Y}),\]
+
+where the neighborhoods are taken with respect to the metric on \(X(G,\mathcal{P},S)\).
+
+Let \(x\) and \(y\) be vertices of \(Y\) and let \(\gamma\) be a geodesic between them. We will show that the vertices of \(\gamma\) are contained in the \(M\)-neighborhood of \(Y\), where \(M\) is a constant independent of \(x\), \(y\), and \(\gamma\). We divide the proof into five (not necessarily disjoint) cases.
+
+**Case 1****.**: The points \(x\) and \(y\) lie deeper than \(10\delta\) in the same horoball.
+
+By recalling some easily verified properties of the geometry of horoballs, we will show that \(\gamma\) is contained in the \(M_{1}\)-neighborhood of \(Y\), where
+
+\[M_{1}=6.\]
+
+To begin with, the \(10\delta\)–horoball containing \(x\) and \(y\) is convex (see [14, Lemma 3.26]). Second, any geodesic with the same endpoints as \(\gamma\) is Hausdorff distance at most \(4\) from \(\gamma\). Finally, there is a geodesic \(\gamma^{\prime}\) of a particularly nice form with the same endpoints as \(\gamma\). The geodesic \(\gamma^{\prime}\) is a _regular geodesic_, which means that all its edges are vertical, except for at most three consecutive horizontal edges at maximum depth (see [14, Lemma 3.10]). Since the vertical subsegments of \(\gamma^{\prime}\) start at points in \(Y\) and are vertical, they stay in \(Y\), and so \(\gamma^{\prime}\) stays in a \(2\)–neighborhood of \(Y\). As \(\gamma\) is contained in a \(4\)–neighborhood of \(\gamma^{\prime}\), we have \(\gamma\) contained in a \(6\)–neighborhood of \(Y\).
+
+**Case 2****.**: The points \(x\) and \(y\) are elements of \(H\), they are in the neighborhood of radius \(\mu\) of a horoball \(\mathcal{H}(tP)\), and the geodesic \(\gamma\) penetrates the horoball \(\mathcal{H}(tP)\) to depth larger than \(100\delta+R\).
+
+In this case, we will approximate \(\gamma\) by a regular geodesic inside \(\mathcal{H}(tP)\) with (possibly different) endpoints in \(Y\). Without loss of generality, assume that \(x\) is the identity, and so \(d(1,t)\leq\mu\).
+
+By Proposition A.6, the intersection \(H\cap P^{t}\) is infinite. It follows that \(H\cap P^{t}=D^{s}\) for some \(D\in\mathcal{D}\) and \(s\in H\). We claim \(s\) can be chosen so that \(d(1,s)<K\) for a constant \(K\) independent of \(x\), \(y\), and \(\gamma\). Indeed, we observe that the set
+
+\[W=\{(r,P)\in G\times\mathcal{P}\mid d(1,r)\leq\mu,\#(H\cap P^{r})=\infty\}\]
+
+is finite. For each \(w=(r,P)\in W\) choose \(u_{w}\in H\) so that \(H\cap P^{r}=D^{u}\) for some \(D\in\mathcal{D}\); we let \(K\) be the maximum of \(d(1,u_{w})\) over all \(w\in W\).
+
+We further claim that there is an element \(y^{\prime}\in H\cap P^{t}\) such that \(d(y,y^{\prime})\leq L\), for a constant \(L\) independent of \(x\), \(y\), and \(\gamma\). Indeed, for each \(w=(r,P)\) in the set \(W\) defined above, Proposition A.7 implies we can find an \(L_{w}>0\) so that
+
+\[H\cap\mathcal{N}_{\mu}({rP})\subseteq\mathcal{N}_{L_{w}}({H\cap P^{r}});\]
+
+we let \(L\) be the maximum \(L_{w}\) over all \(w\in W\).
+
+Recall that to define \(\check{\iota}\colon\, X(H,\mathcal{D},T)^{0}\longrightarrow X(G,\mathcal{P},S)^{0}\), an element \(c_{i}\in G\) was chosen for each \(D_{i}\in\mathcal{D}\) so that \(D_{i}\subset c_{i}P_{j_{i}}c_{i}^{-1}\). Let
+
+(5) \[C=\max\{d(1,c_{i})\mid D_{i}\in\mathcal{D}\}.\]
+
+The subgroup \(D\) is equal to \(D_{i}\) for some \(i\), and we set \(c=c_{i}\) for the same \(i\).
+
+Consider the elements \((sD,s,10\delta)\) and \((sD,y^{\prime}s,10\delta)\) of \(X(H,\mathcal{D},T)\) and their corresponding images in \(Y\) given by \((scP,sc,10\delta)\) and \((scP,y^{\prime}sc,10\delta)\). The points \((scP,sc,10\delta)\) and \((scP,y^{\prime}sc,10\delta)\) belong to the same \(10\delta\)–horoball, which is convex in \(X(G,\mathcal{P},S)\), as we noted in Case 1. Also as noted in Case 1, there is a regular geodesic \(\gamma^{\prime}\) joining the points \((scP,sc,10\delta)\) and \((scP,y^{\prime}sc,10\delta)\); since the endpoints lie in \(Y\), the geodesic \(\gamma^{\prime}\) is contained in the \(2\)–neighborhood of \(Y\).
+
+On the other hand, the endpoints of the geodesics \(\gamma\) and \(\gamma^{\prime}\) are close, namely,
+
+\[d(1,(scP,sc,10\delta))\leq d(1,s)+d(1,c)+10\delta\leq K+C+10\delta,\]
+
+and
+
+\[d(y,(scP,y^{\prime}sc,10\delta))\leq d(y,y^{\prime})+d(y^{\prime},(scP,y^{\prime}sc,10\delta))\leq L+K+C+10\delta.\]
+
+Since \(X(G,\mathcal{P},S)\) is \(\delta\)–hyperbolic, the Hausdorff distance between \(\gamma^{\prime}\) and \(\gamma\) is at most the distance between endpoints plus \(2\delta\). Thus if
+
+\[M_{2}=2\delta+K+L+C+10\delta,\]
+
+then \(\gamma\) is contained in the \(M_{2}\)-neighborhood of \(Y\). This completes this case.
+
+**Case 3****.**: Suppose \(x\) and \(y\) are elements of \(H\).
+
+We split \(\gamma\) into subsegments \(\gamma_{1},\gamma_{2},\dots,\gamma_{k}\) such that no \(\gamma_{i}\) contains any group element (depth \(0\) vertex) in its interior, but the endpoints of each \(\gamma_{i}\) are group elements. Observe that each \(\gamma_{i}\) is either a single edge or a geodesic segment contained in a \(0\)–horoball. Furthermore, since \(H\) is (QC-H) relatively quasiconvex, the endpoints of each \(\gamma_{i}\) are contained in the \(\mu\)-neighborhood of \(H\). We claim that each \(\gamma_{i}\) is contained in the \(M_{3}\)-neighborhood of \(Y\), where
+
+\[M_{3}=110\delta+R+3\mu+2+M_{2}.\]
+
+If \(\gamma_{i}\) is an edge, then the claim is immediate, so we suppose \(\gamma_{i}\) is contained in a \(0\)–horoball \(\mathcal{A}\). First, suppose \(\gamma_{i}\) does not penetrate \(\mathcal{A}\) to depth \(110\delta+R+2\mu\). An easy argument shows that the length of a geodesic in a combinatorial horoball is at most twice its maximum depth plus \(4\), so we have \(|\gamma_{i}|<220\delta+2R+2\mu+4\), and \(\gamma_{i}\) is therefore contained in the \((110\delta+R+3\mu+2)\)–neighborhood of \(H\). In particular, \(\gamma_{i}\) is contained in \(M_{3}\)–neighborhood of \(Y\).
+
+Suppose on the other hand that \(\gamma_{i}\) penetrates the horoball \(\mathcal{A}\) to depth \(110\delta+R+2\mu\). Let \(h_{1}\) and \(h_{2}\) be elements of \(H\) which are at distance at most \(\mu\) from the endpoints of \(\gamma_{i}\), and let \(\alpha\) be a geodesic between them. Since \(X(G,\mathcal{P},S)\) is \(\delta\)–hyperbolic, the Hausdorff distance between \(\gamma_{i}\) and \(\alpha\) is at most \(2\delta+\mu\). It follows that \(\alpha\) penetrates the horoball \(\mathcal{A}\) to depth \(100\delta+R+\mu\), and hence it satisfies the condition of Case 2. Therefore, \(\gamma_{i}\) is in the \((2\delta+\mu+M_{2})\)–neighborhood (and hence in the \(M_{3}\)–neighborhood) of \(Y\).
+
+**Case 4****.**: Suppose \(x\) and \(y\) lie at depth no more than \(50\delta\) in \(X(G,\mathcal{P},S)\).
+
+If \(x\in Y\) lies in a \(1\)–horoball, then \(x=(tP,hc_{i},n)\) for some \(P\in\mathcal{P}\), some \(h\in H\), some \(i\in\{1,\ldots,m\}\), and some \(n\leq 50\delta\); otherwise, \(x\in H\). In any case, there is an element \(h_{1}\in H\) such that \(d(x,h_{1})\leq 50\delta+C\), where \(C\) is the constant defined in (5). By the same argument, there is an element \(h_{2}\in H\) such that \(d(x,h_{2})\leq 50\delta+C\). Since \(X(G,\mathcal{P},S)\) is \(\delta\)–hyperbolic, the Hausdorff distance between \(\gamma\) and any geodesic \(\gamma^{\prime}\) between \(h_{1}\) and \(h_{2}\) is at most \(52\delta+C\). We may apply Case 3 to \(\gamma^{\prime}\), and deduce that \(\gamma\) is contained in the \(M_{4}\)–neighborhood of \(Y\), where
+
+\[M_{4}=52\delta+C+M_{3}.\]
+
+**Case 5****.**: Suppose either \(x\) or \(y\) lies inside a \(50\delta\)–horoball, but we are not in Case 1.
+
+Here we follow the proof of the last case of [2, Proposition 3.12]. If \(x\) or \(y\) lies in a horoball, it is connected by a vertical path to a point in the right coset \(Hc_{i}\) at depth \(0\) in \(X(G,\mathcal{P},S)\). It is therefore possible to modify \(\gamma\) (by appending and deleting (mostly) vertical paths lying in a \(3\)–neighborhood of \(Y\)) to a \(10\delta\)–local geodesic \(\gamma^{\prime}\) with endpoints within \(C\) of \(H\); the geodesic \(\gamma\) is contained in a \(3\)–neighborhood of \(\gamma^{\prime}\cup Y\). By [7, III.H.1.13(3)], \(\gamma^{\prime}\) is a \((\frac{7}{3},2)\)–quasi-geodesic. Since quasi-geodesics track geodesics, there is a constant \(L_{Q}\) depending only on \(\delta\) and \(C\), and a geodesic \(\gamma^{\prime\prime}\) with endpoints in \(H\) such that the Hausdorff distance between \(\gamma^{\prime}\) and \(\gamma^{\prime\prime}\) is at most \(L_{Q}\). By Case 3, \(\gamma^{\prime\prime}\) is contained in the \(M_{2}\)-neighborhood of \(Y\). Let
+
+\[M_{5}=3+L_{Q}+M_{2},\]
+
+and observe that \(\gamma\) is contained in the \(M_{5}\)-neighborhood of \(Y\).
+
+Finally, we set \(M=\max\{M_{1},\ldots,M_{5}\}\), and note that \(M\) does not depend on the vertices \(x\) and \(y\) of \(Y\), or on the geodesic \(\gamma\) joining them. It follows that \(Y=\check{\iota}(X(H,\mathcal{D},T)^{0})\) is \(M\)–quasiconvex in \(X(H,\mathcal{D},T)\), and so \(H\) is (QC-AGM) relatively quasiconvex in \((G,\mathcal{P})\). ∎
+
+Applying the main result of Hruska [18] on the equivalence of various definitions of relative quasiconvexity (our Definition 3.4 is Hruska’s (QC-5), and our Definition A.3 (QC-H) is Hruska’s (QC-3)), we obtain the following useful fact.
+
+**Corollary A.11****.**: _Let \(G\) be relatively hyperbolic, relative to \(\mathcal{P}\), and let \(H\) be a finitely generated subgroup of \(G\). The following are equivalent:_
+
+1. (1)\(H\) _is a relatively quasiconvex subgroup of_ \(G\)_, in the sense of Definition_ 3.4_._
+2. (2)\(H\) _is a relatively quasiconvex subgroup of_ \(G\)_, in the sense of Definition_ A.2_._
+
+## Appendix B On extending the main result in the presence of torsion
+
+In this section, we give some idea of the changes necessary to prove Theorem 1.9 (and therefore Theorem 1.2) in the presence of torsion. In this section, \(G\) is a relatively hyperbolic group, hyperbolic relative to a finite collection \(\mathcal{P}\) of LERF and slender subgroups, and \(H\) is some relatively quasiconvex subgroup of \(G\).
+
+The main difference is that we must deal with the possibility that our relatively quasiconvex subgroup has finite but non-trivial maximal parabolic subgroups. Since a finite subgroup of a relatively hyperbolic group, may intersect arbitrary collections of parabolic subgroups, we have to ignore these intersections. This is already handled in the arguments of Section 3 by only amalgamating with parabolic subgroups which have infinite intersection with \(H\) to obtain the fully quasiconvex subgroup \(Q\).
+
+In Section 4, it is necessary to modify the definition of \(H\)–filling as follows:
+
+**Definition B.1****.**: (Alternate definition in the presence of torsion.) Let \(H\) be a relatively quasiconvex subgroup of \(G\). A filling \(G\longrightarrow G(N_{1},\dots,N_{m})\) is an _\(H\)–filling_ if whenever \(H\cap P_{i}^{g}\) is infinite, \(N_{i}^{g}\subset P_{i}^{g}\cap H\).
+
+With the new definition, we must check that the results from [2, Section 4.2] still hold. (We do not know how to prove the result about height from Section 4.3 of [2] in this more general setting, but we do not need it for our argument.) Examining the proofs from [2], the reader may check that it suffices to extend the technical [2, Lemma 4.2].
+
+We sketch how to do so briefly, for the experts: In [2, Lemma 4.2], the hypothesis of an \(H\)–filling is used to deduce the existence of a nontrivial element of \(H\) which is also in a conjugate of a filling kernel fixing a certain horoball from the fact that a geodesic between elements of \(H\) penetrates that horoball deeply. The heart of the argument is showing that if the geodesic penetrates the horoball deeply, the intersection of \(H\) with the horoball stabilizer is infinite. In the torsion-free setting, it suffices to show that the intersection is nontrivial. The proof in the presence of torsion is given in the previous appendix as A.6. With this proposition, one can prove the extended version of [2, Lemma 4.2] in a straightforward manner, choosing slightly different constants to take the constant \(R\) from Proposition A.6 into account.
+
+The proofs of Propositions 4.3 and 4.5 of [2] go through in exactly the same way, and we obtain the same statement as Theorem 4.4 above, but with the new meaning of \(H\)–filling. Using Osin’s Dehn filling result in place of Theorem 4.2, the rest of the proof of Theorem 1.9 goes through as written, with the exception that each mention of a condition of the form “\(A\cap B\neq\{1\}\)” for \(A\) and \(B\) subgroups of \(G\) should be replaced by “\(A\cap B\) is infinite”.
+
+## References
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+

arxiv_ground_truth/0902.1533.md

@@ -0,0 +1,123 @@
+# The Brera Multi-scale Wavelet _Chandra_ Survey. I. Serendipitous source catalogue
+
+ ,_a_ S. Campana,_b_ R.P. Mignani,_c_ A. Moretti,_b_ M. Mottini,_d_ M.R. Panzera,_b_ G. Tagliaferri_b_
+_a_INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica,
+Via U. La Malfa 153, I-90146 Palermo, Italy
+_b_INAF, Osservatorio Astronomico di Brera,
+Via E. Bianchi 46, I-23807 Merate, Italy
+_c_Mullard Space Science Laboratory, University College London,
+Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
+_d_European Southern Observatory,
+Schwarzschild Straße 2, 85740 Garching bei München, Germany
+E-mail
+
+###### Abstract:
+
+We present the Brera Multi-scale Wavelet _Chandra_ (BMW-_Chandra_) source catalogue drawn from essentially all _Chandra_ ACIS-I pointed observations with an exposure time in excess of 10 ks public as of March 2003 (136 observations). Using the wavelet detection algorithm developed by Lazzati et al. (1999) and Campana et al. (1999), which can characterise both point-like and extended sources, we identified 21325 sources. Among them, 16758 are serendipitous, i.e. not associated with the targets of the pointings. This makes our catalogue the largest compilation of _Chandra_ sources to date. The 0.5–10 keV absorption corrected fluxes of these sources range from \(\sim 3\times 10^{-16}\) to \(9\times 10^{-12}\) erg cm_-2_ s_-1_ with a median of \(7\times 10^{-15}\) erg cm_-2_ s_-1_. The catalogue consists of count rates and relative errors in three energy bands (total, 0.5–7 keV; soft, 0.5–2 keV; and hard, 2–7 keV), where the detection was performed, and source positions relative to the highest signal-to-noise detection among the three bands. The wavelet algorithm also provides an estimate of the extension of the source. We include information drawn from the headers of the original files, as well, and extracted source counts in four additional energy bands, SB1 (0.5–1 keV), SB2 (1–2 keV), HB1 (2–4 keV), and HB2 (4–7 keV). We computed the sky coverage for the full catalogue and for a subset at high Galactic latitude (\(\mid b\mid\,>20^{\circ}\)). Our sky coverage in the soft band (0.5–2 keV, S/N =3) is \(\sim 8\) deg² at a limiting flux of \(\sim 10^{-13}\) erg cm_-2_ s_-1_, and \(\sim 2\) deg² at a limiting flux of \(\sim 10^{-15}\) erg cm_-2_ s_-1_.
+
+## 1 Introduction
+
+The Brera Multi-scale Wavelet (BMW, [4, 1]) algorithm, which was developed to analyse _ROSAT_ High Resolution Imager (HRI) images [6], was modified to support the analysis of _Chandra_ Advanced CCD Imaging Spectrometer (ACIS) images [5], and subsequently led to interesting results on the nature of the cosmic X-ray background [2]. Differently from other WT-based algorithms, the BMW automatically characterises each source through a multi-source \(\chi^{2}\) fitting with respect to a Gaussian model in the wavelet space, and has therefore proven to perform well in crowded fields and in conditions of very low background [4]. Given the reliability and versatility of the BMW, we decided to apply it to a large sample of _Chandra_ ACIS-I images, to take full advantage of the superb spatial resolution of _Chandra_ [\(\sim 0.5\)” point-spread function (PSF) on-axis]. We thus produced the Brera Multi-scale Wavelet _Chandra_ Survey [7] and here we present a pre-release of this catalogue, which is based on a subset of the whole _Chandra_ ACIS observations dataset, roughly corresponding to the first three years of operations. Our catalogue provides source positions, count rates, extensions and relative errors.
+
+## 2 Method
+
+### Sample selection
+
+We chose the _Chandra_ fields which maximised the sky area not occupied by the pointed targets, that is the fields where the original PI was interested in a single, possibly point-like object centred in the field. Our criteria were the following:
+
+1. 1.All ACIS-I [no grating, no High Resolution Camera (HRC) fields and in Timed Exposure mode] fields with exposure time in excess of 10 ks available by 2003 March were considered. Data from all four front-illuminated (FI) CCDs (I0, I1, I2, I3) were used.
+2. 2.We excluded fields dominated by extended sources [covering more than 1/9 of the field of view (FOV)].
+3. 3.We excluded planet observations and supernova remnant observations.
+4. 4.We also excluded fields with bright point-like or high-surface brightness extended sources.
+5. 5.We put no limit on Galactic latitude, but we selected sub-samples based on latitude at a later time.
+
+The exclusion of bright point-like or high-surface brightness extended sources was dictated by the nature of our detection algorithm, which leads to an excessive number of spurious detections at the periphery of the bright source, which is a common problem to most detection algorithms. Therefore, each field was visually inspected to check for such effects (see Figure 1); when found, a conservatively large portion of the field was flagged. Of the 147 fields analysed, 11 (\(\sim 7\)%) were discarded because of problems at various stages of the pipeline execution. As a result of our selection, we retained 136 fields. Figure 2 (left) shows the Aitoff projection in Galactic coordinates of their positions. We note that several fields were observed more than once. These fields were considered as different pointings, so that the number of distinct fields is 94.
+
+Figure 1: Example of detection. **(a)** The \(\eta\) Carinae full field at half resolution. Note the complicated extended structure at the centre and the spurious detections along a readout streak (green arrow). **(b)** Central portion of the field at full resolution. Crosses mark sources that the detection algorithm classifies as extended (e.g. left-bottom corner and along readout streak). **(c)** Example of manual cleaning. The spurious sources along the readout streak were eliminated and **(d)** the sources in the central portion of the image (contained within the box and not shown) were flagged for later inspection.
+
+The data input in our pipeline are Level 2 (L2) data generated by the _Chandra_ X-ray Center (CXC) standard data processing in a uniform fashion, which were filtered to only include the standard event grades, and then corrected for aspect offsets. We applied energy filters to these event lists, and created soft (SB, 0.5–2.0 keV), hard (HB, 2.0–7.0 keV) and total (FB, 0.5–7.0 keV) band event files. The upper limit on our hard and total energy bands was chosen at 7 keV because at higher energy the background increases and the effective area decreases, producing lower signal-to-noise (S/N) data. Our results in the 0.5–10 keV band are then extrapolations from our findings in the 0.5–7 keV range.
+
+### The algorithm
+
+The main steps of the BMW algorithm can be summarised as follows (full details in [3]; [4]; [1]). The first step is the creation of the WT of the input image; the BMW WT is based on the discrete multi-resolution theory and on the “à trous” algorithm, which differs from continuous–WT-based algorithms which can sample more scales at the cost of a longer computing time. We used a Mexican hat mother-wavelet, which can be analytically approximated by the difference of two Gaussian functions. The WT decomposes each image into a set of sub-images, each of them carrying the information of the original image at a given scale. This property makes the WT well suited for the analysis of X-ray images, where the scale of sources is not constant over the field of view, because of the dependence of the PSF on the off-axis angle. We used 7 WT scales \(a=[1,2,4,8,16,32,64]\) pixels, to cover a wide range of source sizes, where \(a\) is the scale of the WT [4].
+
+Candidate sources are identified as local maxima above the significance threshold in the wavelet space at each scale, so that a list is obtained at each scale, and then a cross-match is performed among the 7 lists to merge them. At the end of this step, we have a first estimate of source positions (the pixel with the highest WT coefficient), source counts (the local maximum of the WT) and a guess of the source extension (the scale at which the WT is maximized). A critical parameter is the detection threshold which, in the context of WT algorithms, is usually fixed arbitrarily by the user in terms of expected spurious detections per field [3]. The number of expected spurious detections as a function of the threshold value and for each scale was calculated by means of Monte Carlo simulations [5].
+
+The final step is the characterisation of the sources by means of a multi-source \(\chi^{2}\) minimization with respect to a Gaussian model source in the WT space. In order to fit the model on a set of independent data, the WT coefficients are decimated according to a scheme described in full in [4].
+
+We ran the detection algorithm on the source images rebinned by a factor of 2 (1 pixel \(\sim 0.98\)”), and then in their inner \(512\times 512\) part at the full resolution, using 7 scales. We applied corrections to the source counts for vignetting and PSF modelling (i.e. for using a Gaussian to approximate the PSF function to fit the sources in wavelet space). We excluded the \(480\times 480\) pixel central part in the analysis at rebin 2, then cross-correlated the positions of the sources found at rebin 1 and 2 to exclude common double entries. We repeated this procedure for each of the three energy bands, and cross-correlated the resulting source coordinates to form the definitive list (for coincident sources, the coordinates of the highest S/N one were kept). We ran the detection algorithm with a single significance threshold that corresponds to \(\sim 0.1\) spurious detections per scale, hence (with 7 scales) \(\sim 0.7\) spurious detections per field for each band in which we performed the detection. Given our sample of 136 fields, we expect a total of \(\sim 95\) spurious sources in the catalogue, or a percentage of 2.7% (considering the three energy bands, the two images over which the detection was run, and the total number of detected sources, see Sect. 2.3). An example of the results of the detection is shown in Figure 1.
+
+### The catalogue
+
+The wavelet detection produced a catalogue of source positions, count rates, counts, extensions, and relative errors in three bands, as well as the additional information drawn from the headers of the original files for a total of 21325 sources. We also extracted source counts within a box centered around the positions determined with the detection algorithm, with a side which is the 90% encircled energy diameter at 1.50 keV. For the SB, HB, and FB bands the background counts were extracted from the same box from the background image. We extracted source counts in the four additional bands: SB1 (0.5–1.0 keV), SB2 (1.0–2.0 keV), HB1 (2.0–4.0 keV), and HB2 (4.0–7.0 keV). We calculated the 0.5–10 keV absorption corrected fluxes by converting the count rates in fluxes assuming a Crab spectrum, i.e. a power law with photon index 2.0, modified with the absorption by Galactic \(N_{\rm H}\) relative to each field; we also provide the 0.5–10 keV observed fluxes (simple Crab spectrum and \(N_{\rm H}=0\)). The catalogue lists the 0.5–10 keV observed flux, the absorption corrected one and the corresponding conversion factors [see Figure 2 (right)]. Problematic portions (such as extended pointed objects) and pointed objects (within a radius of 30 arcsec from the target position) were flagged.
+
+Figure 3 (left) shows the distribution of the source off-axis angle, which presents a steep increase with collecting area, and a gentler decrease with decreasing sensitivity with off-axis angles. Differently from what found with the BMW-HRI catalogue [6], our distribution does not present a peak at zero off-axis due to pointed sources. To characterise the source extension, which is one of the main features of the WT method, one cannot simply compare the WT width with the instrumental PSF at a given off-axis angle. Thus, we use a \(\sigma\)-clipping algorithm which divides the distribution of source extensions as a function of off-axis angle in bins of 1’ width. The mean and standard deviation are calculated within each bin and all sources which width exceeds 3\(\sigma\) the mean value are discarded. The procedure is repeated until convergence is reached. The advantage of this method is that it effectively eliminates truly extended sources, while providing a value for the mean and standard deviation in each bin [4]. The mean value plus the 3\(\sigma\) dispersion provides the line discriminating the source extension, but we conservatively classify as extended only the sources that lie 2\(\sigma\) above this limit. Combining this threshold with the 3\(\sigma\) on the intrinsic dispersion, we obtain a \(\sim 4.5\)\(\sigma\) confidence level for the extension classification.
+
+The full catalogue contains 21325 sources, 16834 of which are not associated with bright and/or extended sources, including the pointed ones. Of these, 11124 are detections in the total band, 12631 in the soft, 9775 in the hard band; 4203 sources were only detected in the hard band (see Table 1).
+
+It is particularly important for cosmological studies to have a sample which is not biased toward bright objects. To this end, we constructed the **BMW-**_Chandra_ **Serendipitous Source Catalogue that contains 16758 sources not associated with pointed objects**, by excluding sources within a radius of 30 arcsec from the target position. Their sky coverage is shown in Figure 3 (right).
+
+Figure 2: **Left**: Aitoff Projection in Galactic coordinates of the selected 136 Chandra ACIS-I fields. The thick lines are the limits for the high latitude sub-sample. **Right**: Distribution of the absorption corrected 0.5–10 keV flux in the full sample, high latitude sample (7401), and low latitude sample (9433).
+
+## 3 Catalogue Exploitation
+
+Among the avenues of scientific exploitation are:
+1) Characterization of the sources based on X-ray colours alone;
+2) Cross-correlation with other catalogues (FIRST, IRAS, 2MASS, GSC2) allowed the identification of radio-to-optical counterparts; sub-samples of promising sources for optical follow-up include:
+
+_i)_ blank fields (sources without counterparts at other wavelengths; Mignani et al. in prep.);
+
+_ii)_ heavily absorbed sources (the 4203 only detected in the hard X-ray band);
+3) Analysis of a sample of 300 extended sources (Fig. 3), which constitutes a list of X-ray selected galaxy cluster candidates, to confirm optically (Romano et al. in prep.);
+4) Temporal and spectral variability:
+
+_i)_ autocorrelation of the catalogue allows study of long-term variability of sources observed more than once (Israel et al. in prep.);
+
+_ii)_ intra-observation variability: search for periodicities in the light curves.
+
+## 4 BMW-_Chandra_ online
+
+The current version of the BMW-_Chandra_ source catalogue, (as well as additional information and data) is available at the Brera Observatory and at the INAF-IASF Palermo mirror sites,
+
+http://www.brera.inaf.it/BMC/bmc_home.html
+http://www.ifc.inaf.it/~romano/BMC/bmc_home.html
+The distributed version can also be found at the Centre de Données astronomiques de Strasbourg (Vizier) and at the HEASARC sites.
+
+Figure 3: **Left**: Extension of the BMW _Chandra_ sources as a function of off-axis angle. The solid line is the PSF function, the dashed line is the 3\(\sigma\) limit for point sources. **Right**: Solid angle versus flux limit for S/N \(=3\) for the soft (solid line) and hard (dotted line) bands. This sky coverage was constructed using 94 independent fields (no fields covered the same sky area).
+
+\begin{table}
+\begin{tabular}{l l r}
+\hline \hline
+Source Sample & & Number \\
+\hline
+detected & & 21325 \\
+good_a_ & & 16834 \\
+serendipitous & & 16758 \\
+independent & (within 3”) & 12135 \\
+ & (within 4”.5) & 11954 \\
+detected in total band & & 11124 \\
+detected in soft band & & 12631 \\
+detected in hard band & & 9775 \\
+only detected in hard band & & 4203 \\
+serendipitous extended & & 316 \\ \hline
+\end{tabular}
+
+* _a_Sources which do not require a more in-depth, non-automated analysis (i.e. not associated with bright and/or extended sources at the centre of the field), including the target ones.
+
+\end{table}
+Table 1: BMW-C in short.
+
+###### Acknowledgments.
+
+This work was supported through Consorzio Nazionale per l’Astronomia e l’Astrofisica (CNAA) and Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) grants. We thank A. Mistò for his help with the database software. RPM acknowledges STFC for support through its Rolling Grant programme. This publication makes use of data products from the _Chandra_ Data Archive, the FIRST, IRAS, 2MASS, GSC2 surveys.
+
+## References
+
+* [1] Campana, S., Lazzati, D., Panzera, M. R., Tagliaferri, G. 1999, _ApJ_, **524**, 423
+* [2] Campana, S., Moretti, A., Lazzati, D., Tagliaferri, G. 2001, _ApJ_, **560**, L19
+* [3] Lazzati, D., Campana, S., Rosati, P., Chincarini, G., Giacconi, R. 1998, _A&A_, **331**, 41
+* [4] Lazzati, D., Campana, S., Rosati, P., Panzera, M. R., Tagliaferri, G. 1999, _ApJ_, **524**, 414
+* [5] Moretti, A., Lazzati, D., Campana, S., Tagliaferri, G. 2002, _ApJ_, **570**, 502
+* [6] Panzera, M. R., Campana, S., Covino, S., et al., 2003, _A&A_, **399**, 351
+* [7] Romano, P., Campana, S., Mignani, R.P., et al., 2008, _A&A_, **488**, 1221
+

arxiv_ground_truth/0902.1555.md

@@ -0,0 +1,46 @@
+# Correlation Analysis of Mode Frequencies with Activity Proxies at Different Phases of the Solar Cycle
+
+Kiran Jain, S. C. Tripathy, and F. Hill
+
+###### Abstract
+
+We analyze intermediate degree _p_- and _f_-mode eigenfrequencies measured by GONG and MDI/SOHO for a complete solar cycle to study their correlation with solar activity. We demonstrate that the frequencies do vary linearly with the activity, however the degree of correlation differs from phase to phase of the cycle. During the rising and the declining phases, the mode frequencies are strongly correlated with the activity proxies whereas during the low- and high-activity periods, the frequencies have significantly lower correlation with all the activity proxies considered here.
+
+_National Solar Observatory, 950 N. Cherry Avenue, Tucson, AZ 85719, USA. Email: kjain@noao.edu_
+
+## 1. Introduction
+
+The linear variation of mode frequencies with the changing solar magnetic activity is well established (Jain & Bhatnagar 2003; Chaplin et al. 2007). However, detailed studies based on high quality uniform data indicate the complexity of the relationship between mode frequencies and solar activity. For example, Howe, Komm & Hill (1999) have shown that there is a latitudinal variation in frequencies and splitting coefficients. In addition, Tripathy et al. (2007) found a year-wise distribution in linear-regression slopes (i e. the change in frequency per unit change in activity) and the degree of correlation. Thus, with the availability of continuous data for a complete solar cycle, it is important to study correlations of oscillation frequencies with different measures of the solar activity in order to understand the source of the variability.
+
+## 2. Analysis and Results
+
+The analysis presented here uses oscillation data sets obtained from the Global Oscillation Network Group (GONG) and Michelson Doppler Imager (MDI) _onboard Solar and Helipspheric Observatory (SOHO)_ and solar activity data. It covers a period of about 13 years, i.e. a complete solar cycle including both minimum phases at the beginning and end of solar cycle 23. The 128 108-day GONG data sets spanning over the period from 1995 May 7 to 2008 Feb 27 are continuous while the 61 72-day MDI data sets have two large gaps in 1998-1999. The MDI data cover the period from 1996 May 1 to 2008 Sept 30. The activity data used here are: the integrated emission from the solar disk at 10.7 cm wavelength (\(F_{10}\)), the line-of sight magnetic field strength from Kitt Peak Observatory (KPMI), the international sunspot number (\(R_{I}\)), the Mt. Wilson sunspot index (MWSI), the magnetic plage strength index (MPSI) from Mt. Wilson and the total solar irradiance (TSI). Details of the various data sets are given in (Jain, Tripathy & Hill 2008).
+
+The number of _p-_ modes analysed here are 479 for GONG and 876 for MDI data sets in the frequency range 1.5 \(\leq\nu\leq\) 4.0 mHz. These modes are observed in all data sets of GONG or MDI. We also analyze 76 _f_-modes observed in all MDI data sets. The frequency shifts (\(\delta\nu\)) are calculated with respect to the reference frequency which is determined by taking an average of the frequencies of a particular multiplet (\(n,\ell\)).
+
+Figure 1.: Temporal evolution of the GONG _p_-mode frequency shifts (symbols) with different activity proxies (filled regions).
+
+The temporal variation of GONG frequency shifts (\(\delta\nu\)) with various measures of solar activity (\(I\)) are shown in Figure 1. It is evident that the frequency shifts follow the general trend of the solar activity. The correlation coefficients between \(\delta\nu\) and \(I\) obtained in all cases are comparable. However, we find significant different correlation coefficients when we divide the activity cycle into four phases as shown in Figure 2; the periods of minimum activity at the beginning and end of the solar cycle (_Phase I_), rising activity (_Phase II_), high-acivity (_Phase III_), and declining activity (_Phase IV_). In the right panel of Figure 2, we compare the phase-wise Pearson’s linear correlation coefficients (\(r_{P}\)) for both GONG and MDI data sets with those obtained for the complete cycle. It is interesting to note that the correlation between \(\delta\nu\) and solar activity changes significantly from phase to phase; the rising and declining phases are better correlated than the low- and high-activity phases. The frequencies during _Phase I_ do not correlate well with any of the proxies. During _Phase III_, we obtain significant correlations for \(F_{10}\) and KPMI while a substantial decrease is noticed for \(R_{I}\), MWSI and TSI.
+
+Figure 3 (left panel) shows the temporal variation of _f_-mode frequencies. We notice two distinct features in frequency shifts; the persisting strong 1-year periodicity as discussed by several authors (Antia et al. 2001; Jain & Bhatnagar 2003; Dziembowski & Goode 2005) and the frequencies at current solar minimum (2007-2008) are lower than those at the previous minimum (1996). The correlation coefficients obtained with the original and smoothed frequency shifts are also shown in the right panel. The smoothed frequency shifts are the running mean of five points to minimize the effect of 1-year periodicity. It is seen that smoothing enhances the correlation in all cases. The variation in correlation coefficients for _f_-modes at different phases are consistent with those for _p_-modes. In both cases, MDI data sets have good correlation with TSI at the low-activity phase that requires a detailed investigation.
+
+Figure 2.: (Left) Different phases of the solar activity cycle: Phase I shown by filled regions at the begining and the end of curve corresponds to low activity period, Phase II the rising activity, Phase III the high activity and Phase IV the declining activity periods. (Right) Bar-chart showing phase-wise variation of the Pearson’s linear correlation coefficient between _p_-mode frequency shifts and different activity indices. Each activity phase has values for GONG (filled) and MDI (hatched) data sets. The missing values for KPMI are due to unavailablity of sufficient activity data points. Correlation coefficients for all data sets are also shown here.
+
+## 3. Summary
+
+In summary, the improved and continuous measurements of intermediate-degree mode frequencies for a complete solar cycle demonstrate that, while the frequencies vary in phase with the solar activity, the degree of correlation between frequencies and activity indices differs from one activity phase to another. Although there is a strong correlation during rising and declining activity phases, we find a significant decrease in correlation at the low- and high-activity phases.
+
+Figure 3.: (Left) Temporal evolution of _f_-mode frequency shifts (circles) and solar activity (filled regions). (Right) Linear correlation coefficients between _f_-mode frequency shifts and activity indices for different phases of the solar cycle. Hatched bars show the correlation coefficients between actual values of frequency shifts and activity proxies while open bars are for correlation coefficients between smoothed value of frequency shifts and activity proxies.
+
+Acknowledgments.This work utilizes data obtained by the Global Oscillation Network Group (GONG) project, managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísico de Canarias, and Cerro Tololo Interamerican Observatory. It also utilises data from the Solar Oscillations Investigation/Michelson Doppler Imager on the Solar and Heliospheric Observatory. SOHO is a mission of international cooperation between ESA and NASA. NSO/Kitt Peak magnetic used here are produced cooperatively by NSF/NOAO; NASA/GSFC and NOAA/SEL. This study also includes data from the synoptic program at the 150-Foot Solar Tower of the Mt. Wilson Observatory, operated by UCLA, with funding from NASA, ONR and NSF, under agreement with the Mt. Wilson Institute. The unpublished solar irradiance dataset (version v6_001_0804) was obtained from VIRGO Team through PMOD/WRC, Davos, Switzerland. This work was supported by NASA grants NNG05HL41I and NNG08EI54I.
+
+## References
+
+* Antia et al. (2001) Antia, H. M., Basu, S., Pintar, J., & Schou, J. 2001, in ESA SP-464,: Proceedings of SOHO 10/GONG 2000 Workshop: Helio- and asteroseismology at the dawn of the millennium, ed. A. Wilson & P. L. Palle (Noordwijk: ESA Publications) 27
+* Chaplin et al. (2007) Chaplin, W. J., Elsworth, Y., Miller, B. A., & Verner, G. A. 2007, ApJ, 659, 1760
+* Dziembowski & Goode (2005) Dziembowski, W. A., & Goode, P. R. 2005, ApJ, 625, 548
+* Howe, Komm & Hill (1999) Howe, R., Komm, R., & Hill, F. 1999, ApJ, 524, 1084
+* Jain & Bhatnagar (2003) Jain, K., & Bhatnagar, A. 2003, Solar Phys., 213, 257
+* Jain, Tripathy & Hill (2008) Jain, K., Tripathy, S. C., & Hill, F. 2008, ApJ, submitted
+* Tripathy et al. (2007) Tripathy, S. C., Hill, F., Jain, K., & Leibacher, J. W. 2007, Solar Phys., 243, 105
+

arxiv_ground_truth/0902.2084.md

@@ -0,0 +1,115 @@
+# The Strange Prospects for Astrophysics
+
+I Sagert†, M Hempel†, G Pagliara‡, J Schaffner-Bielich‡⁵
+
+Footnote 5: invited talk given at the International Conference on Strangeness in Quark Matter (SQM2008), Beijing, China, October 6-10, 2008.
+
+ † Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany
+‡ Institut für Theoretische Physik, Ruprecht-Karls-Universität, Philosophenweg 16, 69120 Heidelberg, Germany
+
+T Fischer§, A Mezzacappa\(\|\), F-K Thielemann§ and M Liebendörfer
+
+ § Department of Physics, University of Basel, Klingelbergstr. 82, 4056 Basel, Switzerland
+\(\|\) Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
+schaffner@thphys.uni-heidelberg.de
+
+###### Abstract
+
+The implications of the formation of strange quark matter in neutron stars and in core-collapse supernovae is discussed with special emphasis on the possibility of having a strong first order QCD phase transition at high baryon densities. If strange quark matter is formed in core-collapse supernovae shortly after the bounce, it causes the launch of a second outgoing shock which is energetic enough to lead to a explosion. A signal for the formation of strange quark matter can be read off from the neutrino spectrum, as a second peak in antineutrinos is released when the second shock runs over the neutrinosphere.
+
+## 1 Introduction
+
+The exploration of the QCD phase diagram is not only a task for heavy-ion physics as there are strong relations to astrophysics of extremely dense matter and even to cosmology. The conditions in the early universe are similar to the ones probed at the heavy-ion collisions at RHIC and LHC, high temperatures and low net baryon densities. Supernova matter and neutron star matter is located in the QCD phase diagram at moderate temperatures and high net baryon densities. In this region of the QCD phase diagram one expects to have a strong first order phase transition which is related to the restoration of chiral symmetry¹. QCD matter at extreme baryon densities will be investigated by heavy-ion experiments with FAIR at GSI Darmstadt.
+
+Footnote 1: In the following, we will refer to the new phase at high densities generically to ’quark matter’ although matter is not necessarily deconfined as the phase transition is related to chiral symmetry restoration.
+
+Figure 1: The QCD phase diagram, the lines denote first order phase transition which are due to chiral symmetry breaking and/or the formation of color superconducting quark matter (taken from [1]).
+
+The QCD equation of state (EoS) is an essential input in astrophysical systems. In core-collapse supernovae simulations temperatures of about \(T=10-20\) MeV with densities slightly above normal nuclear matter density are reached at core bounce, the newly born proto-neutron star is heated up to \(T=50\) MeV with densities a few times normal nuclear matter densities, the final cold neutron stars then has central densities of up to ten times normal nuclear matter densities for a soft equation of state. Finally, neutron star mergers simulations can achieve temperatures of typically \(T=30\) MeV, even higher temperatures have been noticed for some equations of states. Note, that the dynamical timescales involved are usually not much less than around a few 100 microseconds so that they are always much larger than the timescale to establish equilibrium with respect to weak interactions involving strange particles of about \(10^{-8}\) seconds or less. Hence, the matter for astrophysical applications is necessarily in weak equilibrium with respect to strangeness and always includes strange matter! Except for cold neutron stars, there is a subtlety here, as protons and neutrons are not in weak equilibrium. Hence, for dynamical astrophysical scenarios, the matter is characterized by a given temperature \(T\), baryon density \(n\) and proton fraction \(Y_{p}\).
+
+For hunting down strange quark matter in the heavens several signals have been suggested in the literature as exotic mass-radius relation of compact stars, rapidly rotating pulsars due to r-mode _stability_ window, enhanced cooling of neutron stars, and gamma-ray bursts by transition to strange quark matter. Let us first concentrate on strange quark matter in neutron star before we discuss the implications of the formation of strange quark matter for core-collapse supernovae.
+
+## 2 Strange Quark Matter in Neutron Stars
+
+Neutron stars are produced in core-collapse supernova explosions and are extremely compact, massive objects with radii of \(\approx\) 10 km and masses of \(1-2M_{\odot}\), involving extreme densities, several times nuclear density: \(n\gg n_{0}=0.16\) fm_-3_.
+
+More than 1700 pulsars, rotation-powered neutron stars, are presently known. The best determined mass is the one of the Hulse-Taylor pulsar, \(M=(1.4414\pm 0.0002)M_{\odot}\)[2], the smallest known mass is \(M=(1.18\pm 0.02)M_{\odot}\) for the pulsar J1756-2251 [3]. Note, that the mass of the pulsar J0751+1807 has been corrected from \(M=2.1\pm 0.2M_{\odot}\) to \(M=1.14-1.40M_{\odot}\)[4]. We note that the extremely large neutron star masses extracted from pulsars found in globular clusters [5] can only give an upper bound. Only the periastron advance of the pulsar PSR J1748-2021B has been determined so far, the inclination angle \(i\) of the orbital plane is still unknown. A statistical analysis for that angle is not really appropriate for one pulsar. For an inclination angle of \(i=4-5\) degrees, two neutron stars with a mass of \(M\sim 1.4M_{\odot}\) are possible. A measurement of a second effect from general relativity is needed to draw a firm final conclusions.
+
+The spectral analysis of the closest isolated neutron star known, RX J1856.5-3754, hints at more complex surface properties than initially expected. Fits with a two-component blackbody as well as with a condensed surface and a small layer of hydrogen result in rather large radiation radii \(R_{\infty}=R/\sqrt{1-2GM/R}=17(d/140pc)\) km. With an inferred gravitational redshift of \(z_{g}\approx 0.22\), the neutron star would have a true radius of \(R\approx 14\) km and a mass of \(M\approx 1.55M_{\odot}\)[6]. A large uncertainty resides in the still not well known distance \(d\), but clearly more data and analysis is needed to understand the atmosphere and the radiation of neutron stars.
+
+In binary systems of a neutron star with an ordinary star accreting material falling onto the neutron star ignites nuclear burning which is observable as an x-ray burster. The analysis of Özel [7] for the x-ray burster EXO 0748–676 arrived at mass-radius constraints of \(M\geq 2.10\pm 0.28M_{\odot}\) and \(R\geq 13.8\pm 1.8\) km. The values derived have to be taken with great care, as a multiwavelength analysis of Pearson et al. [8] concludes that the data is more consistent with a mass of \(M=1.35M_{\odot}\) than with \(M=2.1M_{\odot}\). Even if such large masses and radii are taken for granted, quark matter can still be present in the interior of neutron stars as demonstrated by Alford et al. [9]. The limits would rule out soft equations of states, not quark stars or hybrid stars, compact stars with a hadronic mantle and a quark matter core.
+
+Future telescopes and detectors will probe compact stars in more details as the International X-ray Observatory IXO, the James Webb Space Telescope JWST, the Square Kilometre Array SKA, LISA and UNO, an underground neutrino observatory. With the future x-ray satellites one can measure the profile of the burst oscillations which is modified from the space-time warpage around the compact star. By this method a model independent measurement of the mass and radius of the compact star can be extracted. It was claimed that one could determine the mass-radius ratio to within 5% with Constellation-X with this method [10].
+
+Figure 2: The phase structure of hybrid stars within the MIT bag model and using a HDL approximation for different values of the MIT bag constant. HP: Hadronic phase, MP: mixed phase, QP: Quark phase. Reprinted from [11], Copyright (2000), with permission from Elsevier.
+
+Quark matter in neutron stars has been widely described by using the MIT bag model with basically one free parameter, the MIT bag constant \(B\). The onset of the mixed phase from the hadronic to the quark phase occurs between \((1-2)n_{0}\) even for large values of the bag constant \(B\) and then sufficiently high densities are reached in the core of a \(1.3M_{\odot}\) compact star to have quark matter (see Figure 2). Corrections from hard thermal loop calculations do not change these numbers significantly.
+
+So called hybrid stars consist of hadronic matter and quark matter and there are three phases possible: a hadronic phase, a mixed phase and a pure quark phase. The composition depends crucially on the parameters as the bag constant \(B\) and the interaction strength \(g\) between quarks as well as the total mass of the compact star.
+
+In addition, there exists a third solution to the TOV equations besides the one for the white dwarfs and the one for ordinary neutron stars, which is stabilized by the presence of a pure quark matter phase [12, 13, 14, 15, 11, 16]. The third family of compact stars is generically more compact than ordinary neutron stars and is possible for any first order phase transition.
+
+## 3 Strange Quark Matter in Supernovae
+
+Stars with a mass of more than 8 solar masses end in a core-collapse supernova (type II, Ib or Ic). New generation of simulation codes now have multidimensional treatments and improved neutrino transport. Still, until recently, no explosions could be achieved (see e.g. [17]) suggesting missing physics either with respect to neutrino transport or to the nuclear equation of state. Only after sufficiently long simulation runs with a quasi unconstrained geometry a standing accretion shock instability could develop which leads to an explosion after 600ms [18].
+
+The conditions of core-collapse supernova matter at bounce are as follows: energy densities slightly above normal nuclear matter density \(\epsilon\sim(1-1.5)\epsilon_{0}\), temperatures of \(T\sim 10-20\) MeV and a proton fraction of \(Y_{p}\sim 0.2-0.3\). The standard lore for the onset of the quark phase in core-collapse supernovae is that it happens during the evolution of the proto-neutron star and not at bounce. The timescale for quark matter to appear would then be typically \((5-20)\) s after bounce [19], which is, however, due to using rather large bag constants of \(B>180\) MeV. Such a large bag constant would hardly allow for a pure quark matter phase to develop in the core of cold neutron stars with a mass of \(1.3M_{\odot}\) (see figure 2). The appearance of quark matter would then be well after the supernova explosion itself. So the question is, can it be possible to produce quark matter much earlier with a smaller bag constant?
+
+Figure 3: Left plot: The critical density for the phase transition for different proton-to-baryon ratios \(Y_{p}\) for supernova conditions; thin lines: onset of mixed phase, thick lines: end of mixed phase. Right plot: The phase transition line as a function of the baryon chemical potential and temperature for supernova conditions.
+
+Figure 3 shows the phase transition line to quark matter for a given temperature versus the net baryon density (left plot) and versus the baryochemical potential (right plot) for different values of the proton fraction \(Y_{p}\). The hadronic phase is modelled by using the EoS of Shen et al. [20], the quark phase by the MIT bag model with a bag constant of \(B^{1/4}=165\) MeV and a strange quark mass of 100 MeV, the phase transition by a Maxwell construction. Interestingly, the critical baryochemical potential is nearly independent on the proton fraction \(Y_{p}\) and bends towards low chemical potentials for high temperatures as envisioned in the sketch of the QCD phase diagram depicted in figure 1. However, the phase transition occurs at lower densities for lower proton fractions and higher temperatures, so that neutron-rich hot supernova material is favourable for the formation of strange quark matter. Note, that strangeness is not conserved in supernova matter contrary to the situation in heavy-ion collisions. Strange quark matter can be formed via the coalescence of hyperons in supernova matter which are thermally excited in weak equilibrium on timescales of \(10^{-8}\) s or less. Hyperon fractions of about 0.1% are already present at bounce, see [21]. The lower critical density in neutron-rich matter is due to the sizable nuclear symmetry energy so that strange quark matter becomes the energetically preferred phase. For supernova material at bounce (\(T=10-20\) MeV, \(Y_{p}=0.2-0.3\), \(n\approx n_{0}\)), one reads from the figure that the immediate production of quark matter is possible!
+
+Figure 4: Mass-radius relation of cold neutron stars for the supernova EoS used for different bag constants. For low bag constants, the maximum mass is again above the mass limit from the Hulse-Taylor pulsar of 1.44 \(M_{\odot}\) which is shown by the horizontal line.
+
+Two checks have to be performed in order to be consistent with neutron star data and heavy-ion phenomenology. First, the maximum mass of a compact star with the adopted EoS should be at least above \(1.44M_{\odot}\). The mass-radius relation is plotted in figure 4 for different values of the MIT bag constant (for simplicity a Maxwell construction is used for the phase transition). For large bag constants, the maximum mass is well above the mass limit from the Hulse-Taylor pulsar. For some intermediate values, here for the cases of \(B^{1/4}=170\) MeV and 175 MeV, the maximum mass is below \(1.44M_{\odot}\) because of the large jump in energy density from one phase to the other [22]. That was the reason why smaller values of the bag constants were rejected in the work on proto-neutron stars in ref. [19]. However, for even smaller bag constants of \(B^{1/4}=165\) MeV and below, the maximum mass is again above \(1.44M_{\odot}\) owing to the increased stability of the pure quark matter core. The maximum masses are \(1.56M_{\odot}\) (\(B^{1/4}=162\) MeV) and \(1.5M_{\odot}\) (\(B^{1/4}=165\) MeV). Note, that the maximum mass for pure quark matter stars, selfbound strange stars, is well known to be about \(2.1M_{\odot}\) for the original value of the MIT bag constant, \(B^{1/4}=145\) MeV [23, 24, 25].
+
+Figure 5: Left plot: The critical density for the phase transition lines for different proton-to-baryon ratios \(Y_{p}\) for heavy-ion conditions (no weak equilibrium for strangeness); thin lines: onset of mixed phase, thick lines: end of mixed phase.
+Right plot: The phase transition line as a function of the baryon chemical potential and temperature for heavy-ion conditions.
+
+Secondly, we have to calculate the phase transition line appropriate for matter produced in relativistic heavy-ion collisions. The timescales are too short to generate weak equilibrium in heavy-ion collisions, so the initially produced quark matter has net strangeness zero (of course thermal production of strange quark pairs is possible but highly suppressed for the low temperatures relevant for supernova explosions). Therefore, the quark matter formed consists mainly out of pure up- and down-quarks and becomes highly unfavoured compared to the case of supernova matter, where strange quark matter is formed. Indeed, we find large critical densities for the phase transition, in particular for isospin-symmetric matter (proton fraction \(Y_{p}=0.5\)) which is relevant for the heavy-ion case. The phase transition occurs at much larger baryon densities, well above five times normal nuclear matter density for low temperatures so that low-energy heavy-ion collisions can not produce quark matter. Also the phase transition line is located at larger chemical potentials for up-down-quark matter. The location of the freeze-out points in statistical approaches is about \(\mu_{f.o.}=700-800\) MeV, \(T_{f.o.}=50-70\) MeV for SIS energies and \(\mu_{f.o.}\sim 500\) MeV, \(T_{f.o.}\sim 120\) MeV for AGS energies. These values are lying below the phase transition line in the corresponding right plot of fig. 5. Note, that the hadronic equation of state has been modelled to suit its applications for supernova simulations. Therefore, temperatures of more than 50 MeV can not be handled appropriately in principle, but are also not relevant for our purposes here. For the application to higher temperatures, one needs to improve the hadronic equation of state by e.g. incorporating pions, kaons, hyperons and resonances which would shift the phase transition line to even higher densities and baryochemical potentials.
+
+Now as the supernova EoS has passed these two tests, let us discuss the implications for core-collapse supernovae [26]. The stiffening of the nuclear EoS above normal nuclear matter densities produces a bounce of the supernova material, a shock wave is moving outwards but stalls around 100 km. Afterwards, quark matter is formed in the core and at a few 100ms the mixed phase collapses and an accretion shock develops on the pure quark matter core. The accretion shock turns into an outgoing shock wave which is so energetic that it runs over the stalled first shock leading to an explosion! Note, that normally 1D supernova simulations are not able to achieve an explosion, only multidimensional codes are presently capable of producing an explosion with the help of the standing accretion shock instability, see [18].
+
+Our supernova simulation runs were performed for different parameters, where the quark core appears at \(t_{\rm pb}=200\) ms to \(500\) ms post bounce. The results (\(t_{\rm pb}\), baryonic mass and explosion energy) are significantly sensitive to the location of the QCD phase transition (i.e. the bag constant in our case). Heavier progenitor masses can lead to the formation of a black hole which could be circumvented by stiffening the quark EoS in order to explain the rather long emission of neutrinos from SN1987A.
+
+Figure 6: The neutrino spectrum without a phase transition (thick lines) and with a phase transition (thin lines). The case with a phase transition to strange quark matter results in a second peak in antineutrinos. The average energies of the emitted neutrinos increases also. Reprinted figure with permission from [26]. Copyright (2009) by the American Physical Society.
+
+Most interestingly, we find that the temporal profile of the emitted neutrinos out of the supernova reflects the features of the QCD phase transition. Figure 6 shows the neutrino luminosity and the mean energy as a function of time. The first peak in electron neutrinos is due to the first shock. When the QCD phase transition is included we find a second peak in electron _anti_-neutrinos at about the time when the strange quark matter core is created. The pronounced second peak of anti-neutrinos is due to the protonization of the material when the second shock front runs over the neutrinosphere. We note that the location of the second peak and its height is controlled by the critical density and strength of the QCD phase transition!
+
+## 4 Summary
+
+The QCD phase transition to strange quark matter leads to a rich variety of astrophysical signals involving compact stars and supernovae. Neutron stars with a core of strange quark matter are compatible with present neutron star data. Strange quark matter can be formed in supernovae, even shortly after the first bounce. A second outgoing shock is generated which has enough energy to lead to an explosion. The presence of a strong QCD phase transition can be read off from a second peak in the (anti-)neutrino signal. The formation of strange quark matter will certainly have also significant implications for the gravitational wave signal of core-collapse supernovae and for the r-process nucleosynthesis as core-collapse supernovae are considered to be the prime astrophysical site.
+
+This work is supported by the German Research Foundation (DFG) within the framework of the excellence initiative through the Heidelberg Graduate School of Fundamental Physics, the Gesellschaft für Schwerionenforschung mbH Darmstadt, Germany, the Helmholtz Research School for Quark Matter Studies, the Helmholtz Alliance Program of the Helmholtz Association, contract HA-216 ”Extremes of Density and Temperature: Cosmic Matter in the Laboratory”, the Frankfurt Institute for Advanced Studies, the Italian National Institute for Nuclear Physics, the Swiss National Science Foundation under the grant numbers PP002-106627/1 and PP200020-105328/1, and the ESF CompStar program. A.M. is supported at the Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725.
+
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+

arxiv_ground_truth/0903.1810.md

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+# Collapse and Fragmentation of Molecular Cloud Cores. X. Magnetic Braking of Prolate and Oblate Cores.
+
+Alan P. Boss
+
+Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road, NW, Washington, DC 20015-1305
+boss@dtm.ciw.edu
+
+###### Abstract
+
+The collapse and fragmentation of initially prolate and oblate, magnetic molecular clouds is calculated in three dimensions with a gravitational, radiative hydrodynamics code. The code includes magnetic field effects in an approximate manner: magnetic pressure, tension, braking, and ambipolar diffusion are all modelled. The parameters varied for both the initially prolate and oblate clouds are the initial degree of central concentration of the radial density profile, the initial angular velocity, and the efficiency of magnetic braking (represented by a factor \(f_{mb}=10^{-4}\) or \(10^{-3}\)). The oblate cores all collapse to form rings that might be susceptible to fragmentation into multiple systems. The outcome of the collapse of the prolate cores depends strongly on the initial density profile. Prolate cores with central densities 20 times higher than their boundary densities collapse and fragment into binary or quadruple systems, whereas cores with central densities 100 times higher collapse to form single protostars embedded in bars. The inclusion of magnetic braking is able to stifle protostellar fragmentation in the latter set of models, as when identical models were calculated without magnetic braking (Boss 2002), those cores fragmented into binary protostars. These models demonstrate the importance of including magnetic fields in studies of protostellar collapse and fragmentation, and suggest that even when magnetic fields are included, fragmentation into binary and multiple systems remains as a possible outcome of protostellar collapse.
+
+hydrodynamics — ISM: clouds — ISM: kinematics and dynamics — MHD — stars: formation
+
+## 1 Introduction
+
+Fragmentation during protostellar collapse is widely accepted to be the primary mechanism for the formation of binary and multiple star systems (e.g., Lafrenière et al. 2008; Chen et al. 2008). While it is clear that the overall form of the initial mass function for stars is directly tied to the initial conditions for protostellar collapse, i.e., the mass function of dense cloud cores (e.g., Dib et al. 2008; Swift & Williams 2008), fragmentation is necessary for producing binary star systems within these individual dense cores (Lafrenière et al. 2008; Chen et al. 2008).
+
+Three dimensional calculations of the collapse of centrally condensed, rotating cloud cores have been computed for quite some time (e.g., Boss 1993) and continue to attract theoretical attention (e.g., Saigo et al. 2008; Machida 2008; Commercon et al. 2008). These calculations neglected the effects of magnetic fields. However, observations of OH Zeeman splitting in dark cloud cores have shown that magnetic fields are often an important contributer to cloud support against collapse for densities in the range of \(10^{3}-10^{4}\) cm_-3_ (Troland & Crutcher 2008). Given this observational constraint, it is clear that three dimensional hydrodynamical collapse calculations should include magnetic field effects as well as self-gravity and radiative transfer (e.g., Boss 1997, 1999, 2002, 2005, 2007). Magnetic fields are now being included in other three dimensional collapse models as well (e.g., Machida et al. 2004, 2005a,b, 2007, 2008; Kudoh et al. 2007; Price & Bate 2007, 2008). In particular, Price & Bate (2007) found that while magnetic pressure acts to resist fragmentation during collapse, magnetic tension can actually aid fragmentation, confirming the results found by Boss (2002). Machida et al. (2004, 2005a,b, 2007, 2008) generally found that binary fragmentation could still occur provided that the initial magnetic cloud core rotated fast enough.
+
+Magnetic braking is effective at reducing cloud rotation rates during the pre-collapse cloud phase, but has relatively little effect during the collapse phase, according to the two dimensional magnetohydrodynamics models of Basu & Mouschovias (1994, 1995a,b). However, Hosking & Whitworth (2004) found that rotationally-driven fragmentation could be halted by magnetic braking during the collapse phase. Boss (2004) argued that the thermodynamical treatment employed by Hosking & Whitworth (2004) could have been more important for stifling fragmentation than magnetic braking, but did not offer any models of magnetic braking to support this assertion.
+
+Price & Bate (2007) presented models of the collapse of magnetic cloud cores, finding that magnetic pressure was more important for inhibiting fragmentation than either magnetic tension or braking, contrary to the results presented by Hosking & Whitworth (2004) and Fromang et al. (2006), who found no evidence at all for the fragmentation of magnetic clouds. Fromang et al. (2006) assumed ideal magnetohydrodynamics (MHD), i.e., without ambipolar diffusion, a fact that is likely to stifle fragmentation, whereas Hosking & Whitworh (2004), Boss (2004), and Price & Bate (2007) included ambipolar diffusion. However, subsequent ideal MHD collapse calculations by Hennebelle & Fromang (2008) and Hennebelle & Teyssier (2008) found that magnetic clouds could fragment if the initial density perturbation was large enough, and they speculated on what would happen if ambipolar diffusion was included in their models. These fundamental differences in the results of magnetic cloud collapse calculations results highlight the need to compare models where only one parameter at a time is being changed, so that the true effect of changing that one parameter can be discerned. Such a comparison is the major goal of the present study.
+
+Banerjee & Pudritz (2006) considered the collapse of magnetized cloud cores, finding that even though considerable angular momentum was lost from the cloud core by magnetic outflows from the disk’s surface, the cloud was still able to collapse and fragment into a close binary protostar system.
+
+Price & Bate (2007) and Hennebelle & Teyssier (2008) both considered the collapse of spherical, magnetic cloud cores with initially uniform density and uniform magnetic field strengths. Machida et al. (2004, 2005a,b) studied the collapse of initially cylindrical cloud cores in hydrostatic equilibrium. Machida et al. (2008) considered the collapse of cloud cores with density profiles appropriate for Bonnor-Ebert isothermal spheres, similar to the Gaussian radial density profile clouds studied by Boss (1997, 2002) and by this paper. Such centrally-condensed density profiles represent the best guesses as to the radial structure of pre-collapse molecular cloud cores (e.g., Myers et al. 1991; Ward-Thompson, Motte, & André 1999).
+
+Boss (2002) modelled the collapse of initially prolate and oblate cores, including several magnetic field effects, but ignoring magnetic braking. Prolate and oblate cloud shapes have been inferred from observations of suspected pre-collapse molecular cloud cores (e.g., Jones, Basu, & Dubinski 2001; Curry & Stahler 2001). Here we use the magnetic braking approximation developed by Boss (2007), originally applied to filamentary clouds, to examine the importance of magnetic braking for the same prolate and oblate models as those calculated by Boss (2002), allowing a direct comparison between identical protostellar collapse models with and without magnetic braking. These models thus directly address the different outcomes of the models with magnetic braking but without detailed thermodynamics by Hosking & Whitworth (2004) compared to those without magnetic braking but with detailed thermodynamics of Boss (2004): the present models include both effects.
+
+## 2 Numerical Methods
+
+The numerical models are calculated with a three-dimensional hydrodynamics code that calculates finite-difference solutions of the equations of radiative transfer, hydrodynamics, and gravitation for a compressible fluid (Boss & Myhill 1992). The hydrodynamic equations are solved in conservation law form on a contracting spherical coordinate grid, subject to constant volume boundary conditions on the spherical boundary. The code is second-order-accurate in both space and time, with the van Leer-type hydrodynamical fluxes having been modified to improve stability (Boss 1997). Artificial viscosity is not employed. Radiative transfer is handled in the Eddington approximation, including detailed equations of state and dust grain opacities (e.g., Pollack et al. 1994). The code has been tested on a variety of test problems (Boss & Myhill 1992; Myhill & Boss 1993).
+
+The Poisson equation for the cloud’s gravitational potential is solved by a spherical harmonic expansion (\(Y_{lm}\)) including terms up to \(N_{lm}=32\). The computational grid consists of a spherical coordinate grid with \(N_{r}=200\), \(N_{\theta}=22\) for \(\pi/2\geq\theta>0\) (symmetry through the midplane is assumed for \(\pi\geq\theta>\pi/2\)), and \(N_{\phi}=256\) for \(2\pi\geq\phi\geq 0\), i.e., with no symmetry assumed in \(\phi\). The radial grid contracts to follow the collapsing inner regions and to provide sufficient spatial resolution to ensure satisfaction of the four Jeans conditions for a spherical coordinate grid (Truelove et al. 1997; Boss et al. 2000). The innermost 50 radial grid points are kept uniformly spaced during grid contraction, while the outermost 150 are non-uniformly spaced, in order to provide an inner region with uniform spatial resolution in the radial coordinate. The \(\phi\) grid is uniformly spaced, whereas the \(\theta\) grid is compressed toward the midplane, where the minimum grid spacing is \(0.3\) degrees.
+
+## 3 Initial Conditions
+
+Tables 1 and 2 list the initial conditions for the models. The initial models have Gaussian radial density profiles (Boss 1997) of the form
+
+\[\rho_{i}(x,y,z)=\rho_{o}\ exp\biggl{(}-\biggl{(}{x\over r_{a}}\biggr{)}^{2}-\biggl{(}{y\over r_{b}}\biggr{)}^{2}-\biggl{(}{z\over r_{c}}\biggr{)}^{2}\biggr{)},\] (1)
+
+where \(\rho_{0}=2.0\times 10^{-18}\) g cm_-3_ is the initial central density. The prolate clouds with central densities 20 times higher than a reference boundary density have \(r_{a}=1.16R\) and \(r_{b}=r_{c}=0.580R\), where \(R\) is the cloud radius, yielding a axis ratio of 2 to 1. The oblate clouds with the same degree of central concentration have \(r_{a}=r_{b}=1.16R\) and \(r_{c}=0.580R\). For 100 to 1 density contrasts, the prolate clouds have \(r_{a}=0.932R\) and \(r_{b}=r_{c}=0.466R\), while the oblate clouds have \(r_{a}=r_{b}=0.932R\) and \(r_{c}=0.466R\). Random numbers (\(ran(x,y,z)\)) in the range [0,1] are used to add noise to these initial density distributions by multiplying \(\rho_{i}\) from the above equation by the factor \([1+0.1\ ran(x,y,z)]\). The cloud radius is \(R=1.0\times 10^{17}\) cm \(\approx\) 0.032 pc for all models.
+
+The cloud masses are 1.5 \(M_{\odot}\) and 2.1 \(M_{\odot}\), respectively, for the prolate and oblate clouds with a 20:1 density ratio, and 0.96 \(M_{\odot}\) and 1.5 \(M_{\odot}\), for the prolate and oblate clouds with a 100:1 density ratio. With an initial temperature of 10 K, the initial ratio of thermal to gravitational energy is \(\alpha_{i}=0.39\) for the prolate clouds and 0.30 for the oblate clouds with the 20:1 initial density ratio, while \(\alpha_{i}=0.55\) for the prolate clouds and 0.39 for the oblate clouds with the 100:1 initial density ratio.
+
+Solid body rotation is assumed, with the angular velocity about the \(\hat{z}\) axis (short axis) taken to be \(\Omega_{i}\) = \(10^{-14}\), \(3.2\times 10^{-14}\), or \(10^{-13}\) rad s_-1_. These choices of \(\Omega_{i}\) result in initial ratios of rotational to gravitational energy varying from \(\beta_{i}=1.2\times 10^{-4}\) to 0.015 for the prolate clouds and \(\beta_{i}=1.1\times 10^{-4}\) to 0.013 for the oblate clouds. These choices are all consistent with observational constraints on the densities, shapes, and rotation rates of dense cloud cores (e.g., Myers et al. 1991; Goodman et al. 1993; Ward-Thompson, Motte, & André 1999; Jones, Basu, & Dubinski 2001; Curry and Stahler 2001).
+
+As in the previous three-dimensional models (Boss 1997, 1999, 2002, 2005, 2007), the effects of magnetic fields are approximated through the use of several simplifying approximations regarding magnetic pressure, tension, braking, and ambipolar diffusion (see Boss 2007 for a derivation of these approximations). All models assumed an ambipolar diffusion time scale \(t_{ad}\) = 10 \(t_{ff}\), where the free fall time \(t_{ff}=(3\pi/32G\rho_{0})^{1/2}=4.7\times 10^{4}\) yr for a central density \(\rho_{0}=2.0\times 10^{-18}\) g cm_-3_. The reference magnetic field strength is assumed to be \(B_{oi}=200\mu\)G. The magnetic braking factor \(f_{mb}\) is taken to be either 0.0001 or 0.001. Based on the models of Basu & Mouschovias (1994), Boss (2007) estimated that \(f_{mb}\sim 0.0001\). The models with \(f_{mb}=0.001\) are thus intended to attempt to overestimate the effects of magnetic braking.
+
+Note, however, that the calculations of Basu & Mouschovias (1994) stopped at central densities of \(3\times 10^{9}\) cm_-3_, i.e., before the clouds became optically thick, whereas the present models are continued well in the optically thick regime. Hence the implicit assumption that the trends found in the Basu & Mouschovias (1994) models and used by Boss (2007) to derive the \(f_{mb}\) approximation should continue indefinitely may not be warranted, though the trends do persist over the previous six orders of magnitude increase in central density of their models (see Figure 7 of Basu & Mouschovias 1994). The magnetic braking studied by Basu & Mouschovias (1994) is similar to the magnetic braking produced by disk jets and outflows in the models by Banerjee & Pudritz (2006), Hennebelle & Fromang (2008), and Hennebelle & Teyssier (2008). A superior treatment of magnetic braking beyond the \(f_{mb}\) approximation of Boss (2007) will require a true MHD code.
+
+With the field strength \(B_{oi}=200\mu\)G, the prolate or oblate clouds with 20:1 density contrasts have initial ratios of magnetic to gravitational energy of \(\gamma_{i}=0.58\) or 0.43, respectively. For density ratios of 100:1, \(\gamma_{i}=0.81\) for the prolate clouds and 0.57 for the oblate clouds. The mass to flux ratio of these clouds is less than the critical mass to flux ratio, making all the clouds formally magnetostatically stable and hence magnetically subcritical. Protostellar collapse cannot occur until ambipolar diffusion leads to sufficient loss of magnetic field support for sustained contraction to begin.
+
+## 4 Results
+
+Tables 1 and 2 list the initial conditions as well as the basic outcome of each model, namely the final time to which the cloud was advanced \(t_{f}\) (in units of the initial cloud free fall time) and whether the cloud collapsed to form a quadruple system (Q), binary (B), binary-bar (BB), single-bar (SB), ring (R), or did not collapse (NC). For convenience, the results of the corresponding models by Boss (2002) without magnetic braking are shown as well.
+
+Figure 1 shows the initial equatorial density distribution for the prolate cores with a 20:1 density contrast. Given the stability of the initial models, evolutions consist of the clouds oscillating about the initial configurations, waiting for sufficient time to elapse for ambipolar diffusion to reduce the magnetic field support enough to allow collapse to proceed, as in the previous magnetic cloud models (e.g., Boss 1997). Figure 2 shows the result for model P2BB, which collapsed to form a quadruple protostellar system, though in this case with an additional central density maximum. Without magnetic braking, this core collapsed to form a binary protostar system, so in this case magnetic braking has led to an increased degree of fragmentation. The off-axis clump evident in Figure 2 at 9 o’clock has a maximum density of \(2.5\times 10^{-12}\) g cm_-3_ and an average temperature of 20 K. Considering regions with a density at least 0.1 that of this maximum density, the clump’s mass is 2.6 Jupiter masses, greater than the Jeans mass at that density and temperature of 1.9 Jupiter masses. The ratio of thermal to gravitational energy for the clump is 0.49, showing that it is gravitationally bound. The ratio of rotational to gravitational energy is 0.39, so the clump is in rapid rotation. These fragment properties are quite similar to those found in the Boss (2002) models.
+
+Model P2BA behaved in much the same way as P2BB, even with stronger magnetic braking (\(f_{mb}=0.001\) for P2BA compared to \(f_{mb}=0.0001\) for P2BB).
+
+Figure 3 shows the outcome of model P2BD, which started with a lower initial angular velocity than model P2BB but was otherwise identical. In this case, a well-defined binary protostar system forms. A similar outcome resulted for model P2BC with a higher degree of magnetic braking. With even lower initial angular velocity than P2BC and P2BD, model P2BE collapsed to form a binary-bar system (Figure 4), i.e., a binary with its members connected by a bar of rotating gas. In the case of model P2BE, there are two density maxima in the bar near the center of the system, possible evidence for further fragments, though none of the density maxima are as well-defined as those of the binary in Figure 3. In contrast, when magnetic braking was neglected (Boss 2002), a core identical to model P2BE fragmented into a well-defined binary system, so in this case magnetic braking has reduced the degree of fragmentation.
+
+The prolate core models with 100:1 initial density contrasts all behaved in the same manner and formed single central protostars embedded in bar-like structures. Figure 5 shows the result for model P1BB, termed a single-bar. Figure 6 shows the equatorial temperature contours for this model, emphasizing the highly non-uniform temperature distribution that results from including radiative transfer effects (e.g., Boss et al. 2000). The temperature field is strongest in regions where the infalling gas is forming shock-like density corrugations, yielding an x-shaped pattern in the cloud’s midplane. The formation of single-bars in all five of the prolate 100:1 models compared to the formation of binaries in the corresponding Boss (2002) models shows that when the initial cores are highly centrally condensed, magnetic braking is able to frustrate fragmentation.
+
+Figure 7 shows the initial conditions for the oblate cores with 20:1 initial density contrasts, while Figure 8 depicts the outcome of model O2BB: a well-defined ring. While the ring shows no particular tendency to fragment over the time scale of the calculation, such a configuration is expected to fragment eventually. With the exception of model O2BA, which did not collapse significantly, all of the oblate cores collapsed to form rings, for both 20:1 and 100:1 density contrasts, though the rings were more pronounced for the cores with higher initial rotation rates, such as O2BB in Figure 8. In comparison, the corresponding Boss (2002) models formed a combination of rings or rings which showed a tendency to fragment into quadruple systems. Hence the oblate cloud models indicate that the inclusion of magnetic braking had only a mild tendency to inhibit their fragmentation.
+
+Tables 1 and 2 show that the prolate clouds tended to take considerably longer to undergo collapse than the oblate clouds, implying a considerably longer period in the pre-collapse, quasi-equilibrium phase where thermal and magnetic support dominate. As a result, the effect of magnetic braking through the \(f_{mb}\) approximation should be stronger in the models which took the longest time to collapse, i.e., the models in Table 1 with \(\rho_{0}/\rho_{R}=100:1\) and \(t_{f}/t_{ff}\sim 9\). Prolonged magnetic braking will tend to suppress rotationally-driven fragmentation, consistent with the formation of single-bars for the prolate models in Table 1 with \(t_{f}/t_{ff}\sim 9\). Model P2bb, for example, lost about 0.4% of its total angular momentum during its evolution to \(t_{f}/t_{ff}=4.683\), compared to a loss of about 1% for model P1Bb within \(t_{f}/t_{ff}=9.071\). Note though that for the oblate models in Table 2, no such effect is evident, perhaps because all of those models collapsed within \(t_{f}/t_{ff}<5\). In fact, a comparision of the corresponding oblate models shows that model O2bb lost about 0.4% of its total angular momentum during its evolution to \(t_{f}/t_{ff}=2.456\), compared to a loss of about 1% for model O1Bb within \(t_{f}/t_{ff}=4.636\). Both models O2BB and O1BB collapse to form rings. Their percentage angular momentum losses are identical to those for models P2BB and P1BB, which formed a quadruple and single-bar, respectively, implying that the initial cloud shape and degree of central concentration do have an important effect on the fragmentation process of magnetic clouds.
+
+## 5 Conclusions
+
+These pseudo-magnetohydrodynamics calculations have explored the possibly deleterious effects of magnetic braking on protostellar fragmentation, an issue explored by Hosking & Whitworth (2004). A degree of inhibition of fragmentation caused by magnetic braking of both prolate and oblate, dense cloud cores has been identified in these models through a direct comparison with otherwise identical models calculated by Boss (2002) without magnetic braking. Nevertheless, a cursory examination of the figures and tables reveals that there is a still a large portion of initial conditions space that appears to be permissive of fragmentation of magnetic clouds when the approximate effects of magnetic pressure, tension, and braking are all included. The present models thus suggest that binary and multiple stars may well form from the collapse and fragmentation of magnetic, as well as non-magnetic, dense cloud cores, though perhaps not quite so readily.
+
+Given the approximate nature of the present calculations (Boss 2007), it will be important to try to confirm these results with a true magnetohydrodynamics code. This could be accomplished by adding a numerical solution of the magnetic induction equation to the Boss & Myhill (1992) code. Alternatively, these calculations could be repeated using publically available MHD codes, such as the FLASH adaptive mesh refinement code (e.g., Duffin & Pudritz 2008), though FLASH does not at present include Eddington approximation radiative transfer, unlike the Boss & Myhill (1992) code. Attempting such true MHD calculations stands as a challenge for future work on the question of protostellar collapse and fragmentation.
+
+The numerical calculations were performed on the Carnegie Alpha Cluster, the purchase of which was partially supported by the National Science Foundation under grant MRI-9976645. I thank Sandy Keiser for system management of the cluster, and the referee for numerous helpful suggestions.
+
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+
+\begin{table}
+\begin{tabular}{c c c c c c c}
+\hline \hline
+ model &  \(\rho_{0}/\rho_{R}\) &  \(\Omega_{i}\) &  \(f_{mb}\) &  \(t_{f}\)/\(t_{ff}\) &  result &  Boss(2002) \\
+\hline
+P2BA & 20:1 & \(1.0\times 10^{-13}\) & 0.001 & 4.649 & Q & B \\
+P2BB & 20:1 & \(1.0\times 10^{-13}\) & 0.0001 & 4.683 & Q & B \\
+P2BC & 20:1 & \(3.2\times 10^{-14}\) & 0.001 & 4.602 & B & B \\
+P2BD & 20:1 & \(3.2\times 10^{-14}\) & 0.0001 & 4.603 & B & B \\
+P2BE & 20:1 & \(1.0\times 10^{-14}\) & 0.001 & 4.833 & BB & B \\
+P1BA & 100:1 & \(1.0\times 10^{-13}\) & 0.001 & 9.065 & SB & B \\
+P1BB & 100:1 & \(1.0\times 10^{-13}\) & 0.0001 & 9.071 & SB & B \\
+P1BC & 100:1 & \(3.2\times 10^{-14}\) & 0.001 & 8.942 & SB & B \\
+P1BD & 100:1 & \(3.2\times 10^{-14}\) & 0.0001 & 8.945 & SB & B \\
+P1BE & 100:1 & \(1.0\times 10^{-14}\) & 0.001 & 8.918 & SB & B \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Initial conditions and results for initially prolate cores. In this table and the following, \(\rho_{0}/\rho_{R}\) is the initial ratio of the central to the boundary density, the units for the initial angular velocity \(\Omega_{i}\) are rad s_-1_, and the magnetic braking factor \(f_{mb}\) is dimensionless. The final times \(t_{f}\) are given in units of the initial free fall time \(t_{ff}=(3\pi/32G\rho_{0})^{1/2}=4.7\times 10^{4}\) yr. Q denotes a core that collapses to form a quadruple system, B denotes a binary outcome, BB a binary-bar, and SB a single-bar. The results obtained by Boss (2002) for identical cores but without magnetic braking are shown as well.
+
+\begin{table}
+\begin{tabular}{c c c c c c c}
+\hline \hline
+ model &  \(\rho_{0}/\rho_{R}\) &  \(\Omega_{i}\) &  \(f_{mb}\) &  \(t_{f}\)/\(t_{ff}\) &  result &  Boss(2002) \\
+\hline
+O2BA & 20:1 & \(1.0\times 10^{-13}\) & 0.001 & 4.565 & NC & R \\
+O2BB & 20:1 & \(1.0\times 10^{-13}\) & 0.0001 & 2.456 & R & R \\
+O2BC & 20:1 & \(3.2\times 10^{-14}\) & 0.001 & 2.127 & R & Q \\
+O2BD & 20:1 & \(3.2\times 10^{-14}\) & 0.0001 & 2.134 & R & Q \\
+O2BE & 20:1 & \(1.0\times 10^{-14}\) & 0.001 & 2.109 & R & Q \\
+O1BA & 100:1 & \(1.0\times 10^{-13}\) & 0.001 & 4.656 & R & R \\
+O1BB & 100:1 & \(1.0\times 10^{-13}\) & 0.0001 & 4.636 & R & R \\
+O1BC & 100:1 & \(3.2\times 10^{-14}\) & 0.001 & 4.521 & R & R \\
+O1BD & 100:1 & \(3.2\times 10^{-14}\) & 0.0001 & 4.495 & R & R \\
+O1BE & 100:1 & \(1.0\times 10^{-14}\) & 0.001 & 4.422 & R & Q \\ \hline
+\end{tabular}
+\end{table}
+Table 2: Initial conditions and results for initially oblate cores. R denotes a ring formed, while NC means no significant collapse occurred.
+
+Figure 1: Initial equatorial density contours for prolate core models with 20:1 density contrasts. Maximum density is \(2.0\times 10^{-18}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(1.0\times 10^{17}\) cm in radius.
+
+Figure 2: Equatorial density contours for model P2BB at a time of 4.683 \(t_{ff}\). Maximum density is \(1.0\times 10^{-11}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(4.0\times 10^{14}\) cm in radius. A possible quintuple protostellar system has formed: a central density maximum surrounded by four density maxima.
+
+Figure 3: Equatorial density contours for model P2BD at a time of 4.603 \(t_{ff}\). Maximum density is \(1.3\times 10^{-12}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(4.0\times 10^{14}\) cm in radius. A binary protostellar system has formed.
+
+Figure 4: Equatorial density contours for model P2BE at a time of 4.833 \(t_{ff}\). Maximum density is \(1.6\times 10^{-12}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(4.0\times 10^{14}\) cm in radius. A binary-bar system with four different local density maxima has formed.
+
+Figure 5: Equatorial density contours for model P1BB at a time of 9.069 \(t_{ff}\). Maximum density is \(6.3\times 10^{-12}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(4.0\times 10^{14}\) cm in radius. A single-bar system has formed.
+
+Figure 6: Equatorial temperature contours for model P1BB at a time of 9.069 \(t_{ff}\). Maximum temperature is 25 K. Contours represent changes by a factor of 1.3 in temperature. Region shown is \(4.0\times 10^{14}\) cm in radius.
+
+Figure 7: Initial equatorial density contours for oblate core models with 20:1 density contrasts. Maximum density is \(2.0\times 10^{-18}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(1.0\times 10^{17}\) cm in radius.
+
+Figure 8: Equatorial density contours for model O2BB at a time of 2.456 \(t_{ff}\). Maximum density is \(2.0\times 10^{-12}\) g cm_-3_. Contours represent changes by a factor of 1.3 in density. Region shown is \(2.3\times 10^{15}\) cm in radius. A ring has formed: a strong density minimum occurs at the center of the core.
+

arxiv_ground_truth/0903.4097.md

@@ -0,0 +1,214 @@
+# The Least-Perimeter Partition of a Sphere into Four Equal Areas
+
+Max Engelstein
+
+Department of Mathematics, Yale University, New Haven, CT 06520
+max.engelstein@yale.edu
+
+(Date: July 5, 2024)
+
+###### Abstract.
+
+We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.
+
+Key words and phrases: Minimal partitions, isoperimetric problem, tetrahedral partition 1991 Mathematics Subject Classification: 53C42
+
+###### Contents
+
+1. 1 Introduction
+2. 2 Background and definitions
+3. 3 Area bounds
+4. 4 \(R_{2}\) and \(R_{3}\) contain large triangles
+5. 5 The tetrahedral partition is minimizing
+
+## 1. Introduction
+
+The spherical partition problem asks for the least-perimeter partition of \(\mathbb{S}^{2}\) into \(n\) regions of equal area. The corresponding planar “Honeycomb Conjecture,” open since antiquity and finally proven by Hales [5] in 2001, states that the regular hexagonal tiling provides a least-perimeter way to partition the plane into unit areas. There are five analogous partitions of the sphere into congruent, regular spherical polygons meeting in threes (see Figure 1; for why the edges must meet in threes, see Theorem 2.1), three of which have already been proven minimizing: \(n=2\), a great circle (Bernstein [2]), \(n=3\), three great semi-circles meeting at 120 degrees at antipodal points (Masters [9]), and \(n=12\), a dodecahedral arrangement (Hales [6]). The other two, the \(n=4\) tetrahedral and the \(n=6\) cubical partitions, were conjectured to be minimizing. In this paper we prove the \(n=4\) conjecture:
+
+**Theorem 5.2.**_The least-perimeter partition of the sphere into four equal areas is the tetrahedral partition._
+
+The main difficulty is that in principle each region may have many components. Earlier results by Fejes Tóth [4], Quinn [12], and Engelstein _et al_. [3] required additional assumptions to avoid a proliferation of cases.
+
+Our approach starts with easy estimates to show that each region must have one component that encloses the bulk of the area in that region (Proposition 3.5). Examination of the curvature of the interfaces leads to the result that three of the four regions must contain a triangle with large area (Corollary 3.7, Propositions 4.2 and 4.5). Finally in Section 5 we examine the fourth region and conclude that the tetrahedral partition is minimizing (Theorem 5.2).
+
+Our proof requires little background knowledge beyond what is discussed in Section 2. With the exception of the Gauss-Bonnet theorem, all the ideas and techniques presented are covered by an introductory calculus course.
+
+**Acknowledgements:** This work began with the 2007 Williams College National Science Foundation SMALL undergraduate research Geometry Group. The author thanks the NSF and Williams College for their support of the program. He thanks the 2007 Geometry Group (Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard). He thanks Conor Quinn and Edward Newkirk for their continuing involvement and Nate Harman for his completion of the proof of Proposition 3.5. He thanks Sean Howe and Kestutis Cesnavicius for their helpful comments on earlier drafts. He thanks Yale University for travel support. Finally the author would like to thank Frank Morgan, under whose guidance this research began and without whose patience and persistence its completion would not have been possible.
+
+## 2. Background and definitions
+
+Before delving into the particulars of the \(n=4\) case we recall more general results on the existence and regularity of minimizers.
+
+**Theorem 2.1******(Existence: [11], Thm. 2.3 and Cor. 3.3)**.**: _Given a smooth compact Riemannian surface \(M\) and finitely many positive areas \(A_{i}\) summing to the total area of \(M\), there is a least-perimeter partition of \(M\) into regions of area \(A_{i}\). It is given by finitely many constant-curvature curves meeting in threes at 120 degrees at finitely many points._
+
+It is important to note here that the edges of a minimizing partition are not assumed to be geodesic. In fact, Lamarle [8] and Heppes [7] proved that there are only ten nets of geodesics meeting in threes at 120 degrees on the sphere. These nets are depicted in Figure 1 and include the previously proved minimizers for \(n=2\), \(n=3\), and \(n=12\) and the conjectured minimizers for \(n=4\) and \(n=6\). For other values of \(n\) the solution cannot be geodesic polygons. However, Maurmann _et al._[10] did show that, asymptotically, the perimeter of the solution to the spherical partition problem approaches that of the hexagonal tiling on the plane as \(n\) approaches infinity.
+
+Figure 1. The ten partitions of the sphere by geodesics meeting in threes at \(120\) degrees (picture originally from Almgren and Taylor [1], ©1976 Scientific American).
+
+A further regularity condition involves the concept of pressure:
+
+**Theorem 2.2******([12], Prop. 2.5)**.**: _In a perimeter-minimizing partition each region has a pressure, defined up to addition of a constant, so that the difference in pressure between regions \(A\) and \(B\) is the sum of the (signed) curvatures crossed by any path from the interior of \(B\) to the interior of \(A\)._
+
+**Definition 2.3****.**: Following Quinn [12], we refer to one highest-pressure region as \(R_{1}\), and then in order of decreasing pressure \(R_{2},R_{3}\) and \(R_{4}\). Let \(\kappa_{ij}\) be equal to the pressure of \(R_{i}\) minus the pressure of \(R_{j}\). Note that \(\kappa_{ij}\geq 0\) if \(i<j\) and \(\kappa_{ij}=-\kappa_{ji}\). Theorems 2.1 and 2.2 imply that every edge between \(R_{i}\) and \(R_{j}\) has (signed) curvature \(\kappa_{ij}\).__
+
+With such strong combinatorial and geometric restrictions on perimeter-minimizing partitions it may be tempting to dismiss the spherical partition problem as a simple exercise in case analysis. The crux of the difficulty (as we mentioned in the introduction) is that disconnected regions are allowed. That regions can, _a priori_, have a finite arbitrary number of components renders a näive case analysis almost impossible. On the other hand, under the strong assumption that each region is convex, Fejes Tóth [4] proved that each of the partitions in Figure 1 is minimizing for the areas that it encloses (this also follows easily from the classification of geodesic nets of Figure 1). For the case of \(n=4\), Conor Quinn proved the following, stronger result:
+
+**Theorem 2.4******([12] see Thm. 5.2)**.**: _In a perimeter-minimizing partition of the sphere into four equal areas, if \(R_{1}\) is connected, then that partition is tetrahedral._
+
+This suggests suggest a more focused analysis on the components of \(R_{1}\). In order to avoid confusion we clarify some of our terminology in this manner.
+
+**Definition 2.5****.**: In this paper an_\(m\)-gon_ refers to a spherical polygon with \(m\) edges, each with constant curvature. We write _digon_ instead of \(2\)-gon and often use the colloquial triangle, quadrilateral, or pentagon for \(3\)-gon, \(4\)-gon, or \(5\)-gon. Finally we may abuse terminology and use \(m\)-gon to refer to both the polygon and the region bounded by that polygon (allowing us to refer to the “area” of an \(m\)-gon).__
+
+Before we delve into the analysis let us recall two more results. The first is due to Quinn [12] and is a corollary of Theorem 2.1.
+
+**Corollary 2.6******([12] Lemma 2.11)**.**: _A perimeter-minimizing partition of the sphere does not contain a set of components whose union is a digon, with distinct incident edges._
+
+From this it easily follows that in a non-tetrahedral partition no two triangles share an edge. Our second result hinges on the easy observation that no two components of the same region may share an edge.
+
+**Lemma 2.7****.**: _In a perimeter-minimizing partition of the sphere into four equal areas any component with an odd number of sides is incident to at least one component from every other region._
+
+Specifically a triangle is adjacent to exactly one component from every other region. With all this in mind we can now move on to the numerical analysis of Section 3.
+
+## 3. Area bounds
+
+In this section we show that every region must consist of one large component and then perhaps several small components (Proposition 3.5). Using the isoperimetric inequality on the sphere and the length of the tetrahedral partition we are able to establish strict upper bounds on the perimeter of any one region in a potential minimizer. Our starting point is the famous isoperimetric inequality of Bernstein.
+
+**Lemma 3.1****.**: _[_2_]_ _For given area \(0<A<4\pi\), a curve enclosing area \(A\) on the unit sphere has perimeter \(P\geq B(A)=\sqrt{A(4\pi-A)}\), with equality only for a single circle._
+
+Note that Lemma 3.1 gives a lower bound for the perimeter of a region with area \(A\) even when the region is comprised of several connected components.
+
+**Corollary 3.2****.**: _Given a partition of the sphere into \(n\) equal areas, the total perimeter of the partition is greater than \(2\pi\sqrt{n-1}\)._
+
+Proof.: Each region contains area \(4\pi/n\). By Lemma 3.1 each region must have perimeter at least \(B(4\pi/n)\). Multiply by \(n\) for the number of regions and divide by two (as each edge is incident to at most two regions). Simplifying yields the desired result. ∎
+
+For \(n=4\), Corollary 3.2 yields that the least-perimeter way to partition a sphere into four equal areas must have perimeter at least \(2\pi\sqrt{3}>10.88\), whereas the tetrahedral partition has perimeter \(6\arccos(-1/3)<11.47\) (given by trigonometry). This yields an immediate upper bound on the size of any one region.
+
+**Corollary 3.3****.**: _In a perimeter-minimizing partition of the sphere into four equal areas every region has perimeter less than \(6.62\)._
+
+Proof.: Let \(x\) be the perimeter of some region. By Lemma 3.1 the perimeter \(P\) of the entire partition satisfies \(P>(1/2)(x+3\pi\sqrt{3})\). Yet if the partition is minimizing then we have \(P<11.47\). Numerics yield \(x<6.62\), the desired result. ∎
+
+We will now prove and apply an inequality which will force any region to have one large component (Proposition 3.5).
+
+**Lemma 3.4****.**: _The function (for fixed \(0<k\leq 2\pi\))_
+
+\[f_{k}(t)=\sqrt{t(4\pi-t)}+\sqrt{(k-t)(4\pi-k+t)}\]
+
+_defined on the interval \([0,k]\) is symmetric about the point \(t=k/2\), and \(f_{k}^{\prime}(t)>0\) for all \(0<t<k/2\)._
+
+Proof.: It is evident that the function is symmetric about \(t=k/2\). The radicands are downward parabolas (in \(t\)), so the sum of their square roots is a concave down function. Symmetry implies the desired result. ∎
+
+**Proposition 3.5****.**: _In a perimeter-minimizing partition of the sphere into four equal areas, every region must contain a component with area at least \(23\pi/25\)._
+
+Proof.: Let \(t\) be the area of the largest component in the given region. By Lemma 3.1 we have the inequality \(P(t)\geq B(t)+B(\pi-t),\) where \(P\) is the perimeter of the region. Corollary 3.3 yields \(6.62>B(t)+B(\pi-t)\). On the other hand, setting \(t=23\pi/25\) gives
+
+\[P(\frac{23\pi}{25})\geq B(\frac{23\pi}{25})+B(\frac{2\pi}{25})=\frac{\pi}{25}(\sqrt{23\cdot 77}+14)\approx 7>6.62.\]
+
+So Lemma 3.4 says \(t>23\pi/25\) or \(t<2\pi/25\).
+
+Suppose \(t<2\pi/25\). By Lemma 3.1 when \(A=2\pi/25\), the region must have perimeter greater than \(14\pi/25=7A\). As \(B(x)\) is concave down we have \(B(x)\geq 7x\) for \(x<2\pi/25\). Therefore the perimeter of the region is greater than 7 times the area of the region. So \(t<2\pi/25\) implies that the perimeter of the region is at least \(7\pi\approx 21.99>6.62\), a clear contradiction of Corollary 3.3. ∎
+
+The following Lemma 3.6 due to Quinn [12] will produce a large triangle in \(R_{1}\) (Corollary 3.7).
+
+**Lemma 3.6******([12], Lemma 5.12)**.**: _In the highest-pressure region of a perimeter-minimizing partition, (1) a triangle must have area less than or equal to \(\pi\), (2) a square must have area less than or equal to \(2\pi/3\), (3) a pentagon must have area less than or equal to \(\pi/3\), and (4) all other polygons cannot exist. Equality can only occur when the polygon is geodesic._
+
+Proof.: The result follows directly from Gauss-Bonnet and the convexity of the components of \(R_{1}\). ∎
+
+**Corollary 3.7****.**: _In a perimeter-minimizing partition of the sphere into four equal areas, \(R_{1}\) must contain a triangle, and this triangle must have area at least \(23\pi/25\)._
+
+Proof.: The result is immediate from Lemma 3.6 and Proposition 3.5. ∎
+
+## 4. \(R_{2}\) and \(R_{3}\) contain large triangles
+
+The goals of this section are Propositions 4.2 and 4.5: that \(R_{2}\) and \(R_{3}\) each contains a triangle with area at least \(23\pi/25\). The possibility that the components of \(R_{2}\) or \(R_{3}\) are not convex prohibits us from using Lemma 3.6 and necessitates a closer look at the curvature of the interfaces. We start off by bounding \(\kappa_{12}\).
+
+**Lemma 4.1****.**: _In a least-perimeter partition of the sphere into four equal areas, \(\kappa_{12}<1/21\)._
+
+Proof.: By Corollary 3.7, \(R_{1}\) has a triangle, \(T\), of area \(A_{T}\geq 23\pi/25\). By Gauss-Bonnet, the perimeter \(P\) and exterior angles \(\alpha_{i}\), of this triangle satisfy
+
+\[2\pi=A_{T}+\int_{\partial T}\kappa ds+\sum\alpha_{i}\geq\frac{23\pi}{25}+P\kappa_{12}+\pi,\]
+
+so \(P\kappa_{12}\leq 2\pi/25.\) By Lemma 3.1, \(P\geq B(23\pi/25)\). Therefore \(\kappa_{12}<1/21\). ∎
+
+Now we are able to establish an analogue to Corollary 3.7 for \(R_{2}\).
+
+**Proposition 4.2****.**: _In a least-perimeter partition of the sphere into four equal areas, \(R_{2}\) contains a triangle of area at least \(23\pi/25\)._
+
+Proof.: Proposition 3.5 says that \(R_{2}\) must have a component with area no less than \(23\pi/25\). Assume by way of contradiction that this component has at least four sides. Then Gauss-Bonnet gives
+
+\[\frac{23\pi}{25}-\kappa_{12}P_{12}+\kappa_{r}P_{r}+\frac{4\pi}{3}\leq 2\pi,\]
+
+where \(P_{12}\) is the perimeter between \(R_{1}\) and \(R_{2}\). \(\kappa_{r}P_{r}\) represents the (positive) contribution of curvature from lower pressure regions. Combining terms we get: \(\pi(23/25-2/3)\leq\kappa_{12}P_{12}\). Bounding \(\kappa_{12}\) using Lemma 4.1 yields \(19\pi/75\leq P_{12}/21.\) Isolating \(P_{12}\) gives \(P_{12}\geq 133\pi/25>12\), obviously contradicting Corollary 3.3. ∎
+
+In order to establish an analogue to Lemma 4.1 for \(R_{3}\), we must insure that \(R_{2}\) does not occupy too much of the perimeter of \(R_{1}\)’s large triangle. A quick corollary bounds the length of the side of \(R_{1}\)’s large triangle which is incident to \(R_{2}\) (a side we know exists by Lemma 2.7).
+
+**Corollary 4.3****.**: _In a least-perimeter partition of the sphere into four equal areas, let \(P\) be the perimeter of the large triangle in \(R_{1}\), and let \(l\) be the length of the side incident to \(R_{2}\) in that triangle. Then \(l\leq P/3\)._
+
+Proof.: If the partition in question is tetrahedral, then the statement is trivial. Assume that it is not tetrahedral and, to obtain a contradiction, that \(l>P/3\). By Corollary 2.6 no two triangles are incident to one another in a non-tetrahedral minimizing partition. Therefore the perimeter of \(R_{2}\) is at least \(P_{2}+l\), where \(P_{2}\) is the perimeter of the large triangle in \(R_{2}\) (whose existence is established in Proposition 4.2). By Lemma 3.1 we have that the perimeter of \(R_{2}\) is at least \((4/3)B(23\pi/25)>7\), which contradicts the partition’s minimality by Corollary 3.3. ∎
+
+Now we proceed as in the \(R_{2}\) case; first we bound \(\kappa_{13}\).
+
+**Lemma 4.4****.**: _In a least-perimeter partition of the sphere into four equal areas, \(\kappa_{13}<1/14\)._
+
+Proof.: By Corollary 3.7\(R_{1}\) has a large triangle. Let \(P\) be the perimeter of this triangle and \(P^{\prime}\) the lengths of the side of the triangle which are not incident to \(R_{2}\). Then Corollary 4.3 states that \(P\leq 3P^{\prime}/2\). Using Lemma 3.1 to obtain a lower bound for \(P\) we write \(P^{\prime}\geq(2/3)B(23\pi/25).\) Applying Gauss-Bonnet to this large triangle gives the inequality \(23\pi/25+\kappa_{13}P^{\prime}\leq\pi.\) Substituting the bound on \(P^{\prime}\) and simplifying results in the desired inequality \(\kappa_{13}\leq 3/\sqrt{23\cdot 77}<1/14\). ∎
+
+In the same vein as Proposition 4.2 we now prove that \(R_{3}\) must contain a triangle with area at least \(23\pi/25\).
+
+**Proposition 4.5****.**: _In a least-perimeter partition of the sphere into four equal areas, \(R_{3}\) contains a triangle with area at least \(23\pi/25\)._
+
+Proof.: By Proposition 3.5\(R_{3}\) must contain some component with area at least \(23\pi/25\). For the sake of contradiction, assume that component has at least four sides. Let \(P\) be the perimeter of this component, then Gauss-Bonnet yields
+
+\[\frac{23\pi}{25}-\kappa_{13}P+\frac{4\pi}{3}\leq 2\pi,\]
+
+or \(\kappa_{13}P\geq 19\pi/75.\) By Lemma 4.4 we get that \(\kappa_{13}<1/14\), which means that \(P>266\pi/75>11\) which contradicts the partition’s minimality by Corollary 3.3. ∎
+
+Now we can establish an analogue to Corollary 4.3 for \(R_{3}\).
+
+**Corollary 4.6****.**: _In a least-perimeter partition of the sphere into four equal areas, let \(P\) be the perimeter of the large triangle in \(R_{1}\), and let \(l\) be the length of the side incident to \(R_{3}\) in that triangle. Then \(l\leq P/3\)._
+
+Proof.: If the partition is tetrahedral, then the statement is trivial. Assume that it is not tetrahedral and, to obtain a contradiction, that \(l>P/3\). By Corollary 2.6 no two triangles are incident to one another in a non-tetrahedral minimizing partition. Therefore the perimeter of \(R_{3}\) is at least \(P_{3}+l\), where \(P_{3}\) is the perimeter of the large triangle in \(R_{3}\) (whose existence is established in Proposition 4.5). By Lemma 3.1 we have that the perimeter of \(R_{3}\) is at least \((4/3)B(23\pi/25)>7\), which contradicts the partition’s minimality by Corollary 3.3. ∎
+
+## 5. The tetrahedral partition is minimizing
+
+In this section we reach our goal in Theorem 5.2, which states that the perimeter-minimizing partition of the sphere into four equal areas is the tetrahedral partition. We require only one lemma, a lower bound for \(\kappa_{14}\).
+
+**Lemma 5.1****.**: _In a non-tetrahedral perimeter-minimizing partition of the sphere into four equal areas we have \(\kappa_{14}>1/2\)._
+
+Proof.: Let \(P\) be the perimeter of the large triangle in \(R_{1}\) (which we know exists by Corollary 3.7) and \(P_{r}\) be the rest of the perimeter of \(R_{1}\). Corollary 3.3 then gives
+
+\[6.62>P+P_{r}\geq B(23\pi/25)+P_{r}\]
+
+where the second inequality is Lemma 3.1. This yields \(P_{r}<1.34\).
+
+Since the partition is non-tetrahedral, by Theorem 2.4\(R_{1}\) must have another component, and this component has at most five sides (Lemma 3.6). Use Gauss-Bonnet on this component to obtain a second inequality on \(P_{r}\):
+
+\[\kappa_{14}P_{r}\geq\frac{\pi}{3}-\frac{2\pi}{25}\]
+
+and combine the two inequalities to get that \(\kappa_{14}>1/2\). ∎
+
+Now we reach our ultimate goal.
+
+**Theorem 5.2****.**: _The least-perimeter partition of the sphere into four equal areas is the tetrahedral partition._
+
+Proof.: Assume that \(R_{1}\) is non-tetrahedral. Let \(P\) be the perimeter of the large triangle in \(R_{1}\) (which exists by Corollary 3.7), and let \(l_{4}\) be the length of the side of this large triangle incident to \(R_{4}\). By Corollaries 4.3 and 4.6 and Lemma 3.1 we have \(l_{4}\geq P/3\geq(1/3)B(23\pi/25)\).
+
+Apply Gauss-Bonnet to the large triangle in \(R_{1}\) to get
+
+\[\frac{23\pi}{25}+\frac{\kappa_{14}B(\frac{23\pi}{25})}{3}\leq\pi.\]
+
+Simplify and isolate \(\kappa_{14}\) to obtain \(\kappa_{14}\leq 6/\sqrt{23\cdot 77}<1/7\), a clear contradiction of Lemma 5.1. ∎
+
+## References
+
+* [AT76] F.J. Almgren Jr. and J.E. Taylor, Geometry of soap films. _Sci. Amer._**235** (1976), 82-93.
+* [B05] F. Bernstein, Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. _Math. Ann._**60** (1905), 117 - 136.
+* [EMM08] M. Engelstein, Q. Maurmann, A. Marcuccio, and T. Pritchard, Sphere partitions and density problems. _NY J. Math._**15** (2009), 97-123.
+* [FT64] L. Fejes Tóth. ”Regular Figures.” Pergamom Press, New York, 1964.
+* [H01] T. Hales, The honeycomb conjecture. _Disc. Comput. Geom._**25** (2001), 1-22.
+* [H02] T. Hales, The honeycomb conjecture on the sphere. Arxiv.org, 2002.
+* [He95] A. Heppes, On surface-minimizing polyhedral decompositions. _Disc. Comp. Geom._**13** (1995), 529-539.
+* [L64] E. Lemarle, Sur la stabilité des systémes liquides en lames minces. _Mém. Acad. R. Belg._**35** (1864), 3-104.
+* [Ma96] J. Masters, The perimeter-minimizing enclosure of two areas in \({\bf S}^{2}\). _Real Anal. Exchange_**22** (1996), 1-10.
+* [MEM08] Q. Maurmann, M. Engelstein, A. Marcuccio, and T. Pritchard, Asymptotics of perimeter-minimizing partitions. _Can. Math. Bull._, to appear.
+* [Mo92] F. Morgan, Soap bubbles in \({\bf R}^{2}\) and in surfaces. _Pacific J. of Math._**165** (1994), 347-361.
+* [Q07] C. Quinn, Least-perimeter partitions of the sphere. _Rose-Hulman Und. Math. J._**8** (2007), no. 2.
+

arxiv_ground_truth/0903.4561.md

@@ -0,0 +1,599 @@
+To appear in _The Oxford Handbook on Nanoscience and Nanotechnology: Frontiers and Advances, eds. A.V. Narlikar and Y.Y. Fu, Vol. I, Ch. 21_
+ Sergei Sergeenkov
+
+Departamento de Física, Universidade Federal da Paraíba, João Pessoa, Brazil
+
+**2D ARRAYS OF JOSEPHSON NANOCONTACTS AND NANOGRANULAR SUPERCONDUCTORS**
+ ABSTRACT
+
+By introducing a realistic model of nanogranular superconductors (NGS) based on 2D arrays of Josephson nanocontacts (created by a network of twin-boundary dislocations with strain fields acting as insulating barriers between hole-rich domains), in this Chapter we present some novel phenomena related to mechanical, magnetic, electric and transport properties of NGS in underdoped single crystals. In particular, we consider chemically induced magnetoelectric effects and flux driven temperature oscillations of thermal expansion coefficient. We also predict a giant enhancement of the nonlinear thermal conductivity of NGS reaching up to \(500\%\) when the intrinsically induced chemoelectric field (created by the gradient of the chemical potential due to segregation of hole producing oxygen vacancies) closely matches the externally produced thermoelectric field. The estimates of the model parameters suggest quite an optimistic possibility to experimentally realize these promising and important for applications effects in non-stoichiometric NGS and artificially prepared arrays of Josephson nanocontacts.
+
+1. INTRODUCTION
+
+Inspired by new possibilities offered by the cutting-edge nanotechnologies, the experimental and theoretical physics of increasingly sophisticated mesoscopic quantum devices, heavily based on Josephson junctions (JJs) and their arrays (JJAs), is becoming one of the most exciting and rapidly growing areas of modern science (for reviews on charge and spin effects in mesoscopic 2D JJs and quantum-state engineering with Josephson devices, see, e.g., Newrock et al. 2000, Makhlin et al. 2001, Krive et al. 2004, Sergeenkov 2006, Beloborodov et al. 2007). In particular, a remarkable increase of the measurements technique resolution made it possible to experimentally detect such interesting phenomena as flux avalanches (Altshuler and Johansen 2004) and geometric quantization (Sergeenkov and Araujo-Moreira 2004) as well as flux dominated behavior of heat capacity (Bourgeois et al. 2005) both in JJs and JJAs.
+
+Recently, it was realized that JJAs can be also used as quantum channels to transfer quantum information between distant sites (Makhlin et al. 2001, Wendin and Shumeiko 2007) through the implementation of the so-called superconducting qubits which take advantage of both charge and phase degrees of freedom.
+
+Both granular superconductors and artificially prepared JJAs proved useful in studying the numerous quantum (charging) effects in these interesting systems, including Coulomb blockade of Cooper pair tunneling (Iansity et al. 1988), Bloch oscillations (Haviland et al. 1991), propagation of quantum ballistic vortices (van der Zant 1996), spin-tunneling related effects using specially designed \(SFS\)-type junctions (Ryazanov et al. 2001, Golubov et al. 2002), novel Coulomb effects in \(SINIS\)-type nanoscale junctions (Ostrovsky and Feigel’man 2004), and dynamical AC reentrance (Araujo-Moreira et al. 1997, Barbara et al. 1999, Araujo-Moreira et al. 2005).
+
+At the same time, given a rather specific magnetostrictive (Sergeenkov and Ausloos 1993) and piezomagnetic (Sergeenkov 1998b, Sergeenkov 1999) response of Josephson systems, one can expect some nontrivial behavior of the thermal expansion (TE) coefficient in JJs as well (Sergeenkov et al. 2007). Of special interest are the properties of TE in applied magnetic field. For example, some superconductors like \(Ba_{1-x}K_{x}BiO_{3}\), \(BaPb_{x}Bi_{1-x}O_{3}\) and \(La_{2-x}Sr_{x}CuO_{4}\) were found (Anshukova et al. 2000) to exhibit anomalous temperature behavior of both magnetostriction and TE which were attributed to the field-induced suppression of the superstructural ordering in the oxygen sublattices of these systems.
+
+The imaging of the granular structure in underdoped \(Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}\) crystals (Lang et al. 2002) revealed an apparent segregation of its electronic structure into superconducting domains (of the order of a few nanometers) located in an electronically distinct background. In particular, it was found that at low levels of hole doping (\(\delta<0.2\)), the holes become concentrated at certain hole-rich domains. (In this regard, it is interesting to mention a somewhat similar phenomenon of ”chemical localization” that takes place in materials, composed of atoms of only metallic elements, exhibiting metal-insulator transitions, see, e.g., Gantmakher 2002.) Tunneling between such domains leads to intrinsic nanogranular superconductivity (NGS) in high-\(T_{c}\) superconductors (HTS). Probably one of the first examples of NGS was observed in \(YBa_{2}Cu_{3}O_{7-\delta}\) single crystals in the form of the so-called ”fishtail” anomaly of magnetization (Daeumling et al. 1990). The granular behavior has been related to the 2D clusters of oxygen defects forming twin boundaries (TBs) or dislocation walls within \(CuO\) plane that restrict supercurrent flow and allow excess flux to enter the crystal. Indeed, there are serious arguments to consider the TB in HTS as insulating regions of the Josephson SIS-type structure. An average distance between boundaries is essentially less than the grain size. In particular, the networks of localized grain boundary dislocations with the spacing ranged from \(10nm\) to \(100nm\) have been observed (Daeumling et al. 1990) which produce effectively continuous normal or insulating barriers at the grain boundaries. It was also verified that the processes of the oxygen ordering in HTS leads to the continuous change of the lattice period along TB with the change of the oxygen content. Besides, a destruction of bulk superconductivity in these non-stoichiometric materials with increasing the oxygen deficiency parameter \(\delta\) was found to follow a classical percolation theory (Gantmakher et al. 1990).
+
+In addition to their importance for understanding the underlying microscopic mechanisms governing HTS materials, the above experiments can provide rather versatile tools for designing chemically-controlled atomic scale JJs and JJAs with pre-selected properties needed for manufacturing the modern quantum devices (Sergeenkov 2001, Araujo-Moreira et al. 2002, Sergeenkov 2003, Sergeenkov 2006). Moreover, as we shall see below, NGS based phenomena can shed some light on the origin and evolution of the so-called paramagnetic Meissner effect (PME) which manifests itself both in high-\(T_{c}\) and conventional superconductors (Geim et al. 1998, De Leo and Rotoli 2002, Li 2003) and is usually associated with the presence of \(\pi\)-junctions and/or unconventional (\(d\)-wave) pairing symmetry.
+
+In this Chapter we present numerous novel phenomena related to the magnetic, electric, elastic and transport properties of Josephson nanocontacts and NGS. The paper is organized as follows. In Section 1, a realistic model of NGS is introduced which is based on 2D JJAs created by a regular network of twin-boundary dislocations with strain fields acting as an insulating barrier between hole-rich domains (like in underdoped crystals). In Section 2, we consider some phase-related phenomena expected to occur in NGS, such as Josephson chemomagnetism and magnetoconcentration effect. Section 3 is devoted to a thorough discussion of charge-related polarization phenomena in NGS, including such topics as chemomagnetoelectricity, magnetocapacitance, charge analog of the ”fishtail” (magnetization) anomaly, and field-tuned weakening of the chemically-induced Coulomb blockade. In Section 4 we present our latest results on the influence of an intrinsic chemical pressure (created by the gradient of the chemical potential due to segregation of hole producing oxygen vacancies) on temperature behavior of the nonlinear thermal conductivity (NLTC) of NGS. In particular, our theoretical analysis (based on the inductive model of 2D JJAs) predicts a giant enhancement of NLTC reaching up to \(500\%\) when the intrinsically induced chemoelectric field \({\bf E}_{\mu}=\frac{1}{2e}\nabla\mu\) closely matches thermoelectric field \({\bf E}_{T}=S_{T}\nabla T\). And finally, by introducing a concept of thermal expansion (TE) of Josephson contact (as an elastic response of JJ to an effective stress field), in Section 5 we consider the temperature and magnetic field dependence of the TE coefficient \(\alpha(T,H)\) in a small single JJ and in a single plaquette (a prototype of the simplest JJA). In particular, we found that in addition to expected _field_ oscillations due to Fraunhofer-like dependence of the critical current, \(\alpha\) of a small single junction also exhibits strong flux driven _temperature_ oscillations near \(T_{C}\). The condition under which all the effects predicted here can be experimentally realized in artificially prepared JJAs and NGS are also discussed. Some important conclusions of the present study are drawn in Section 6.
+
+2. MODEL OF NANOSCOPIC JOSEPHSON JUNCTION ARRAYS
+
+As is well-known, the presence of a homogeneous chemical potential \(\mu\) through a single JJ leads to the AC Josephson effect with time dependent phase difference \(\partial\phi/\partial t=\mu/\hbar\). In this Section, we will consider some effects in dislocation induced JJ caused by a local variation of excess hole concentration \(c({\bf x})\) under the chemical pressure (described by inhomogeneous chemical potential \(\mu({\bf x})\)) equivalent to presence of the strain field of 2D dislocation array \(\epsilon({\bf x})\) forming this Josephson contact.
+
+To understand how NGS manifests itself in non-stoichiometric crystals, let us invoke an analogy with the previously discussed dislocation models of twinning-induced superconductivity (Khaikin and Khlyustikov 1981) and grain-boundary Josephson junctions (Sergeenkov 1999). Recall that under plastic deformation, grain boundaries (GBs) (which are the natural sources of weak links in HTS), move rather rapidly via the movement of the grain boundary dislocations (GBDs) comprising these GBs. At the same time, observed (Daeumling et al. 1990, Lang et al. 2002, Yang et al. 1993, Moeckley et al. 1993) in HTS single crystals regular 2D dislocation networks of oxygen depleted regions (generated by the dissociation of \(<110>\) twinning dislocations) with the size \(d_{0}\) of a few Burgers vectors, forming a triangular lattice with a spacing \(d\geq d_{0}\) ranging from \(10nm\) to \(100nm\), can provide quite a realistic possibility for the existence of 2D Josephson network within \(CuO\) plane. Recall furthermore that in a \(d\)-wave orthorhombic \(YBCO\) crystal TBs are represented by tetragonal regions (in which all dislocations are equally spaced by \(d_{0}\) and have the same Burgers vector \({\bf a}\) parallel to \(y\)-axis within \(CuO\) plane) which produce screened strain fields (Gurevich and Pashitskii 1997) \(\epsilon({\bf x})=\epsilon_{0}e^{-{\mid{{\bf x}}\mid}/d_{0}}\) with \({\mid{{\bf x}}\mid}=\sqrt{x^{2}+y^{2}}\).
+
+Though in \(YBa_{2}Cu_{3}O_{7-\delta}\) the ordinary oxygen diffusion \(D=D_{0}e^{-U_{d}/k_{B}T}\) is extremely slow even near \(T_{C}\) (due to a rather high value of the activation energy \(U_{d}\) in these materials, typically \(U_{d}\simeq 1eV\)), in underdoped crystals (with oxygen-induced dislocations) there is a real possibility to facilitate oxygen transport via the so-called osmotic (pumping) mechanism (Girifalco 1973, Sergeenkov 1995) which relates a local value of the chemical potential (chemical pressure) \(\mu({\bf x})=\mu(0)+\nabla\mu\cdot{\bf x}\) with a local concentration of point defects as follows \(c({\bf x})=e^{-\mu({\bf x})/k_{B}T}\). Indeed, when in such a crystal there exists a nonequilibrium concentration of vacancies, dislocation is moved for atomic distance \(a\) by adding excess vacancies to the extraplane edge. The produced work is simply equal to the chemical potential of added vacancies. What is important, this mechanism allows us to explicitly incorporate the oxygen deficiency parameter \(\delta\) into our model by relating it to the excess oxygen concentration of vacancies \({\bf c_{v}}\equiv c(0)\) as follows \(\delta=1-{\bf c_{v}}\). As a result, the chemical potential of the single vacancy reads \(\mu_{v}\equiv\mu(0)=-k_{B}T\log(1-\delta)\simeq k_{B}T\delta\). Remarkably, the same osmotic mechanism was used by Gurevich and Pashitskii (1997) to discuss the modification of oxygen vacancies concentration in the presence of the TB strain field. In particular, they argue that the change of \(\epsilon({\bf x})\) under an applied or chemically induced pressure results in a significant oxygen redistribution producing a highly inhomogeneous filamentary structure of oxygen-deficient nonsuperconducting regions along GB (Moeckley et al. 1993) (for underdoped superconductors, the vacancies tend to concentrate in the regions of compressed material). Hence, assuming the following connection between the variation of mechanical and chemical properties of planar defects, namely \(\mu({\bf x})=K\Omega_{0}\epsilon({\bf x})\) (where \(\Omega_{0}\) is an effective atomic volume of the vacancy and \(K\) is the bulk elastic modulus), we can study the properties of TB induced JJs under intrinsic chemical pressure \(\nabla\mu\) (created by the variation of the oxygen doping parameter \(\delta\)). More specifically, a single \(SIS\) type junction (comprising a Josephson network) is formed around TB due to a local depression of the superconducting order parameter \(\Delta({\bf x})\propto\epsilon({\bf x})\) over distance \(d_{0}\) producing thus a weak link with (oxygen deficiency \(\delta\) dependent) Josephson coupling \(J(\delta)=\epsilon({\bf x})J_{0}=J_{0}(\delta)e^{-{\mid{{\bf x}}\mid}/d_{0}}\) where \(J_{0}(\delta)=\epsilon_{0}J_{0}=(\mu_{v}/K\Omega_{0})J_{0}\) (here \(J_{0}\propto\Delta_{0}/R_{n}\) with \(R_{n}\) being a resistance of the junction). Thus, the present model indeed describes chemically induced NGS in underdoped systems (with \(\delta\neq 0\)) because, in accordance with the observations, for stoichiometric situation (when \(\delta\simeq 0\)), the Josephson coupling \(J(\delta)\simeq 0\) and the system loses its explicitly granular signature.
+
+To adequately describe chemomagnetic properties of an intrinsically granular superconductor, we employ a model of 2D overdamped Josephson junction array which is based on the well known Hamiltonian
+
+\[{\cal H}=\sum_{ij}^{N}J_{ij}(1-\cos\phi_{ij})+\sum_{ij}^{N}\frac{q_{i}q_{j}}{C_{ij}}\] (1)
+
+and introduces a short-range interaction between \(N\) junctions (which are formed around oxygen-rich superconducting areas with phases \(\phi_{i}(t)\)), arranged in a two-dimensional (2D) lattice with coordinates \({\bf x_{i}}=(x_{i},y_{i})\). The areas are separated by oxygen-poor insulating boundaries (created by TB strain fields \(\epsilon({\bf x}_{ij})\)) producing a short-range Josephson coupling \(J_{ij}=J_{0}(\delta)e^{-{\mid{{\bf x}_{ij}}\mid}/d}\). Thus, typically for granular superconductors, the Josephson energy of the array varies exponentially with the distance \({\bf x}_{ij}={\bf x}_{i}-{\bf x}_{j}\) between neighboring junctions (with \(d\) being an average junction size). As usual, the second term in the rhs of Eq.(1) accounts for Coulomb effects where \(q_{i}=-2en_{i}\) is the junction charge with \(n_{i}\) being the pair number operator. Naturally, the same strain fields \(\epsilon({\bf x}_{ij})\) will be responsible for dielectric properties of oxygen-depleted regions as well via the \(\delta\)-dependent capacitance tensor \(C_{ij}(\delta)=C[\epsilon({\bf x}_{ij})]\).
+
+If, in addition to the chemical pressure \(\nabla\mu({\bf x})=K\Omega_{0}\nabla\epsilon({\bf x})\), the network of superconducting grains is under the influence of an applied frustrating magnetic field \({\bf B}\), the total phase difference through the contact reads
+
+\[\phi_{ij}(t)=\phi^{0}_{ij}+\frac{\pi w}{\Phi_{0}}({\bf x}_{ij}\wedge{\bf n}_{ij})\cdot{\bf B}+\frac{\nabla\mu\cdot{\bf x}_{ij}t}{\hbar},\] (2)
+
+where \(\phi^{0}_{ij}\) is the initial phase difference (see below), \({\bf n}_{ij}={\bf X}_{ij}/{\mid{{\bf X}_{ij}}\mid}\) with \({\bf X}_{ij}=({\bf x}_{i}+{\bf x}_{j})/2\), and \(w=2\lambda_{L}(T)+l\) with \(\lambda_{L}\) being the London penetration depth of superconducting area and \(l\) an insulator thickness which, within the discussed here scenario, is simply equal to the TB thickness (Sergeenkov 1995).
+
+To neglect the influence of the self-field effects in a real material, the corresponding Josephson penetration length \(\lambda_{J}=\sqrt{\Phi_{0}/2\pi\mu_{0}j_{c}w}\) must be larger than the junction size \(d\). Here \(j_{c}\) is the critical current density of superconducting (hole-rich) area. As we shall see below, this condition is rather well satisfied for HTS single crystals.
+
+Within our scenario, the sheet magnetization **M** of 2D granular superconductor is defined via the average Josephson energy of the array
+
+\[<{\cal H}>=\int_{0}^{\tau}\frac{dt}{\tau}\int\frac{d^{2}x}{s}{\cal H}({\bf x},t)\] (3)
+
+as follows
+
+\[{\bf M}({\bf B},\delta)\equiv-\frac{\partial<{\cal H}>}{\partial{\bf B}},\] (4)
+
+where \(s=2\pi d^{2}\) is properly defined normalization area, \(\tau\) is a characteristic Josephson time, and we made a usual substitution \(\frac{1}{N}\sum_{ij}A_{ij}(t)\to\frac{1}{s}\int d^{2}xA({\bf x},t)\) valid in the long-wavelength approximation (Sergeenkov 2002).
+
+To capture the very essence of the superconducting analog of the chemomagnetic effect, in what follows we assume for simplicity that a _stoichiometric sample_ (with \(\delta\simeq 0\)) does not possess any spontaneous magnetization at zero magnetic field (that is \({\bf M}(0,0)=0\)) and that its Meissner response to a small applied field **B** is purely diamagnetic (that is \({\bf M}({\bf B},0)\simeq-{\bf B}\)). According to Eq.(4), this condition implies \(\phi_{ij}^{0}=2\pi m\) for the initial phase difference with \(m=0,\pm 1,\pm 2,..\).
+
+Taking the applied magnetic field along the \(c\)-axis (and normal to the \(CuO\) plane), we obtain finally
+
+\[{\bf M}({\bf B},\delta)=-{\bf M}_{0}(\delta)\frac{{\bf b}-{\bf b}_{\mu}}{(1+{\bf b}^{2})(1+({\bf b}-{\bf b}_{\mu})^{2})}\] (5)
+
+for the chemically-induced sheet magnetization of the 2D Josephson network. Here \({\bf M}_{0}(\delta)=J_{0}(\delta)/{\bf B}_{0}\) with \(J_{0}(\delta)\) defined earlier, \({\bf b}={\bf B}/{\bf B}_{0}\), and \({\bf b}_{\mu}={\bf B}_{\mu}/{\bf B}_{0}\simeq(k_{B}T\tau/\hbar)\delta\) where \({\bf B}_{\mu}(\delta)=(\mu_{v}\tau/\hbar){\bf B}_{0}\) is the chemically-induced contribution (which disappears in optimally doped systems with \(\delta\simeq 0\)), and \({\bf B}_{0}=\Phi_{0}/wd\) is a characteristic Josephson field.
+
+Fig. 1: The susceptibility as a function of applied magnetic field for different values of oxygen deficiency parameter: \({\bf\delta}\simeq 0\) (solid line), \({\bf\delta}=0.05\) (dashed line), and \({\bf\delta}=0.1\) (dotted line).
+
+Fig. 2: The oxygen deficiency induced susceptibility for different values of applied magnetic field (chemomagnetism).
+
+Fig.1 shows changes of the initial (stoichiometric) diamagnetic susceptibility \(\chi({\bf B},\delta)=\partial{\bf M}({\bf B},\delta)/\partial{\bf B}\) (solid line) with oxygen deficiency \(\delta\). As is seen, even relatively small values of \(\delta\) parameter render a low field Meissner phase strongly paramagnetic (dotted and dashed lines). Fig.2 presents concentration (deficiency) induced susceptibility \(\chi({\bf B},\delta)/\chi_{0}(0)\) for different values of applied magnetic field \({\bf b}={\bf B}/{\bf B}_{0}\) including a true _chemomagnetic_ effect (solid line). According to Eq.(5), the initially diamagnetic Meissner effect turns paramagnetic as soon as the chemomagnetic contribution \({\bf B}_{\mu}(\delta)\) exceeds an applied magnetic field \({\bf B}\). To see whether this can actually happen in a real material, let us estimate a magnitude of the chemomagnetic field \({\bf B}_{\mu}\). Typically (Daeumling et al. 1990, Gurevich and Pashitskii 1997), for HTS single crystals \(\lambda_{L}(0)\approx 150nm\) and \(d\simeq 10nm\), leading to \({\bf B}_{0}\simeq 0.5T\). Using \(\tau\simeq\hbar/\mu_{v}\) and \(j_{c}=10^{10}A/m^{2}\) as a pertinent characteristic time and the typical value of the critical current density, respectively, we arrive at the following estimate of the chemomagnetic field \({\bf B}_{\mu}(\delta)\simeq 0.5B_{0}\) for \(\delta=0.05\). Thus, the predicted chemically induced PME should be observable for applied magnetic fields \({\bf B}\simeq 0.5B_{0}\simeq 0.25T\) which are actually much higher than the fields needed to observe the previously discussed piezomagnetism and stress induced PME in high-\(T_{c}\) ceramics (Sergeenkov 1999). Notice that for the above set of parameters, the Josephson length \(\lambda_{J}\simeq 1\mu m\), which means that the assumed here small-junction approximation (with \(d\ll\lambda_{J}\)) is valid and the so-called ”self-field” effects can be safely neglected.
+
+Fig. 3: Magnetic field dependence of the oxygen vacancy concentration (magnetoconcentration effect).
+
+So far, we neglected a possible field dependence of the chemical potential \(\mu_{v}\) of oxygen vacancies. However, in high enough applied magnetic fields \({\bf B}\), the field-induced change of the chemical potential \(\Delta\mu_{v}({\bf B})\equiv\mu_{v}({\bf B})-\mu_{v}(0)\) becomes tangible and should be taken into account. As is well-known (Abrikosov 1988, Sergeenkov and Ausloos 1999), in a superconducting state \(\Delta\mu_{v}({\bf B})=-{\bf M}({\bf B}){\bf B}/n\), where \({\bf M}({\bf B})\) is the corresponding magnetization, and \(n\) is the relevant carriers number density. At the same time, within our scenario, the chemical potential of a single oxygen vacancy \(\mu_{v}\) depends on the concentration of oxygen vacancies (through deficiency parameter \(\delta\)). As a result, two different effects are possible related respectively to magnetic field dependence of \(\mu_{v}({\bf B})\) and to its dependence on magnetization \(\mu_{v}({\bf M})\). The former is nothing else but a superconducting analog of the so-called _magnetoconcentration_ effect which was predicted and observed in inhomogeneously doped semiconductors (Akopyan et al.1990) with field-induced creation of oxygen vacancies \({\bf c_{v}}({\bf B})={\bf c_{v}}(0)\exp(-\Delta\mu_{v}({\bf B})/k_{B}T)\), while the latter results in a ”fishtail”-like behavior of the magnetization. Let us start with the magnetoconcentration effect. Fig.3 depicts the predicted field-induced creation of oxygen vacancies \({\bf c_{v}}({\bf B})\) using the above-obtained magnetization \({\bf M}({\bf B},\delta)\) (see Fig.1 and Eq.(5)). We also assumed, for simplicity, a complete stoichiometry of the system in a zero magnetic field (with \({\bf c_{v}}(0)=1\)). Notice that \({\bf c_{v}}({\bf B})\) exhibits a maximum at \({\bf c_{m}}\simeq 0.23\) for applied fields \({\bf B}={\bf B}_{0}\) (in agreement with the classical percolative behavior observed in non-stoichiometric \(YBa_{2}Cu_{3}O_{7-\delta}\) samples (Daeumling et al. 1990, Gantmakher et al. 1990, Moeckley et al. 1993). Finally, let us show that in underdoped crystals the above-discussed osmotic mechanism of oxygen transport is indeed much more effective than a traditional diffusion. Using typical \(YBCO\) parameters (Gurevich and Pashitskii 1997), \(\epsilon_{0}=0.01\), \(\Omega_{0}=a_{0}^{3}\) with \(a_{0}=0.2nm\), and \(K=115GPa\), we have \(\mu_{v}(0)=\epsilon_{0}K\Omega_{0}\simeq 1meV\) for a zero-field value of the chemical potential in HTS crystals, which leads to creation of excess vacancies with concentration \({\bf c_{v}}(0)=e^{-\mu_{v}(0)/k_{B}T}\simeq 0.75\) (equivalent to a deficiency value of \(\delta(0)\simeq 0.25\)) at \(T=T_{C}\), while the probability of oxygen diffusion in these materials (governed by a rather high activation energy \(U_{d}\simeq 1eV\)) is extremely low under the same conditions because \(D\propto e^{-U_{d}/k_{B}T_{C}}\ll 1\). On the other hand, the change of the chemical potential in applied magnetic field can reach as much as (Sergeenkov and Ausloos 1999) \(\Delta\mu_{v}({\bf B})\simeq 0.5meV\) for \({\bf B}=0.5T\), which is quite comparable with the above-mentioned zero-field value of \(\mu_{v}(0)\).
+
+Fig. 4: A ”fishtail”-like behavior of magnetization in applied magnetic field in the presence of magnetoconcentration effect (with field-induced oxygen vacancies \({\bf c_{v}}({\bf B})\), see Fig.3) for three values of field-free deficiency parameter: \(\delta(0)\simeq 0\) (solid line), \(\delta(0)=0.1\) (dashed line), and \(\delta(0)=0.2\) (dotted line).
+
+Let us turn now to the second effect related to the magnetization dependence of the chemical potential \(\mu_{v}({\bf M}({\bf B}))\). In this case, in view of Eq.(2), the phase difference will acquire an extra \({\bf M}({\bf B})\) dependent contribution and as a result the r.h.s. of Eq.(5) will become a nonlinear functional of \({\bf M}({\bf B})\). The numerical solution of this implicit equation for the resulting magnetization \(m_{f}={\bf M}({\bf B},\delta({\bf B}))/{\bf M}_{0}(0)\) is shown in Fig.4 for three values of zero-field deficiency parameter \(\delta(0)\). As is clearly seen, \(m_{f}\) exhibits a field-induced ”fishtail”-like behavior typical for underdoped crystals with intragrain granularity. The extra extremum of the magnetization appears when the applied magnetic field \({\bf B}\) matches an intrinsic chemomagnetic field \({\bf B}_{\mu}(\delta({\bf B}))\) (which now also depends on \({\bf B}\) via the above-discussed magnetoconcentration effect). Notice that a ”fishtail” structure of \(m_{f}\) manifests itself even at zero values of field-free deficiency parameter \(\delta(0)\) (solid line in Fig.3) thus confirming a field-induced nature of intrinsic nanogranularity (Lang et al. 2002, Daeumling et al. 1990, Yang et al. 1993, Gurevich and Pashitskii 1997, Moeckley et al. 1993). At the same time, even a rather small deviation from the zero-field stoichiometry (with \(\delta(0)=0.1\)) immediately brings about a paramagnetic Meissner effect at low magnetic fields. Thus, the present model predicts appearance of two interrelated phenomena, Meissner paramagnetism at low fields and ”fishtail” anomaly at high fields. It would be very interesting to verify these predictions experimentally in non-stoichiometric superconductors with pronounced networks of planar defects.
+
+3. MAGNETIC FIELD INDUCED POLARIZATION EFFECTS IN 2D JJA
+
+In this Section, within the same model of JJAs created by a regular 2D network of twin-boundary (TB) dislocations with strain fields acting as an insulating barrier between hole-rich domains in underdoped crystals, we discuss charge-related effects which are actually dual to the above-described phase-related chemomagnetic effects. Specifically, we consider a possible existence of a non-zero electric polarization \({\bf P}(\delta,{\bf B})\) (chemomagnetoelectric effect) and the related change of the charge balance in intrinsically granular non-stoichiometric material under the influence of an applied magnetic field. In particular, we predict an anomalous low-field magnetic behavior of the effective junction charge \({\bf Q}(\delta,{\bf B})\) and concomitant magnetocapacitance \({\bf C}(\delta,{\bf B})\) in paramagnetic Meissner phase and a charge analog of ”fishtail”-like anomaly at high magnetic fields along with field-tuned weakening of the chemically-induced Coulomb blockade (Sergeenkov 2007).
+
+Recall that a conventional (zero-field) pair polarization operator within the model under discussion reads (Sergeenkov 1997, 2002, 2007)
+
+\[{\bf p}=\sum_{i=1}^{N}q_{i}{\bf x}_{i}\] (6)
+
+In view of Eqs.(1), (2) and (6), and taking into account a usual ”phase-number” commutation relation, \([\phi_{i},n_{j}]=i\delta_{ij}\), it can be shown that the evolution of the pair polarization operator is determined via the equation of motion
+
+\[\frac{d{\bf p}}{dt}=\frac{1}{i\hbar}\left[{\bf p},{\cal H}\right]=\frac{2e}{\hbar}\sum_{ij}^{N}J_{ij}\sin\phi_{ij}(t){\bf x}_{ij}\] (7)
+
+Resolving the above equation, we arrive at the following net value of the magnetic-field induced longitudinal (along \(x\)-axis) electric polarization \({\bf P}(\delta,{\bf B})\) and the corresponding effective junction charge
+
+\[{\bf Q}(\delta,{\bf B})=\frac{2eJ_{0}}{\hbar\tau d}\int\limits_{0}^{\tau}dt\int\limits_{0}^{t}dt^{\prime}\int\frac{d^{2}x}{S}\sin\phi({\bf x},t^{\prime})xe^{-{\mid{{\bf x}}\mid}/d},\] (8)
+
+where \(S=2\pi d^{2}\) is properly defined normalization area, \(\tau\) is a characteristic time (see below), and we made a usual substitution \(\frac{1}{N}\sum_{ij}A_{ij}(t)\to\frac{1}{S}\int d^{2}xA({\bf x},t)\) valid in the long-wavelength approximation (Sergeenkov 2002).
+
+To capture the very essence of the superconducting analog of the chemomagnetoelectric effect, in what follows we assume for simplicity that a _stoichiometric sample_ (with \(\delta\simeq 0\)) does not possess any spontaneous polarization at zero magnetic field, that is \({\bf P}(0,0)=0\). According to Eq.(8), this condition implies \(\phi_{ij}^{0}=2\pi m\) for the initial phase difference with \(m=0,\pm 1,\pm 2,..\).
+
+Taking the applied magnetic field along the \(c\)-axis (and normal to the \(CuO\) plane), we obtain finally
+
+\[{\bf Q}(\delta,{\bf B})={\bf Q}_{0}(\delta)\frac{2{\tilde{\bf b}}+{\bf b}(1-{\tilde{\bf b}}^{2})}{(1+{\bf b}^{2})(1+{\tilde{\bf b}}^{2})^{2}}\] (9)
+
+for the magnetic field behavior of the effective junction charge in chemically induced granular superconductors.
+
+Fig. 5: A variation of effective junction charge with an applied magnetic field (chemomagnetoelectric effect) for different values of oxygen deficiency parameter: \(\delta\simeq 0\) (solid line), \(\delta=0.1\) (dashed line), and \(\delta=0.2\) (dotted line).
+
+Here \({\bf Q}_{0}(\delta)=e\tau J_{0}(\delta)/\hbar\) with \(J_{0}(\delta)\) defined earlier, \({\bf b}={\bf B}/{\bf B}_{0}\), \({\tilde{\bf b}}={\bf b}-{\bf b}_{\mu}\), and \({\bf b}_{\mu}={\bf B}_{\mu}/{\bf B}_{0}\simeq(k_{B}T\tau/\hbar)\delta\) where \({\bf B}_{\mu}(\delta)=(\mu_{v}\tau/\hbar){\bf B}_{0}\) is the chemically-induced contribution (which disappears in optimally doped systems with \(\delta\simeq 0\)), and \({\bf B}_{0}=\Phi_{0}/wd\) is a characteristic Josephson field.
+
+Fig.5 shows changes of the initial (stoichiometric) effective junction charge \(\Delta{\bf Q}(\delta,{\bf B})={\bf Q}(\delta,{\bf B})-{\bf Q}(\delta,0)\) (solid line) with oxygen deficiency \(\delta\). According to Eq.(9), the effective charge \({\bf Q}\) changes its sign at low magnetic fields (driven by non-zero values of \(\delta\)) as soon as the chemomagnetic contribution \({\bf B}_{\mu}(\delta)\) exceeds an applied magnetic field \({\bf B}\). This is nothing else but a charge analog of chemically induced PME.
+
+Fig. 6: A variation of the _chemomagnetoelectric_ effect with concentration (deficiency) for different values of the applied magnetic field.
+
+At the same time, Fig.6 presents a variation of the _chemomagnetoelectric_ effect with concentration (deficiency) for different values of the applied magnetic field. Notice that a zero-field contribution (which is a true _chemoelectric_ effect) exhibits a maximum around \(\delta_{c}\simeq 0.2\), in agreement with the classical percolative behavior observed in non-stoichiometric \(YBa_{2}Cu_{3}O_{7-\delta}\) samples (Gantmakher et al. 1990).
+
+It is of interest also to consider the magnetic field behavior of the concomitant effective flux capacitance \({\bf C}\equiv\tau d{\bf Q}(\delta,{\bf B})/d\Phi\) which in view of Eq.(9) reads
+
+\[{\bf C}(\delta,{\bf B})={\bf C}_{0}(\delta)\frac{1-3{\bf b}{\tilde{\bf b}}-3{\tilde{\bf b}}^{2}+{\bf b}{\tilde{\bf b}}^{3}}{(1+{\bf b}^{2})(1+{\tilde{\bf b}}^{2})^{3}},\] (10)
+
+where \(\Phi=SB\), and \({\bf C}_{0}(\delta)=\tau{\bf Q}_{0}(\delta)/\Phi_{0}\).
+
+Fig.7 depicts the behavior of the effective flux capacitance \(\Delta{\bf C}(\delta,{\bf B})={\bf C}(\delta,{\bf B})-{\bf C}(\delta,0)\) in applied magnetic field for different values of oxygen deficiency parameter: \(\delta\simeq 0\) (solid line), \(\delta=0.1\) (dashed line), and \(\delta=0.2\) (dotted line). Notice a decrease of magnetocapacitance amplitude and its peak shifting with increase of \(\delta\) and sign change at low magnetic fields which is another manifestation of the charge analog of chemically induced PME (Cf. Fig.5).
+
+Fig. 7: The effective flux capacitance as a function of applied magnetic field for different values of oxygen deficiency parameter: \(\delta\simeq 0\) (solid line), \(\delta=0.1\) (dashed line), and \(\delta=0.2\) (dotted line).
+
+Up to now, we neglected a possible field dependence of the chemical potential \(\mu_{v}\) of oxygen vacancies. Recall, however, that in high enough applied magnetic fields \({\bf B}\), the field-induced change of the chemical potential \(\Delta\mu_{v}({\bf B})\equiv\mu_{v}({\bf B})-\mu_{v}(0)\) becomes tangible and should be taken into account (Abrikosov 1988, Sergeenkov and Ausloos 1999). As a result, we end up with a superconducting analog of the so-called _magnetoconcentration_ effect (Sergeenkov 2003) with field induced creation of oxygen vacancies \({\bf c_{v}}({\bf B})={\bf c_{v}}(0)\exp(-\Delta\mu_{v}({\bf B})/k_{B}T)\) which in turn brings about a ”fishtail”-like behavior of the high-field chemomagnetization (see Section 2 for more details).
+
+Fig. 8: A ”fishtail”-like behavior of an effective charge in applied magnetic field in the presence of magnetoconcentration effect (with field-induced oxygen vacancies \(\delta({\bf B})\)) for three values of field-free deficiency parameter (from top to bottom): \(\delta(0)\simeq 0\) (solid line), \(\delta(0)=0.1\) (dashed line), and \(\delta(0)=0.2\) (dotted line).
+
+Fig.8 shows the field behavior of the effective junction charge in the presence of the above-mentioned magnetoconcentration effect. As it is clearly seen, \({\bf Q}(\delta({\bf B}),{\bf B})\) exhibits a ”fishtail”-like anomaly typical for previously discussed (Sergeenkov 2003) chemomagnetization in underdoped crystals with intragrain granularity. This more complex structure of the effective charge appears when the applied magnetic field \({\bf B}\) matches an intrinsic chemomagnetic field \({\bf B}_{\mu}(\delta({\bf B}))\) (which now also depends on \({\bf B}\) via the magnetoconcentration effect). Notice that a ”fishtail” structure of \({\bf Q}(\delta({\bf B}),{\bf B})\) manifests itself even at zero values of field-free deficiency parameter \(\delta(0)\) (solid line in Fig.8) thus confirming a field-induced nature of intrinsic granularity.
+
+Fig. 9: The behavior of the effective flux capacitance in applied magnetic field in the presence of magnetoconcentration effect for three values of field-free deficiency parameter: \(\delta(0)\simeq 0\) (solid line), \(\delta(0)=0.1\) (dashed line), and \(\delta(0)=0.2\) (dotted line).
+
+Likewise, Fig.9 depicts the evolution of the effective flux capacitance \(\Delta{\bf C}(\delta({\bf B}),{\bf B})={\bf C}(\delta({\bf B}),{\bf B})-{\bf C}(\delta(0),0)\) in applied magnetic field \({\bf B}\) in the presence of magnetoconcentration effect (Cf. Fig.7).
+
+Thus, the present model predicts appearance of two interrelated phenomena dual to the previously discussed behavior of chemomagnetizm (see Section 2), namely a charge analog of Meissner paramagnetism at low fields and a charge analog of ”fishtail” anomaly at high fields. To see whether these effects can be actually observed in a real material, let us estimate an order of magnitude of the main model parameters.
+
+Using typical for HTS single crystals values of \(\lambda_{L}(0)\simeq 150nm\), \(d\simeq 10nm\), and \(j_{c}\simeq 10^{10}A/m^{2}\), we arrive at the following estimates of the characteristic \({\bf B}_{0}\simeq 0.5T\) and chemomagnetic \({\bf B}_{\mu}(\delta)\simeq 0.5B_{0}\) fields, respectively. So, the predicted charge analog of PME should be observable for applied magnetic fields \({\bf B}<0.25T\). Notice that, for the above set of parameters, the Josephson length is of the order of \(\lambda_{J}\simeq 1\mu m\), which means that the small-junction approximation assumed in this paper is valid and the ”self-field” effects can be safely neglected.
+
+Furthermore, the characteristic frequencies \(\omega\simeq\tau^{-1}\) needed to probe the effects suggested here are related to the processes governed by tunneling relaxation times \(\tau\simeq\hbar/J_{0}(\delta)\). Since for oxygen deficiency parameter \(\delta=0.1\) the chemically-induced zero-temperature Josephson energy in non-stoichiometric \(YBCO\) single crystals is of the order of \(J_{0}(\delta)\simeq k_{B}T_{C}\delta\simeq 1meV\), we arrive at the required frequencies of \(\omega\simeq 10^{13}Hz\) and at the following estimates of the effective junction charge \({\bf Q}_{0}\simeq e=1.6\times 10^{-19}C\) and flux capacitance \({\bf C}_{0}\simeq 10^{-18}F\). Notice that the above estimates fall into the range of parameters used in typical experiments for studying the single-electron tunneling effects both in JJs and JJAs (Makhlin et al. 2001, van Bentum et al. 1988) suggesting thus quite an optimistic possibility to observe the above-predicted field induced effects experimentally in non-stoichiometric superconductors with pronounced networks of planar defects or in artificially prepared JJAs. It is worth mentioning that a somewhat similar behavior of the magnetic field induced charge and related flux capacitance has been observed in 2D electron systems (Chen et al. 1994).
+
+And finally, it can be easily verified that, in view of Eqs.(6)-(8), the field-induced Coulomb energy of the oxygen-depleted region within our model is given by
+
+\[E_{C}(\delta,{\bf B})\equiv\left<\sum_{ij}^{N}\frac{q_{i}q_{j}}{2C_{ij}}\right>=\frac{{\bf Q}^{2}(\delta,{\bf B})}{2{\bf C}(\delta,{\bf B})}\] (11)
+
+with \({\bf Q}(\delta,{\bf B})\) and \({\bf C}(\delta,{\bf B})\) defined by Eqs. (9) and (10), respectively.
+
+A thorough analysis of the above expression reveals that in the PME state (when \({\bf B}\ll{\bf B}_{\mu}\)) the chemically-induced granular superconductor is in the so-called Coulomb blockade regime (with \(E_{C}>J_{0}\)), while in the ”fishtail” state (for \({\bf B}\geq{\bf B}_{\mu}\)) the energy balance tips in favor of tunneling (with \(E_{C}<J_{0}\)). In particular, we obtain that \(E_{C}(\delta,{\bf B}=0.1{\bf B}_{\mu})=\frac{\pi}{2}J_{0}(\delta)\) and \(E_{C}(\delta,{\bf B}={\bf B}_{\mu})=\frac{\pi}{8}J_{0}(\delta)\). It would be also interesting to check this phenomenon of field-induced weakening of the Coulomb blockade experimentally.
+
+4. GIANT ENHANCEMENT OF THERMAL CONDUCTIVITY IN 2D JJA
+
+In this Section, using a 2D model of inductive Josephson junction arrays (created by a network of twin boundary dislocations with strain fields acting as an insulating barrier between hole-rich domains in underdoped crystals), we study the temperature, \({\bf T}\), and chemical pressure, \(\nabla\mu\), dependence of the thermal conductivity (TC) \(\kappa\) of an intrinsically nanogranular superconductor. Two major effects affecting the behavior of TC under chemical pressure are predicted: decrease of the _linear_ (i.e., \(\nabla{\bf T}\) - independent) TC, and giant enhancement of the _nonlinear_ (i.e., \(\nabla{\bf T}\) - dependent) TC with \([\kappa({\bf T},\nabla{\bf T},\nabla\mu)-\kappa({\bf T},\nabla{\bf T},0)]/\kappa({\bf T},\nabla{\bf T},0)\) reaching \(500\%\) when chemoelectric field \({\bf E}_{\mu}=\frac{1}{2e}\nabla\mu\) matches thermoelectric field \({\bf E}_{T}=S_{T}\nabla{\bf T}\). The conditions under which these effects can be experimentally measured in non-stoichiometric high-\(T_{C}\) superconductors are discussed.
+
+There are several approaches for studying the thermal response of JJs and JJAs based on phenomenology of the Josephson effect in the presence of thermal gradients (see, e.g., van Harlingen et al. 1980, Guttman et al. 1997, Deppe and Feldman 1994, Sergeenkov 2002, Sergeenkov 2007 and further references therein). To adequately describe transport properties of the above-described chemically induced nanogranular superconductor for all temperatures and under a simultaneous influence of intrinsic chemical pressure \(\nabla\mu({\bf x})=K\Omega_{0}\nabla\epsilon({\bf x})\) and applied thermal gradient \(\nabla T\), we employ a model of 2D overdamped Josephson junction array which is based on the following total Hamiltonian (Sergeenkov 2002)
+
+\[{\cal H}(t)={\cal H}_{T}(t)+{\cal H}_{L}(t)+{\cal H}_{\mu}(t),\] (12)
+
+where
+
+\[{\cal H}_{T}(t)=\sum_{ij}^{N}J_{ij}[1-\cos\phi_{ij}(t)]\] (13)
+
+is the well-known tunneling Hamiltonian,
+
+\[{\cal H}_{L}(t)=\sum_{ij}^{N}\frac{\Phi_{ij}^{2}(t)}{2L_{ij}}\] (14)
+
+accounts for a mutual inductance \(L_{ij}\) between grains (and controls the normal state value of the thermal conductivity, see below) with \(\Phi_{ij}(t)=(\hbar/2e)\phi_{ij}(t)\) being the total magnetic flux through an array, and finally
+
+\[{\cal H}_{\mu}(t)=\sum_{i=1}^{N}n_{i}(t)\delta\mu_{i}\] (15)
+
+describes chemical potential induced contribution with \(\delta\mu_{i}={\bf x}_{i}\nabla\mu\), and \(n_{i}\) being the pair number operator.
+
+According to the above-mentioned scenario, the tunneling Hamiltonian \({\cal H}_{T}(t)\) introduces a short-range (nearest-neighbor) interaction between \(N\) junctions (which are formed around oxygen-rich superconducting areas with phases \(\phi_{i}(t)\)), arranged in a two-dimensional (2D) lattice with coordinates \({\bf x_{i}}=(x_{i},y_{i})\). The areas are separated by oxygen-poor insulating boundaries (created by TB strain fields \(\epsilon({\bf x}_{ij})\)) producing a short-range Josephson coupling \(J_{ij}=J_{0}(\delta)e^{-{\mid{{\bf x}_{ij}}\mid}/d}\). Thus, typically for granular superconductors, the Josephson energy of the array varies exponentially with the distance \({\bf x}_{ij}={\bf x}_{i}-{\bf x}_{j}\) between neighboring junctions (with \(d\) being an average grain size). The temperature dependence of chemically induced Josephson coupling is governed by the following expression, \(J_{ij}({\bf T})=J_{ij}(0)F({\bf T})\) where
+
+\[F({\bf T})=\frac{\Delta({\bf T})}{\Delta(0)}\tanh\left[\frac{\Delta({\bf T})}{2k_{B}{\bf T}}\right]\] (16)
+
+and \(J_{ij}(0)=[\Delta(0)/2](R_{0}/R_{ij})\) with \(\Delta({\bf T})\) being the temperature dependent gap parameter, \(R_{0}=h/4e^{2}\) is the quantum resistance, and \(R_{ij}\) is the resistance between grains in their normal state.
+
+By analogy with a constant electric field \({\bf E}\), a thermal gradient \(\nabla{\bf T}\) applied to a chemically induced JJA will cause a time evolution of the phase difference across insulating barriers as follows (Sergeenkov 2002)
+
+\[\phi_{ij}(t)=\phi_{ij}^{0}+\omega_{ij}(\nabla\mu,\nabla T)t\] (17)
+
+Here \(\phi_{ij}^{0}\) is the initial phase difference (see below), and \(\omega_{ij}=2e({\bf E}_{\mu}-{\bf E}_{T}){\bf x}_{ij}/\hbar\) where \({\bf E}_{\mu}=\frac{1}{2e}\nabla\mu\) and \({\bf E}_{T}=S_{T}\nabla{\bf T}\) are the induced chemoelectric and thermoelectric fields, respectively. \(S_{T}\) is the so-called thermophase coefficient (Sergeenkov 1998a) which is related to the Seebeck coefficient \(S_{0}\) as follows, \(S_{T}=(l/d)S_{0}\) (where \(l\) is a relevant sample’s size responsible for the applied thermal gradient, that is \(|\nabla{\bf T}|=\Delta{\bf T}/l\)).
+
+We start our consideration by discussing the temperature behavior of the conventional (that is _linear_) thermal conductivity of a chemically induced nanogranular superconductor paying a special attention to its evolution with a mutual inductance \(L_{ij}\). For simplicity, in what follows we limit our consideration to the longitudinal component of the total thermal flux \({\bf Q}(t)\) which is defined (in a q-space representation) via the total energy conservation law as follows
+
+\[{\bf Q}(t)\equiv\lim_{{\bf q}\to 0}\left[i\frac{{\bf q}}{{\bf q}^{2}}{\dot{\cal H}_{\bf q}}(t)\right],\] (18)
+
+where \({\dot{\cal H}_{\bf q}}=\partial{\cal H}_{\bf q}/\partial t\) with
+
+\[{\cal H}_{\bf q}(t)=\frac{1}{s}\int d^{2}xe^{i{\bf q}{\bf x}}{\cal H}({\bf x},t)\] (19)
+
+Here \(s=2\pi d^{2}\) is properly defined normalization area, and we made a usual substitution \(\frac{1}{N}\sum_{ij}A_{ij}(t)\to\frac{1}{s}\int d^{2}xA({\bf x},t)\) valid in the long-wavelength approximation (\({\bf q}\to 0\)).
+
+In turn, the heat flux \({\bf Q}(t)\) is related to the _linear_ thermal conductivity (LTC) tensor \(\kappa_{\alpha\beta}\) by the Fourier law as follows (hereafter, \(\{\alpha,\beta\}=x,y,z\))
+
+\[\kappa_{\alpha\beta}({\bf T},\nabla\mu)\equiv-\frac{1}{V}\left[\frac{\partial\overline{<{\bf Q}_{\alpha}>}}{\partial(\nabla_{\beta}{\bf T})}\right]_{\nabla{\bf T}=0},\] (20)
+
+where
+
+\[\overline{<{\bf Q}_{\alpha}>}=\frac{1}{\tau}\int_{0}^{\tau}dt<{\bf Q}_{\alpha}(t)>\] (21)
+
+Here \(V\) is sample’s volume, \(\tau\) is a characteristic Josephson tunneling time for the network, and \(<...>\) denotes the thermodynamic averaging over the initial phase differences \(\phi_{ij}^{0}\)
+
+\[<A(\phi_{ij}^{0})>=\frac{1}{Z}\int_{0}^{2\pi}\prod_{ij}d\phi_{ij}^{0}A(\phi_{ij}^{0})e^{-\beta H_{0}}\] (22)
+
+with an effective Hamiltonian
+
+\[H_{0}[\phi_{ij}^{0}]=\int_{0}^{\tau}\frac{dt}{\tau}\int\frac{d^{2}x}{s}{\cal H}({\bf x},t)\] (23)
+
+Here, \(\beta=1/k_{B}{\bf T}\), and \(Z=\int_{0}^{2\pi}\prod_{ij}d\phi_{ij}^{0}e^{-\beta H_{0}}\) is the partition function. The above-defined averaging procedure allows us to study the temperature evolution of the system.
+
+Taking into account that in JJAs (Eichenberger et al. 1996) \(L_{ij}\propto R_{ij}\), we obtain \(L_{ij}=L_{0}\exp({\mid{{\bf x}_{ij}}\mid}/d)\) for the explicit \(x\)-dependence of the weak-link inductance in our model. Finally, in view of Eqs.(12)-(23), and making use of the usual ”phase-number” commutation relation, \([\phi_{i},n_{j}]=i\delta_{ij}\), we find the following analytical expression for the temperature and chemical gradient dependence of the electronic contribution to _linear_ thermal conductivity of a granular superconductor
+
+\[\kappa_{\alpha\beta}({\bf T},\nabla\mu)=\kappa_{0}[\delta_{\alpha\beta}\eta({\bf T},\epsilon)+\beta_{L}({\bf T})\nu({\bf T},\epsilon)f_{\alpha\beta}(\epsilon)]\] (24)
+
+where
+
+\[f_{\alpha\beta}(\epsilon)=\frac{1}{4}\left[\delta_{\alpha\beta}A(\epsilon)-\epsilon_{\alpha}\epsilon_{\beta}B(\epsilon)\right]\] (25)
+
+with
+
+\[A(\epsilon)=\frac{5+3\epsilon^{2}}{(1+\epsilon^{2})^{2}}+\frac{3}{\epsilon}\tan^{-1}\epsilon\] (26)
+
+and
+
+\[B(\epsilon)=\frac{3\epsilon^{4}+8\epsilon^{2}-3}{\epsilon^{2}(1+\epsilon^{2})^{3}}+\frac{3}{\epsilon^{3}}\tan^{-1}\epsilon\] (27)
+
+Here, \(\kappa_{0}=Nd^{2}S_{T}\Phi_{0}/VL_{0}\), \(\beta_{L}({\bf T})=2\pi I_{C}({\bf T})L_{0}/\Phi_{0}\) with \(I_{C}({\bf T})=(2e/\hbar)J({\bf T})\) being the critical current; \(\epsilon\equiv\sqrt{\epsilon_{x}^{2}+\epsilon_{y}^{2}+\epsilon_{z}^{2}}\) with \(\epsilon_{\alpha}={\bf E}_{\mu}^{\alpha}/{\bf E}_{0}\) where \({\bf E}_{0}=\hbar/2ed\tau\) is a characteristic field. In turn, the above-introduced ”order parameters” of the system, \(\eta({\bf T},\epsilon)\equiv<\phi_{ij}^{0}>\) and \(\nu({\bf T},\epsilon)\equiv<\sin\phi_{ij}^{0}>\), are defined as follows
+
+\[\eta({\bf T},\epsilon)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{2}}\left[\frac{I_{2n+1}(\beta_{\mu})}{I_{0}(\beta_{\mu})}\right]\] (28)
+
+and
+
+\[\nu({\bf T},\epsilon)=\frac{\sinh\beta_{\mu}}{\beta_{\mu}I_{0}(\beta_{\mu})},\] (29)
+
+where
+
+\[\beta_{\mu}({\bf T},\epsilon)=\frac{\beta J({\bf T})}{2}\left(\frac{1}{1+\epsilon^{2}}+\frac{1}{\epsilon}\tan^{-1}\epsilon\right)\] (30)
+
+Here \(J({\bf T})\) is given by Eq.(17), and \(I_{n}(x)\) stand for the modified Bessel functions.
+
+Turning to the discussion of the obtained results, we start with a more simple zero-pressure case. The relevant parameters affecting the behavior of the LTC in this particular case include the mutual inductance \(L_{0}\) and the normal state resistance between grains \(R_{n}\). For the temperature dependence of the Josephson energy (see Eq.(17)), we used the well-known (Sergeenkov 2002) approximation for the BCS gap parameter, valid for all temperatures, \(\Delta({\bf T})=\Delta(0)\tanh\left(\gamma\sqrt{\frac{{\bf T_{C}}-{\bf T}}{{\bf T}}}\right)\) with \(\gamma=2.2\).
+
+Fig. 10: Temperature dependence of the zero-pressure (\(\nabla\mu=0\)) _linear_ thermal conductivity for \(r_{n}=0.1\) (left) and \(r_{n}=1\) (right) for different values of the SQUID parameter (from bottom to top): \(\beta_{L}(0)=1,3,5,7\), and \(9\).
+
+Despite a rather simplified nature of our model, it seems to quite reasonably describe the behavior of the LTC for all temperatures. Indeed, in the absence of intrinsic chemical pressure (\(\nabla\mu=0\)), the LTC is isotropic (as expected), \(\kappa_{\alpha\beta}({\bf T},0)=\delta_{\alpha\beta}\kappa_{L}({\bf T},0)\) where \(\kappa_{L}({\bf T},0)=\kappa_{0}[\eta({\bf T},0)+2\beta_{L}({\bf T})\nu({\bf T},0)]\) vanishes at zero temperature and reaches a normal state value \(\kappa_{n}\equiv\kappa_{L}({\bf T_{C}},0)=(\pi/2)\kappa_{0}\) at \({\bf T}={\bf T_{C}}\). Fig.10 shows the temperature dependence of the normalized LTC \(\kappa_{L}({\bf T},0)/\kappa_{n}\) for different values of the so-called SQUID parameter \(\beta_{L}(0)=2\pi I_{C}(0)L_{0}/\Phi_{0}\) (increasing from the bottom to the top) and for two values of the resistance ratio \(r_{n}=R_{0}/R_{n}=0.1\) and \(r_{n}=R_{0}/R_{n}=1\). First of all, with increasing of the SQUID parameter, the LTC evolves from a flat-like pattern (for a relatively small values of \(L_{0}\)) to a low-temperature maximum (for higher values of \(\beta_{L}(0)\)). Notice that the peak temperature \({\bf T}_{p}\) is practically insensitive to the variation of inductance parameter \(L_{0}\) while being at the same time strongly influenced by resistivity \(R_{n}\). Indeed, as it is clearly seen in Fig.10, a different choice of \(r_{n}\) leads to quite a tangible shifting of the maximum. Namely, the smaller is the normal resistance between grains \(R_{n}\) (or the better is the quality of the sample) the higher is the temperature at which the peak is developed. As a matter of fact, the peak temperature \({\bf T}_{p}\) is related to the so-called phase-locking temperature \({\bf T}_{J}\) (which marks the establishment of phase coherence between the adjacent grains in the array and always lies below a single grain superconducting temperature \({\bf T_{C}}\)) which is usually defined via an average (per grain) Josephson coupling energy as \(J({\bf T}_{J},r_{n})=k_{B}{\bf T}_{J}\). Indeed, it can be shown analytically that for \({\bf T}_{J}<{\bf T}<{\bf T_{C}}\), \({\bf T}_{J}(r_{n})\simeq r_{n}{\bf T_{C}}\).
+
+Turning to the discussion of the LTC behavior under chemical pressure, let us assume, for simplicity, that \(\nabla\mu=(\nabla_{x}\mu,0,0)\) with oxygen-deficiency parameter \(\delta\) controlled chemical pressure \(\nabla_{x}\mu\simeq\mu_{v}(\delta)/d\), and \(\nabla{\bf T}=(\nabla_{x}{\bf T},\nabla_{y}{\bf T},0)\). Such a choice of the external fields allows us to consider both parallel \(\kappa_{xx}({\bf T},\nabla\mu)\) and perpendicular \(\kappa_{yy}({\bf T},\nabla\mu)\) components of the LTC corresponding to the two most interesting configurations, \({\bf\nabla\mu}\|\nabla{\bf T}\) and \({\bf\nabla\mu}\bot\nabla{\bf T}\), respectively. Fig.11 demonstrates the predicted chemical pressure dependence of the normalized LTC \(\Delta\kappa_{L}({\bf T},\nabla\mu)=\kappa_{L}({\bf T},\nabla\mu)-\kappa_{L}({\bf T},0)\) for both configurations taken at \({\bf T}=0.2{\bf T_{C}}\) (with \(r_{n}=0.1\) and \(\beta_{L}(0)=1\)). First of all, we note that both components of the LTC are _decreasing_ with increasing of the pressure \({\bf E}_{\mu}/{\bf E}_{0}=\mu_{v}(\delta)\tau/\hbar\). And secondly, the normal component \(\kappa_{yy}\) decreases more slowly than the parallel one \(\kappa_{xx}\), suggesting thus some kind of anisotropy in the system. In view of the structure of Eq.(25), the same behavior is also expected for the temperature dependence of the chemically-induced LTC, that is \(\Delta\kappa_{L}({\bf T},\nabla\mu)/\kappa_{L}({\bf T},0)<0\) for all gradients and temperatures. In terms of the absolute values, for \({\bf T}=0.2{\bf T_{C}}\) and \({\bf E}_{\mu}={\bf E}_{0}\), we obtain \([\Delta\kappa_{L}({\bf T},\nabla\mu)/\kappa_{L}({\bf T},0)]_{xx}=90\%\) and \([\Delta\kappa_{L}({\bf T},\nabla\mu)/\kappa_{L}({\bf T},0)]_{yy}=60\%\) for _attenuation_ of LTC under chemical pressure.
+
+Fig. 11: The dependence of the _linear_ thermal conductivity on the chemical pressure for parallel (\({\bf\nabla\mu}\|\nabla{\bf T}\)) and perpendicular (\({\bf\nabla\mu}\bot\nabla{\bf T}\)) configurations.
+
+Let us turn now to the most intriguing part of this Section and consider a _nonlinear_ generalization of the Fourier law and very unusual behavior of the resulting _nonlinear_ thermal conductivity (NLTC) under the influence of chemical pressure. In what follows, by the NLTC we understand a \(\nabla{\bf T}\)-dependent thermal conductivity \(\kappa_{\alpha\beta}^{NL}({\bf T},{\bf\nabla\mu})\equiv\kappa_{\alpha\beta}({\bf T},{\bf\nabla\mu};\nabla{\bf T})\) which is defined as follows
+
+\[\kappa_{\alpha\beta}^{NL}({\bf T},{\bf\nabla\mu})\equiv-\frac{1}{V}\left[\frac{\partial\overline{<{\bf Q}_{\alpha}>}}{\partial(\nabla_{\beta}{\bf T})}\right]_{\nabla{\bf T}\neq 0}\] (31)
+
+with \(\overline{<{\bf Q}_{\alpha}>}\) given by Eq.(21).
+
+Repeating the same procedure as before, we obtain finally for the relevant components of the NLTC tensor
+
+\[\kappa_{\alpha\beta}^{NL}({\bf T},{\bf\nabla\mu})=\kappa_{0}[\delta_{\alpha\beta}\eta({\bf T},\epsilon_{eff})+\beta_{L}({\bf T})\nu({\bf T},\epsilon_{eff})D_{\alpha\beta}(\epsilon_{eff})],\] (32)
+
+where
+
+\[D_{\alpha\beta}(\epsilon_{eff})=f_{\alpha\beta}(\epsilon_{eff})+\epsilon_{T}^{\gamma}g_{\alpha\beta\gamma}(\epsilon_{eff})\] (33)
+
+with
+
+\[g_{\alpha\beta\gamma}(\epsilon)=\frac{1}{8}[(\delta_{\alpha\beta}\epsilon_{\gamma}+\delta_{\alpha\gamma}\epsilon_{\beta}+\delta_{\gamma\beta}\epsilon_{\alpha})B(\epsilon)+3\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}C(\epsilon)]\] (34)
+
+and
+
+\[C(\epsilon)=\frac{3+11\epsilon^{2}-11\epsilon^{4}-3\epsilon^{6}}{\epsilon^{4}(1+\epsilon^{2})^{4}}-\frac{3}{\epsilon^{5}}\tan^{-1}\epsilon\] (35)
+
+Here, \(\epsilon_{eff}^{\alpha}=\epsilon_{\mu}^{\alpha}-\epsilon_{T}^{\alpha}\) where \(\epsilon_{\mu}^{\alpha}={\bf E}_{\mu}^{\alpha}/{\bf E}_{0}\) and \(\epsilon_{T}^{\alpha}={\bf E}_{T}^{\alpha}/{\bf E}_{0}\) with \({\bf E}_{T}^{\alpha}=S_{T}\nabla_{\alpha}{\bf T}\); other parameters (\(\eta\), \(\nu\), \(B\) and \(f_{\alpha\beta}\)) are the same as before but with \(\epsilon\to\epsilon_{eff}\).
+
+Fig. 12: The dependence of the _nonlinear_ thermal conductivity on the chemical pressure for different values of the applied thermal gradient \(\epsilon_{T}=S_{T}\nabla{\bf T}/{\bf E}_{0}\) (\(\epsilon_{T}=0.2,0.4,0.6,0.8\), and \(1.0\), increasing from bottom to top).
+
+As expected, in the limit \({\bf E}_{T}\to 0\) (or when \({\bf E}_{\mu}\gg{\bf E}_{T}\)), from Eq.(32) we recover all the results obtained in the previous section for the LTC. Let us see now what happens when thermoelectric field \({\bf E}_{T}=S_{T}\nabla{\bf T}\) becomes comparable with chemoelectric field \({\bf E_{\mu}}\). Fig.12 depicts the resulting chemical pressure dependence of the parallel component of the NLTC tensor \(\Delta\kappa_{xx}^{NL}({\bf T},{\bf E}_{\mu})=\kappa_{xx}^{NL}({\bf T},{\bf E}_{\mu})-\kappa_{xx}^{NL}({\bf T},0)\) for different values of the dimensionless parameter \(\epsilon_{T}={\bf E}_{T}/{\bf E}_{0}\) (the other parameters are the same as before). As is clearly seen from this picture, in a sharp contrast with the pressure behavior of the previously considered LTC, its _nonlinear_ analog evolves with the chemoelectric field quite differently. Namely, NLTC strongly _increases_ for small pressure values (with \({\bf E}_{\mu}<{\bf E}_{m}\)), reaches a pronounced maximum at \({\bf E}_{\mu}={\bf E}_{m}=\frac{3}{2}{\bf E}_{T}\), and eventually declines at higher values of _μ_ (with \({\bf E}_{\mu}>{\bf E}_{m}\)). Furthermore, as it directly follows from the very structure of Eq.(32), a similar ”reentrant-like” behavior of the _nonlinear_ thermal conductivity is expected for its temperature dependence as well. Even more remarkable is the absolute value of the pressure-induced enhancement. According to Fig.12, it is easy to estimate that near maximum (with \({\bf E}_{\mu}={\bf E}_{m}\) and \({\bf E}_{T}={\bf E}_{0}\)) one gets \(\Delta\kappa_{xx}^{NL}({\bf T},{\bf E}_{\mu})/\kappa_{xx}^{NL}({\bf T},0)\simeq 500\%\).
+
+To understand the above-obtained rather unusual results, let us take a closer look at the chemoelectric field induced behavior of the Josephson voltage in our system (see Eq.(17)). Clearly, strong heat conduction requires establishment of a quasi-stationary (that is nearly zero-voltage) regime within the array. In other words, the maximum of the thermal conductivity under chemical pressure should correlate with a minimum of the total voltage in the system, \(V(\nabla\mu)\equiv(\frac{\hbar}{2e})<\frac{\partial\phi_{ij}(t)}{\partial t}>=V_{0}(\epsilon-\epsilon_{T})\) where \(\epsilon\equiv{\bf E}_{\mu}/{\bf E}_{0}\) and \(V_{0}={\bf E}_{0}d=\hbar/2e\tau\) is a characteristic voltage. For linear TC (which is valid only for small thermal gradients with \(\epsilon_{T}\equiv{\bf E}_{T}/{\bf E}_{0}\ll 1\)), the average voltage through an array \(V_{L}(\nabla\mu)\simeq V_{0}({\bf E}_{\mu}/{\bf E}_{0})\) has a minimum at zero chemoelectric field (where LTC indeed has its maximum value, see Fig.11) while for nonlinear TC (with \(\epsilon_{T}\simeq 1\)) we have to consider the total voltage \(V(\nabla\mu)\) which becomes minimal at \({\bf E}_{\mu}={\bf E}_{T}\) (in a good agreement with the predictions for NLTC maximum which appears at \({\bf E}_{\mu}=\frac{3}{2}{\bf E}_{T}\), see Fig.12).
+
+To complete our study, let us estimate an order of magnitude of the main model parameters. Starting with chemoelectric fields \({\bf E}_{\mu}\) needed to observe the above-predicted nonlinear field effects in nanogranular superconductors, we notice that according to Fig.12, the most interesting behavior of NLTC takes place for \({\bf E}_{\mu}\simeq{\bf E}_{0}\). Using typical \(YBCO\) parameters, \(\epsilon_{0}=0.01\), \(\Omega_{0}=a_{0}^{3}\) with \(a_{0}=0.2nm\), and \(K=115GPa\), we have \(\mu_{v}=\epsilon_{0}K\Omega_{0}\simeq 1meV\) for an estimate of the chemical potential in HTS crystals, which defines the characteristic Josephson tunneling time \(\tau\simeq\hbar/\mu_{v}\simeq 5\times 10^{-11}s\) and, at the same time, leads to creation of excess vacancies with concentration \(c_{v}=e^{-\mu_{v}/k_{B}{\bf T}}\simeq 0.75\) at \({\bf T}=0.2{\bf T_{C}}\) (equivalent to a deficiency value of \(\delta\simeq 0.25\)). Notice that in comparison with this linear defects mediated channeling (osmotic) mechanism, the probability of the conventional oxygen diffusion in these materials \(D\propto e^{-U_{d}/k_{B}T}\) (governed by a rather high activation energy \(U_{d}\simeq 1eV\)) is extremely low under the same conditions (\(D\ll 1\)).
+
+Furthermore, taking \(d\simeq 10nm\) for typical values of the average ”grain” size (created by oxygen-rich superconducting regions), we get \({\bf E}_{0}=\hbar/2ed\tau\simeq 5\times 10^{5}V/m\) and \(|\nabla\mu|=\mu_{v}/d\simeq 10^{6}eV/m\) for the estimates of the characteristic field and chemical potential gradient (intrinsic chemical pressure), respectively. On the other hand, the maximum of NLTC occurs when this field nearly perfectly matches an ”intrinsic” thermoelectric field \({\bf E}_{T}=S_{T}\nabla{\bf T}\) induced by an applied thermal gradient, that is when \({\bf E}_{\mu}\simeq{\bf E}_{0}\simeq{\bf E}_{T}\). Recalling that \(S_{T}=(l/d)S_{0}\) and using \(S_{0}\simeq 0.5\mu V/K\) and \(l\simeq 0.5mm\) for an estimate of the _linear_ Seebeck coefficient and a typical sample’s size, we obtain \(\nabla{\bf T}\simeq{\bf E}_{0}/S_{T}\simeq 2\times 10^{6}K/m\) for the characteristic value of applied thermal gradient needed to observe the predicted here giant chemical pressure induced effects. Let us estimate now the absolute value of the linear thermal conductivity governed by the intrinsic Josephson junctions. Recall that within our model the scattering of normal electrons is due to the presence of mutual inductance between the adjacent grains \(L_{0}\) which is of the order of \(L_{0}\simeq\mu_{0}d\simeq 1fH\) assuming \(d=10nm\) for an average ”grain” size. In the absence of chemical pressure effects, the temperature evolution of LTC is given by \(\kappa_{L}({\bf T},0)=\kappa_{0}[\eta({\bf T},0)+2\beta_{L}({\bf T})\nu({\bf T},0)]\) where \(\kappa_{0}=Nd^{2}S_{T}\Phi_{0}/VL_{0}\). Assuming \(V\simeq Nd^{2}l\) for the sample’s volume, using the above-mentioned expression for \(S_{T}\), and taking \(\beta_{L}(0)=5\) and \(r_{n}=0.1\) for the value of the SQUID parameter and the resistance ratio, we obtain \(\kappa_{L}(0.2{\bf T_{C}},0)\simeq 1W/mK\) for an estimate of the maximum of the LTC (see Fig.10).
+
+And finally, it is worth comparing the above estimates for inductively coupled grains (Sergeenkov 2002) with the estimates for capacitively coupled grains (Sergeenkov 2007) where the scattering of normal electrons is governed by the Stewart-McCumber parameter \(\beta_{C}({\bf T})=2\pi I_{C}({\bf T})C_{0}R_{n}^{2}/\Phi_{0}\) due to the presence of the normal resistance \(R_{n}\) and mutual capacitance \(C_{0}\) between the adjacent grains. The latter is estimated to be \(C_{0}\simeq 1aF\) using \(d=10nm\) for an average ”grain” size. Furthermore, the critical current \(I_{C}(0)\) can be estimated via the critical temperature \({\bf T_{C}}\) as follows, \(I_{C}(0)\simeq 2\pi k_{B}{\bf T_{C}}/\Phi_{0}\) which gives \(I_{C}(0)\simeq 10\mu A\) (for \({\bf T_{C}}\simeq 90K\)) and leads to \(\beta_{C}(0)\simeq 3\) for the value of the Stewart-McCumber parameter assuming \(R_{n}\simeq R_{0}\) for the normal resistance which, in turn, results in \(q\simeq\Phi_{0}/R_{n}\simeq 10^{-19}C\) and \(E_{C}=q^{2}/2C_{0}\simeq 0.1eV\) for the estimates of the ”grain” charge and the Coulomb energy. Using the above-mentioned expressions for \(S_{0}\) and \(\beta_{C}(0)\), we obtain \(\kappa_{L}\simeq 10^{-3}W/mK\) for the maximum of the capacitance controlled LTC which is actually much smaller than a similar estimate obtained above for inductance controlled \(\kappa_{L}\) (Sergeenkov 2002) but at the same time much higher than phonon dominated heat transport in granular systems (Deppe and Feldman 1994).
+
+5. THERMAL EXPANSION OF A SINGLE JOSEPHSON CONTACT AND 2D JJA
+
+In this Section, by introducing a concept of thermal expansion (TE) of a Josephson junction as an elastic response to an effective stress field, we study (both analytically and numerically) the temperature and magnetic field dependence of TE coefficient \(\alpha\) in a single small junction and in a square array. In particular, we found (Sergeenkov et al. 2007) that in addition to _field_ oscillations due to Fraunhofer-like dependence of the critical current, \(\alpha\) of a small single junction also exhibits strong flux driven _temperature_ oscillations near \({\bf T_{C}}\). We also numerically simulated stress induced response of a closed loop with finite self-inductance (a prototype of an array) and found that \(\alpha\) of a \(5\times 5\) array may still exhibit temperature oscillations if the applied magnetic field \({\bf H}\) is strong enough to compensate for the screening induced effects.
+
+Since thermal expansion coefficient \(\alpha({\bf T},{\bf H})\) is usually measured using mechanical dilatometers (Nagel et al. 2000), it is natural to introduce TE as an elastic response of the Josephson contact to an effective stress field \(\sigma\) (D’yachenko et al. 1995, Sergeenkov 1998b, Sergeenkov 1999). Namely, we define the TE coefficient (TEC) \(\alpha({\bf T},{\bf H})\) as follows:
+
+\[\alpha({\bf T},{\bf H})=\frac{d\epsilon}{d{\bf T}}\] (36)
+
+where an appropriate strain field \(\epsilon\) in the contact area is related to the Josephson energy \(E_{J}\) as follows (\(V\) is the volume of the sample):
+
+\[\epsilon=-\frac{1}{V}\left[\frac{dE_{J}}{d\sigma}\right]_{\sigma=0}\] (37)
+
+For simplicity and to avoid self-field effects, we start with a small Josephson contact of length \(w<\lambda_{J}\) (\(\lambda_{J}=\sqrt{\Phi_{0}/\mu_{0}dj_{c}}\) is the Josephson penetration depth) placed in a strong enough magnetic field (which is applied normally to the contact area) such that \({\bf H}>\Phi_{0}/2\pi\lambda_{J}d\), where \(d=2\lambda_{L}+t\), \(\lambda_{L}\) is the London penetration depth, and \(t\) is an insulator thickness.
+
+The Josephson energy of such a contact in applied magnetic field is governed by a Fraunhofer-like dependence of the critical current (Orlando and Delin 1991):
+
+\[E_{J}=J\left(1-\frac{\sin\varphi}{\varphi}\cos\varphi_{0}\right),\] (38)
+
+where \(\varphi=\pi\Phi/\Phi_{0}\) is the frustration parameter with \(\Phi={\bf H}wd\) being the flux through the contact area, \(\varphi_{0}\) is the initial phase difference through the contact, and \(J\propto e^{-t/\xi}\) is the zero-field tunneling Josephson energy with \(\xi\) being a characteristic (decaying) length and \(t\) the thickness of the insulating layer. The self-field effects (screening), neglected here, will be considered later for an array.
+
+Notice that in non-zero applied magnetic field \({\bf H}\), there are two stress-induced contributions to the Josephson energy \(E_{J}\), both related to decreasing of the insulator thickness under pressure. Indeed, according to the experimental data (D’yachenko et al. 1995), the tunneling dominated critical current \(I_{C}\) in granular high-\(T_{C}\) superconductors was found to exponentially increase under compressive stress, viz. \(I_{C}(\sigma)=I_{C}(0)e^{\kappa\sigma}\). More specifically, the critical current at \(\sigma=9kbar\) was found to be three times higher its value at \(\sigma=1.5kbar\), clearly indicating a weak-links-mediated origin of the phenomenon. Hence, for small enough \(\sigma\) we can safely assume that (Sergeenkov 1999) \(t(\sigma)\simeq t(0)(1-\beta\sigma/\sigma_{0})\) with \(\sigma_{0}\) being some characteristic value (the parameter \(\beta\) is related to the so-called ultimate stress \(\sigma_{m}\) as \(\beta=\sigma_{0}/\sigma_{m}\)). As a result, we have the following two stress-induced effects in Josephson contacts:
+
+(I) amplitude modulation leading to the explicit stress dependence of the zero-field energy
+
+\[J({\bf T},\sigma)=J({\bf T},0)e^{\gamma\sigma/\sigma_{0}}\] (39)
+
+with \(\gamma=\beta t(0)/\xi\), and
+
+(II) phase modulation leading to the explicit stress dependence of the flux
+
+\[\Phi({\bf T},{\bf H},\sigma)={\bf H}wd({\bf T},\sigma)\] (40)
+
+with
+
+\[d({\bf T},\sigma)=2\lambda_{L}({\bf T})+t(0)(1-\beta\sigma/\sigma_{0})\] (41)
+
+Finally, in view of Eqs.(36)-(41), the temperature and field dependence of the small single junction TEC reads (the initial phase difference is conveniently fixed at \(\varphi_{0}=\pi\)):
+
+\[\alpha({\bf T},{\bf H})=\alpha({\bf T},0)\left[1+F({\bf T},{\bf H})\right]+\epsilon({\bf T},0)\frac{dF({\bf T},{\bf H})}{d{\bf T}}\] (42)
+
+where
+
+\[F({\bf T},{\bf H})=\left[\frac{\sin\varphi}{\varphi}+\frac{\xi}{d({\bf T},0)}\left(\frac{\sin\varphi}{\varphi}-\cos\varphi\right)\right]\] (43)
+
+with
+
+\[\varphi({\bf T},{\bf H})=\frac{\pi\Phi({\bf T},{\bf H},0)}{\Phi_{0}}=\frac{{\bf H}}{{\bf H}_{0}({\bf T})}\] (44)
+
+\[\alpha({\bf T},0)=\frac{d\epsilon({\bf T},0)}{d{\bf T}}\] (45)
+
+and
+
+\[\epsilon({\bf T},0)=-\left(\frac{\Phi_{0}}{2\pi}\right)\left(\frac{2\gamma}{V\sigma_{0}}\right)I_{C}({\bf T})\] (46)
+
+Here, \({\bf H}_{0}({\bf T})=\Phi_{0}/\pi wd({\bf T},0)\) with \(d({\bf T},0)=2\lambda_{L}({\bf T})+t(0)\).
+
+Fig. 13: Temperature dependence of the flux driven strain field in a single short contact for different values of the frustration parameter \({\bf f}\) according to Eqs.(36)-(48).
+
+For the explicit temperature dependence of \(J({\bf T},0)=\Phi_{0}I_{C}({\bf T})/2\pi\) we use the well-known (Meservey and Schwartz 1969, Sergeenkov 2002) analytical approximation of the BCS gap parameter (valid for all temperatures), \(\Delta({\bf T})=\Delta(0)\tanh\left(2.2\sqrt{\frac{{\bf T_{C}}-{\bf T}}{{\bf T}}}\right)\) with \(\Delta(0)=1.76k_{B}{\bf T_{C}}\) which governs the temperature dependence of the Josephson critical current
+
+\[I_{C}({\bf T})=I_{C}(0)\left[\frac{\Delta({\bf T})}{\Delta(0)}\right]\tanh\left[\frac{\Delta({\bf T})}{2k_{B}{\bf T}}\right]\] (47)
+
+while the temperature dependence of the London penetration depth is governed by the two-fluid model:
+
+\[\lambda_{L}({\bf T})=\frac{\lambda_{L}(0)}{\sqrt{1-({\bf T}/{\bf T_{C}})^{2}}}\] (48)
+
+Fig. 14: Temperature dependence of flux driven normalized TEC in a single small contact for different values of the frustration parameter \({\bf f}\) (for the same set of parameters as in Fig.13) according to Eqs.(36)-(48).
+
+From the very structure of Eqs.(36)-(44) it is obvious that TEC of a single contact will exhibit _field_ oscillations imposed by the Fraunhofer dependence of the critical current \(I_{C}\). Much less obvious is its temperature dependence. Indeed, Fig.13 presents the temperature behavior of the contact area strain field \(\Delta\epsilon({\bf T},{\bf f})=\epsilon({\bf T},{\bf f})-\epsilon({\bf T},0)\) (with \(t(0)/\xi=1\), \(\xi/\lambda_{L}(0)=0.02\) and \(\beta=0.1\)) for different values of the frustration parameter \({\bf f}={\bf H}/{\bf H}_{0}(0)\). Notice characteristic flux driven temperature oscillations near \({\bf T_{C}}\) which are better seen on a semi-log plot shown in Fig.14 which depicts the dependence of the properly normalized field-induced TEC \(\Delta\alpha({\bf T},{\bf f})=\alpha({\bf T},{\bf f})-\alpha({\bf T},0)\) as a function of \({\bf 1}-{\bf T}/{\bf T_{C}}\) for the same set of parameters.
+
+To answer an important question how the neglected in the previous analysis screening effects will affect the above-predicted oscillating behavior of the field-induced TEC, let us consider a more realistic situation with a junction embedded into an array (rather than an isolated contact) which is realized in artificially prepared arrays using photolithographic technique that nowadays allows for controlled manipulations of the junctions parameters (Newrock et al. 2000). Besides, this is also a good approximation for a granular superconductor (if we consider it as a network of superconducting islands connected with each other via Josephson links). Our goal is to model and simulate the elastic response of such systems to an effective stress \(\sigma\). For simplicity, we will consider an array with a regular topology and uniform parameters (such approximation already proved useful for describing high-quality artificially prepared structures, see, e.g., Sergeenkov and Araujo-Moreira 2004).
+
+Let us consider a planar square array as shown in Fig.15. The total current includes the bias current flowing through the vertical junctions and the induced screening currents circulating in the plaquette (Nakajima and Sawada 1981). This situation corresponds to the inclusion of screening currents only into the nearest neighbors, neglecting thus the mutual inductance terms (Phillips et al. 1993). Therefore, the equation for the vertical contacts will read (horizontal and vertical junctions are denoted by superscripts \(h\) and \(v\), respectively):
+
+Fig. 15: Left: sketch of a regular square array (a single plaquette). Right: electrical scheme of the array with the circulating currents. The bias current is fed via virtual loops external to the array.
+
+\[\frac{\hbar C}{2e}\frac{d^{2}\phi_{i,j}^{v}}{dt^{2}}+\frac{\hbar}{2eR}\frac{d\phi_{i,j}^{v}}{dt}+I_{c}\sin\phi_{i,j}^{v}=\Delta I^{s}_{i,j}+I_{b}\] (49)
+
+where \(\Delta I^{s}_{i,j}=I^{s}_{i,j}-I^{s}_{i-1,j}\) and the screening currents \(I^{s}\) obey the fluxoid conservation condition:
+
+\[-\phi^{v}_{i,j}+\phi^{v}_{i,j+1}-\phi^{h}_{i,j}+\phi^{h}_{i+1,j}=2\pi\frac{\Phi^{ext}}{\Phi_{0}}-\frac{2\pi LI^{s}_{i,j}}{\Phi_{0}}\] (50)
+
+Recall that the total flux has two components (an external contribution and the contribution due to the screening currents in the closed loop) and it is equal to the sum of the phase differences describing the array. It is important to underline that the external flux in Eq.(50), \(\eta=2\pi\Phi^{ext}/\Phi_{0}\), is related to the frustration of the whole array, i.e., this is the flux across the void of the network (Araujo-Moreira et al. 1997, Araujo-Moreira et al. 2005, Grimaldi et al. 1996), and it should be distinguished from the previously introduced applied magnetic field \({\bf H}\) across the junction barrier which is related to the frustration of a single contact \({\bf f}=2\pi{\bf H}dw/\Phi_{0}\) and which only modulates the critical current \(I_{C}({\bf T},{\bf H},\sigma)\) of a single junction while inducing a negligible flux into the void area of the array.
+
+For simplicity, in what follows we will consider only the elastic effects due to a uniform (homogeneous) stress imposed on the array. With regard to the geometry of the array, the deformation of the loop is the dominant effect with its radius \(a\) deforming as follows:
+
+\[a(\sigma)=a_{0}(1-\chi\sigma/\sigma_{0})\] (51)
+
+As a result, the self-inductance of the loop \(L(a)=\mu_{0}aF(a)\) (with \(F(a)\) being a geometry dependent factor) will change accordingly:
+
+\[L(a)=L_{0}(1-\chi_{g}\sigma/\sigma_{0})\] (52)
+
+The relationship between the coefficients \(\chi\) and \(\chi_{g}\) is given by
+
+\[\chi_{g}=\left(1+a_{0}B_{g}\right)\chi\] (53)
+
+where \(B_{g}=\frac{1}{F(a)}\left(\frac{dF}{da}\right)_{a_{0}}\).
+
+Fig. 16: Numerical simulation results for an array \(5\times 5\) (solid line) and a small single contact (dashed line). The dependence of the normalized TEC on the frustration parameter \({\bf f}\) (applied magnetic field \({\bf H}\) across the barrier) for the reduced temperature \({\bf T}/{\bf T_{C}}=0.95\). The parameters used for the simulations: \(\eta=0\), \(\beta=0.1\), \(t(0)/\xi=1\), \(\xi/\lambda_{L}=0.02\), \(\beta_{L}=10\), \(\gamma_{b}=0.95\), and \(\chi_{g}=\chi=0.01\).
+
+It is also reasonable to assume that in addition to the critical current, the external stress will modify the resistance of the contact:
+
+\[R(\sigma)=\frac{\pi\Delta(0)}{2eI_{C}(\sigma)}=R_{0}e^{-\chi\sigma/\sigma_{0}}\] (54)
+
+as well as capacitance (due to the change in the distance between the superconductors):
+
+\[C(\sigma)=\frac{C_{0}}{1-\chi\sigma/\sigma_{0}}\simeq C_{0}(1+\chi\sigma/\sigma_{0})\] (55)
+
+To simplify the treatment of the dynamic equations of the array, it is convenient to introduce the standard normalization parameters such as the Josephson frequency:
+
+\[\omega_{J}=\sqrt{\frac{2\pi I_{C}(0)}{C_{0}\Phi_{0}}}\] (56)
+
+the analog of the SQUID parameter:
+
+\[\beta_{L}=\frac{2\pi I_{C}(0)L_{0}}{\Phi_{0}},\] (57)
+
+and the dissipation parameter:
+
+\[\beta_{C}=\frac{2\pi I_{C}(0)C_{0}R_{0}^{2}}{\Phi_{0}}\] (58)
+
+Combining Eqs.(49) and (50) with the stress-induced effects described by Eqs. (54) and (55) and using the normalization parameters given by Eqs.(56)-(58), we can rewrite the equations for an array in a rather compact form. Namely, the equations for vertical junctions read:
+
+\[\frac{1}{1-\chi\sigma/\sigma_{0}}\ddot{\phi}_{i,j}^{v}+\frac{e^{-\chi\sigma/\sigma_{0}}}{\sqrt{\beta_{C}}}\dot{\phi}_{i,j}^{v}+e^{\chi\sigma/\sigma_{0}}\sin\phi_{i,j}^{v}=\gamma_{b}+\]
+\[\frac{1}{\beta_{L}\left(1-\chi_{g}\sigma/\sigma_{0}\right)}\left[\phi^{v}_{i,j-1}-2\phi^{v}_{i,j}+\phi^{v}_{i,j+1}+\phi^{h}_{i,j}-\phi^{h}_{i-1,j}+\phi^{h}_{i+1,j-1}-\phi^{h}_{i,j-1}\right]\] (59)
+
+Here an overdot denotes the time derivative with respect to the normalized time (inverse Josephson frequency), and the bias current is normalized to the critical current without stress, \(\gamma_{b}=I_{b}/I_{C}(0)\).
+
+The equations for the horizontal junctions will have the same structure safe for the explicit bias related terms:
+
+\[\frac{1}{1-\chi\sigma/\sigma_{0}}\ddot{\phi}_{i,j}^{h}+\frac{e^{-\chi\sigma/\sigma_{0}}}{\sqrt{\beta_{C}}}\dot{\phi}_{i,j}^{h}+e^{\chi\sigma/\sigma_{0}}\sin\phi_{i,j}^{h}=\]
+\[\frac{1}{\beta_{L}\left(1-\chi_{g}\sigma/\sigma_{0}\right)}\left[\phi^{h}_{i,j-1}-2\phi^{h}_{i,j}+\phi^{h}_{i,j+1}+\phi^{v}_{i,j}-\phi^{v}_{i-1,j}+\phi^{v}_{i+1,j-1}-\phi^{v}_{i,j-1}\right]\] (60)
+
+Finally, Eqs.(59) and (60) should be complemented with the appropriate boundary conditions (Binder et al. 2000) which will include the normalized contribution of the external flux through the plaquette area \(\eta=2\pi\Phi^{ext}/\Phi_{0}\).
+
+Fig. 17: Numerical simulation results for an array \(5\times 5\). The influence of the flux across the void of the network \(\eta\) frustrating the whole array on the temperature dependence of the normalized TEC for different values of the barrier field \({\bf f}\) frustrating a single junction for \(\gamma_{b}=0.5\) and the rest of parameters same as in Fig.16.
+
+It is interesting to notice that Eqs.(59) and (60) will have the same form as their stress-free counterparts if we introduce the stress-dependent renormalization of the parameters:
+
+\[\tilde{\omega}_{J}=\omega_{J}e^{\chi\sigma/2\sigma_{0}}\] (61)
+
+\[\tilde{\beta}_{C}=\beta_{C}e^{-3\chi\sigma/\sigma_{0}}\] (62)
+
+\[\tilde{\beta}_{L}=\beta_{L}(1-\chi_{g}\sigma/\sigma_{0})e^{\chi\sigma/\sigma_{0}}\] (63)
+
+\[\tilde{\eta}=\eta(1-2\chi\sigma/\sigma_{0})\] (64)
+
+\[\tilde{\gamma}_{b}=\gamma_{b}e^{-\chi\sigma/\sigma_{0}}\] (65)
+
+Turning to the discussion of the obtained numerical simulation results, it should be stressed that the main problem in dealing with an array is that the total current through the junction should be retrieved by solving self-consistently the array equations in the presence of screening currents. Recall that the Josephson energy of a single junction for an arbitrary current \(I\) through the contact reads:
+
+\[E_{J}({\bf T},{\bf f},I)=E_{J}({\bf T},{\bf f},I_{C})\left[1-\sqrt{1-\left(\frac{I}{I_{C}}\right)^{2}}\right]\] (66)
+
+The important consequence of Eq.(66) is that if no current flows in the array’s junction, such junction will not contribute to the TEC (simply because a junction disconnected from the current generator will not contribute to the energy of the system).
+
+Below we sketch the main steps of the numerical procedure used to simulate the stress-induced effects in the array:
+
+* (1)a bias point \(I_{b}\) is selected for the whole array;
+* (2)the parameters of the array (screening, Josephson frequency, dissipation, etc) are selected and modified according to the intensity of the applied stress \(\sigma\);
+* (3)the array equations are simulated to retrieve the static configuration of the phase differences for the parameters selected in step \(2\);
+* (4)the total current flowing through the individual junctions is retrieved as:
+
+\[I^{v,h}_{i,j}=I_{C}\sin\phi^{v,h}_{i,j}\] (67)
+* (5)the energy dependence upon stress is numerically estimated using the value of the total current \(I^{v,h}_{i,j}\) (which is not necessarily identical for all junctions) found in step \(4\) via Eq.(67);
+* (6)the array energy \(E_{J}^{A}\) is obtained by summing up the contributions of all junctions with the above-found phase differences \(\phi^{v,h}_{i,j}\);
+* (7)the stress-modified screening currents \(I^{s}_{i,j}({\bf T},{\bf H},\sigma)\) are computed using Eq.(50) and inserted into the magnetic energy of the array \(E_{M}^{A}=\frac{1}{2L}\Sigma_{i,j}(I^{s}_{i,j})^{2}\);
+* (8)the resulting strain field and TE coefficient of the array are computed using numerical derivatives based on the finite differences:
+
+\[\epsilon^{A}\simeq\frac{1}{V}\left[\frac{\Delta\left(E_{M}^{A}+E_{J}^{A}\right)}{\Delta\sigma}\right]_{\Delta\sigma\to 0}\] (68)
+
+\[\alpha({\bf T},{\bf H})\simeq\frac{\Delta\epsilon^{A}}{\Delta{\bf T}}\] (69)
+
+The numerical simulation results show that the overall behavior of the strain field and TE coefficient in the array is qualitatively similar to the behavior of the single contact. In Fig.16 we have simulated the behavior of both the small junction and the array as a function of the field across the barrier of the individual junctions in the presence of bias and screening currents. As is seen, the dependence of \(\alpha({\bf T},{\bf f})\) is very weak up to \({\bf f}\simeq 0.5\), showing a strong decrease of about \(50\%\) when the frustration approaches \({\bf f}=1\).
+
+A much more profound change is obtained by varying the temperature for the fixed value of applied magnetic field. Fig.17 depicts the temperature behavior of \(\alpha({\bf T},{\bf f})\) (on semi-log scale) for different field configurations which include barrier field \(f\) frustrating a single junction and the flux across the void of the network \(\eta\) frustrating the whole array. First of all, comparing Fig.17(a) and Fig.14 we notice that, due to substantial modulation of the Josephson critical current \(I_{C}({\bf T},{\bf H})\) given by Eq.(38), the barrier field \({\bf f}\) has similar effects on the TE coefficient of both the array and the single contact including temperature oscillations. However, finite screening effects in the array result in the appearance of oscillations at higher values of the frustration \({\bf f}\) (in comparison with a single contact). On the other hand, Fig.17(b-d) represent the influence of the external field across the void \(\eta\) on the evolution of \(\alpha({\bf T},{\bf f})\). As is seen, in comparison with a field-free configuration (shown in Fig.17(a)), the presence of external field \(\eta\) substantially reduces the magnitude of the TE coefficient of the array. Besides, with \(\eta\) increasing, the onset of temperature oscillations markedly shifts closer to \({\bf T_{C}}\).
+
+6. SUMMARY
+
+In this Chapter, using a realistic model of 2D Josephson junction arrays (created by 2D network of twin boundary dislocations with strain fields acting as an insulating barrier between hole-rich domains in underdoped crystals), we considered many novel effects related to the magnetic, electric, elastic and transport properties of Josephson nanocontacts and nanogranular superconductors. Some of the topics covered here include such interesting phenomena as chemomagnetism and magnetoelectricity, electric analog of the ”fishtai” anomaly and field-tuned weakening of the chemically-induced Coulomb blockade as well as a giant enhancement of nonlinear thermal conductivity (reaching \(500\%\) when the intrinsically induced chemoelectric field \(E_{\mu}\propto|\nabla\mu|\), created by the gradient of the chemical potential due to segregation of hole producing oxygen vacancies, closely matches the externally produced thermoelectric field \(E_{T}\propto|\nabla T|\)). Besides, we have investigated the influence of a homogeneous mechanical stress on a small single Josephson junction and on a plaquette (array of \(5\times 5\) junctions) and have shown how the stress-induced modulation of the parameters describing the junctions (as well as the connecting circuits) produces such an interesting phenomenon as a thermal expansion (TE) in a single contact and two-dimensional array (plaquette). We also studied the variation of the TE coefficient with an external magnetic field and temperature. In particular, near \({\bf T_{C}}\) (due to some tremendous increase of the effective ”sandwich” thickness of the contact) the field-induced TE coefficient of a small junction exhibits clear _temperature_ oscillations scaled with the number of flux quanta crossing the contact area. Our numerical simulations revealed that these oscillations may actually still survive in an array if the applied field is strong enough to compensate for finite screening induced self-field effects.
+
+The accurate estimates of the model parameters suggest quite an optimistic possibility to experimentally realize all of the predicted in this Chapter promising and important for applications effects in non-stoichiometric nanogranular superconductors and artificially prepared arrays of Josephson nanocontacts.
+
+ACKNOWLEDGMENTS
+
+Some of the results presented in Section 5 were obtained in collaboration with Giacomo Rotoli and Giovanni Filatrella. This work was supported by the Brazilian agency CAPES.
+
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+

arxiv_ground_truth/0904.1520.md

@@ -0,0 +1,586 @@
+# Graphene and Boron Nitride Single Layers
+
+Thomas Greber, Physik Institut, Universität Zürich, Switzerland
+Uni Irchel 36K88, +41 44 635 5744, greber@physik.uzh.ch
+
+Chapter in the Handbook of Nanophysics, Taylor and Francis Books, Inc. Editor Klaus Sattler
+
+###### Contents
+
+1. 1 Single layer systems
+2. 2 \(sp^{2}\) single layers
+  1. 2.1 Synthesis
+  1. 2.1.1 Chemical Vapor Deposition (CVD)
+  2. 2.1.2 Segregation
+  2. 2.2 Atomic structure
+  1. 2.2.1 Flat layers: Domain boundaries
+  2. 2.2.2 Corrugated layers: Moiré and dislocation networks
+  3. 2.3 Electronic structure I: Work function
+  1. 2.3.1 Flat layers: Vertical polarization
+  2. 2.3.2 Corrugated layers: Lateral electric fields, dipole rings
+  4. 2.4 Electronic structure II: Bandstructure, Fermisurfaces
+  1. 2.4.1 Flat layers: The generic case
+  2. 2.4.2 Corrugated layers: \(\sigma\) band splitting
+  5. 2.5 Sticking and intercalation
+  6. 2.6 Functionality
+  1. 2.6.1 Flat layers: Tunneling junctions
+  2. 2.6.2 Corrugated layers: Templates
+3. A Atomic and Electronic Structure in Real and Reciprocal Space
+  1. A.1 \(sp^{2}\) hybridization
+  2. A.2 Electronic band structure
+  1. A.2.1 \(\pi\) bands
+  2. A.2.2 \(\sigma\) bands
+
+This Chapter deals with single layers of carbon (graphene) and hexagonal boron nitride on transition metal surfaces. The transition metal substrates take the role of the support and allow due to their catalytic activity the growth of perfect layers by means of chemical vapor deposition. The layers are \(sp^{2}\) hybridized honeycomb networks with strong in plane \(\sigma\) and weaker \(\pi\) bonds to the substrate and to the adsorbates. This hierarchy in bond strength causes anisotropic elastic properties, where the \(sp^{2}\) layers are stiff in plane and soft out of plane. A corrugation of these layers imposes a third hierarchy level in bond energies, with lateral bonding to molecular objects with sizes between 1 and 5 nanometer. This extra bond energies are in the range of thermal energies \(k_{B}T\) at room temperature and are particularly interesting for nanotechnology. The concomitant template function will be discussed in Section 2.6.2. The peculiar bond hierarchy also imposes intercalation as another property of \(sp^{2}\) layer systems (Section 2.5). Last but not least \(sp^{2}\) layer systems are particularly robust, i.e. survive immersion into liquids [1], which is a promise for \(sp^{2}\) layers being useful outside ultra high vacuum.
+
+The Chapter shortly recalls the synthesis, describes the atomic and electronic structure, is followed by a discussion of properties like intercalation and the use of \(sp^{2}\) layers on metals as tunneling junctions or as templates. The Sections are divided into subsections along the sketch in Figure 1, i.e. for flat and corrugated layers. Since there are flat and corrugated layers for graphene (g) as well as hexagonal boron nitride (h-BN) the similarities and differences between C-C and B-N are discussed in every subsection. The Chapter ends with an Appendix that summarizes the basics of atomic and electronic structure of honeycomb lattices.
+
+Of course the Chapter does not cover all aspects of \(sp^{2}\) single layers. Topics like free standing layers [2], edge structures of ribbons [3, 4], topological defects [5], or mechanical and chemical properties were not covered.
+
+## 1 Single layer systems
+
+Figure 1: Schematic side view of a single layer on top of a metal. The layer changes the properties of adsorbed molecules as compared to the pristine surface. a) Flat layer, where molecule-molecule interactions are present (see Figure 4 and 18. b) Corrugated layer, which is a template on which single molecules can be laterally isolated (see Figure 7).
+
+A single layer of an adsorbate strongly influences the physical and chemical properties of a surface. Sticking and bonding of atoms and molecules may change by orders of magnitude, as well as the charge transport properties across and parallel to the interface.
+
+There are many single layer systems like e.g. graphite/graphene [6, 7], hexagonal boron nitride [8, 9], boron carbides [10], molybdenum disulfide [11], sodium chlorides [12, 13], or aluminium oxide [14], copper nitride [15, 16] to name a few. In order to decide whether single layers are ”dielectric” or ”metallic” the electronic structure at the Fermi level has to be studied, where a metallic layer introduces new bands at the Fermi energy, while a dielectric layer does not.
+
+Figure 1 shows a schematic view of the single layer systems, with adsorbed molecules. The beneath conduction electrons of the substrate still have tunneling contact to the molecule, though the bonding is much weaker compared to the bare metal. If the single layer is flat (Figure 1 a)) adsorbed molecules may touch, while in corrugated layer systems (Figure 1 b)) molecules may be trapped also laterally on the surface. The corrugation (see Secton 2.2.2) results from the epitaxial stress due to the lattice misfit between the substrate and the \(sp^{2}\) layer. This results in lateral superstructures with nanometer periodicities and corrugations in the 0.1 nm range. For developments in nanotechnology it is useful to have single layer systems which are inert, remain clean at ambient conditions and are stable up to high temperatures. In this field sp² hybridized graphene and hexagonal boron nitride are outstanding examples. These two intimately related model systems, are discussed in more detail. Both are \(sp^{2}\) hybridized layers with about the same lattice constant, but, while on most transition metals graphene is metallic, h-BN is an insulator.
+
+## 2 \(sp^{2}\) single layers
+
+In this Chapter we restrict ourselves to the \(sp^{2}\) hybridized single layer systems of graphene and hexagonal boron nitride. Except for some considerations in the Appendix we always deal with layers on transition metal supports. The comparison between graphene and boron nitride is particularly helpful for the understanding of their functionality. They are siblings with similarities and differences. Both form a \(sp^{2}\) honeycomb network with similar lattice constant (0.25 nm), with strong \(\sigma\) in plane bonds and soft \(\pi\) bonds to the substrate and the adsorbates. Depending on the substrate, both form flat or corrugated overlayers. The corrugation is a consequence of the lattice mismatch and the anisotropic bonding of the atoms in the honeycomb with the substrate, and leads to the peculiar functionality of molecular trapping [17]. Graphene and h-BN differ in their atomic structure: In the case of h-BN two different atoms constitute the honeycomb unit cell, while the two carbon atoms are equivalent within the honeycomb lattice. This causes most graphene overlayers to be metallic, while the h-BN layers are insulators. It is furthermore the reason for an ”inverted topography” if corrugation occurs (see Figure 8).
+
+The theoretical description of \(sp^{2}\) single layers [18] is easier than the experimental realisation [19]. The first experimental reports on \(sp^{2}\) layer production on supports dates back to early times of surface science [20] and has sometimes been considered to be an annoyance since they poison catalysts [6]. In the field of ultra-thin epitaxial films of graphene and hexagonal boron nitride on solid surfaces the work of the Oshima group has to be highlighted [21]. They studied the production of \(sp^{2}\) layer systems systematically and carefully characterized their electronic and vibronic structures. Today this work is the starting point for numerous ongoing investigations, also because of the rise of graphene [2] i.e. the discovery that devices containing free-standing single layer devices may be realized. Therefore there is a strong demand for production routes for these materials that have, somehow, to start on a solid surface.
+
+### Synthesis
+
+There are several roads to the production of \(sp^{2}\) single layers. Here the Chemical Vapor Deposition (CVD) processes that base on the reactive adsorption of precursor molecules from the gas phase onto the substrate, and the segregation method, where the constituents diffuse from the bulk to the surface, are briefly summarized.
+
+The \(sp^{2}\) layers may also be grown on substrates without C_3v_ symmetry as e.g. present for the (111) surfaces of face centered cubic \(fcc\) crystals. Examples are the growth of h-BN on Mo(110) [22] or on Pd(110) [23].
+
+Although this chapter deals with single \(sp^{2}\) layers only, it could be desirable to grow multilayers. There is e.g. one report on the growth of graphene on h-BN/Ni(111) [24], and for the segregation approach it seems to be easier to grow multilayers, as it was shown for silicon carbide (SiC) substrates [25], since this method has not to overcome the low sticking probability of precursor gases on complete \(sp^{2}\) layers.
+
+#### 2.1.1 Chemical Vapor Deposition (CVD)
+
+Chemical Vapor Deposition processes comprise the adsorption and reaction of the educts i.e. precursor molecules on the surface where a new material shall grow. The process often involves the cracking or decomposition of the precursor molecules and a partial release of products into the gas phase.
+
+The first h-BN single layers have been synthesized on Ru(0001) and Pt(111) surfaces by CVD of benzene like borazine \((HBNH)_{3}\)[8]. The process comprises the hydrogen abstraction from the borazine molecules, the assembly of hexagonal boron nitride and the desorption of \(H_{2}\) gas. The h-BN growth rate drops after the formation of the first layer by several orders of magnitude. This has the practical benefit that it is easy to prepare single layers. In Figure 2 the growth of h-BN on Ni(111) is shown as a function of the exposure to the precursor molecules. Clearly, the growth rate drops by more than two orders of magnitude after the completion of the first layer. This behavior is quite general for \(sp^{2}\) layer systems and similar growth behavior is expected for graphene layers. The drop in growth rate is presumably due to the much lower sticking probability of borazine after the completion of the first h-BN layer, and the much lower catalytic activity of h-BN compared to clean transition metals. However, not much is known on the details of the growth process of h-BN. From the study of h-BN island morphologies on Ni(111) it was suggested that the borazine BN six-ring is opened during the self assembly process [26, 27]. For h-BN layer formation, also trichloroborazine (ClBNH)₃[27], and diborane (B₂H₆) ammonia (NH₃) gas mixtures [28] were successfully used.
+
+Figure 2: Growth of a h-BN layer on Ni(111): Coverage as a function of borazine exposure. 1 Langmuir (L)= \(10^{-6}Torr\cdot s\). Note the drop of growth rate after the completion of the first layer. By courtesy of W, Auwärter.
+
+For the formation of graphene layers many different precursors have been used, as e.g. ethene (\(C_{2}H_{4}\)) [29], acetylene (\(C_{2}H_{2}\)) [24], or propylene (\(C_{3}H_{6}\)) [30]. It turns out that almost all hydrocarbons react in non oxidizing environments on transition metal surfaces to graphene, though there are also growth conditions where diamond films grow [31].
+
+#### 2.1.2 Segregation
+
+An alternative way to the above mentioned CVD processes is the use of segregation. Here the educts are diluted in the bulk of the substrate. If the substrate is heated, they start to diffuse and eventually meet the surface, where they have a larger binding energy than in the bulk. At the surface they may then react to the new material. If the substrate temperature exceeds the stability limit of the \(sp^{2}\) layer, back dissolution into the bulk or desorption may occur. Well known examples are e.g. the formation of graphene on Ni(111) [6, 32], Ru(0001) [33], SiC [25] or the formation of BC₃ layers on NbB₂[10].
+
+### Atomic structure
+
+The atomic structure of the \(sp^{2}\) layer systems is fairly well understood. It bases on a strong \(sp^{2}\) hybridized in plane bonded honeycomb lattice and a, relative to these \(\sigma\) bonds, weak \(\pi\) bonding to the substrate. The \(\pi\) (\(p_{z}\)) bonding depends on the registry to the substrate atoms, where the layer-substrate (\(p_{z}-d_{z^{2}}\)) hybridization causes a tendency for lateral lock-in of the overlayer atoms to the substrate atoms. For systems, where substrates have the same symmetry (\(C_{3v}\)), as the \(sp^{2}\) honeycombs, the lattice mismatch \({M}\) is defined as:
+
+\[{M}=\frac{a_{ovl}-a_{sub}}{a_{sub}}\] (1)
+
+where \(a_{ovl}\) is the lattice constant of the overlayer, and \(a_{sub}\) that of the substrate. Often the sign of the lattice mismatch is not indicated, and the absolute value of the difference between the substrate and the overlayer lattice constant is used. Then it has to be explicitly said whether the mismatch induces compressive (+) or tensile (-) stress in the overlayer, or vice versa, tensile (+) or compressive (-) stress in the substrate. The mismatch plays a key role in the epitaxy of the \(sp^{2}\) layers. Of course, the bonding to the substrate also depends on the substrate type, and in turn may lead to the formation of corrugated super structures with a peculiar template function.
+
+Figure 3: Calculated BN binding energies of h-BN on various transition-metals. The h-BN is strained to the lattice constant of the substrate. The bond strength correlates with the \(d\)-band occupancy, where the bonding is strongest for the 4d metals. The dashed lines are guides to the eyes. Results within the Local Density Approximation (LDA) are shown. Data from [34].
+
+Figure 3 shows the calculated BN bond strength on different transition metals (TM) [34]. The calculations were performed for (1x1) unit cells, where the h-BN was strained to the lattice constant of the corresponding substrate, with nitrogen in on top position. The BN bond energy is determined from the difference of the energy of the (1x1) h-BN/TM system and the strained h-BN + TM system. The trend indicates the importance of the \(d\)-band occupancy of the substrate and it is e.g. similar to the dissociative adsorption energies of ammonia \(NH_{3}\) on transition metals [35]. Since the lock-in energy, i.e. the energy that has to be paid when the nitrogen atoms are moved laterally away from the on top sites, is expected to scale with the adsorption energy, Figure 3 also rationalizes e.g. why h-BN/Pd(111) is less corrugated than h-BN/Rh(111).
+
+This section is divided into two parts, where we distinguish flat and corrugated layers. Flat means that the same types of atoms have the same height above the substrate, which has only to be expected for (1x1) unit cells like that in the h-BN/Ni(111) system. Corrugation occurs due to lattice mismatch and lock-in energy gain, i.e. due to the formation of dislocations.
+
+#### 2.2.1 Flat layers: Domain boundaries
+
+The Ni(111) substrate has \(C_{3v}\) symmetry and a very small lattice mismatch of +0.4 % to the \(sp^{2}\) layers. On Ni(111) h-BN forms perfect single layers. Figure 4 shows scanning tunneling microscopy (STM) images from h-BN/Ni(111). The layers appear to be defect free and flat. The STM may resolve the nitrogen and the boron sublattices, where a Tersoff Haman calculation indicated that the nitrogen atoms map brighter than the boron atoms [36]. The production process of the h-BN layers (see Section 2.1) also leads to the formation of larger terraces compared to uncovered Ni(111), where widths of 200 nm are easily obtained. The stability of the (111) facet was also observed in experiments with stepped Ni(755) [37] and Ni(223) surfaces, where the miscut of these surfaces relative to the [111] direction lead to large (111) facets and step bunches.
+
+Figure 4: Constant current scanning tunneling microscopy (STM) images of a flat layer of hexagonal boron nitride on Ni(111). a) 30x30 nm. The grey scales indicate 3 terraces with different heights. Large defect free terraces form. b) 3x3 nm. The nitrogen (bright) and the boron (grey) sublattices are resolved with different topographical contrast. From [9].
+
+For the case of the h-BN/Ni(111) system the mismatch \({M}\) is +0.4%, i.e. the h-BN is laterally weakly compressed. This small mismatch leads to (1x1) unit cells and atomically flat layers. The atomic structure of h-BN/Ni(111) is well understood and there is good agreement between experiment [38, 9, 39] and theory [36, 40, 41]. Figure 5 shows six (1x1) configurations for h-BN within the Ni(111) unit cell. The nomenclature of the structure (B,N)=(fcc,top) indicates that boron is sitting on the fcc site, i.e. on a site where no atom is found in the second nickel layer, and where the nitrogen atom sits on top of the atom in the first nickel layer. Theory found two stable structures (B,N)=(fcc,top) and (hcp,top) only. In both cases nitrogen is on top [36]. The (fcc,top) structure has the lowest energy, which was consistent with the published experimental structure determinations. The calculated energy difference between the structure with boron on fcc and on hcp differs only by 9 meV, which is reasonable since it indicates interaction of the overlayer with the second nickel layer.
+
+Figure 5: Possible (1x1) configurations for one-monolayer h-BN/Ni(111). The top, fcc hollow, and hcp hollow sites are considered for the position of the boron and nitrogen atoms, respectively. The (B,N)=(fcc,top) registry has the strongest bond, while calculations predict 9 meV lower binding energy for the (B,N)=(hcp,top) configuration. From [36].
+
+The 9 meV are, however, one order of magnitude smaller than the thermal energies \(k_{B}T\) during the synthesis, and from this viewpoint it is not clear why pure (B,N)=(fcc,top) single domain can be grown. Inspection of Figure 5 shows that the (fcc,top) structure is a translation of the (top,hcp) or (hcp,fcc). A transformation between (fcc,top) and (hcp,top) involves, a rotation, or a permutation of B and N and a translation. The growth of the h-BN layers on Ni(111) proceeds via the nucleation and growth of triangular islands that are separated by distances between 10 and 100 nm [26]. Therefore, if (fcc,top) and (hcp,top) nucleation seeds form, two domains are expected since the change of orientation of islands with sizes larger than 10 nm costs too much energy [26]. The concomitant domain boundaries (see Figure 16 a)) have interesting functionalities. They act as collectors for intercalating atoms, and clusters like to grow on these lines. Also, it is expected that such domain boundaries are model systems for \(sp^{2}\) edges. Unfortunately, the (fcc,top):(hcp,top) ratio is not yet under full experimental control. But it is e.g. known that the omission of the oxygen treatment of the Ni(111) surface before h-BN growth causes two domain systems [26].
+
+The vertical position of the nitrogen and the boron atoms is not the same within the (1x1) unit cell. This local corrugation or buckling, where nitrogen is the outmost atom, and boron sits closer to the first Ni plane, was taken as an indication of the compressive stress on the h-BN in the h-BN/Ni(111) system [42, 9]. The comprehensive density functional theory study of Laskowski et al. [34], indicated however that this buckling, i.e. height difference between boron and nitrogen persists also for systems with tensile stress in the overlayer, where the substrate lattice constant is larger than that of the h-BN. B is closer to the first substrate plane for all investigated cases. This is a consequence of the bonding to the substrate, where the boron atoms are attracted and the nitrogen atoms are repelled from the surface [34].
+
+For the corresponding graphene Ni system the same structure i.e the (C_A_,C_B_)=(fcc,top) which is equivalent to the (top,fcc) configuration has been singled out against the (fcc,hcp) structure [38]. There are no reports on (top,fcc)/(top,hcp) domain boundaries, as observed for the h-BN/Ni(111) case [26]. For g/Ir(111), which belongs to the corrugated layer systems, dislocation lines that are terminated by heptagon-pentagon defects were found [5]. Of course these kinds of defects were less likely for BN, since this would imply energetically unfaforable N-N or B-B bonds in the BN network.
+
+#### 2.2.2 Corrugated layers: Moiré and dislocation networks
+
+When the lattice mismatch \({M}\) of the laterally rigid sp² networks exceeds a critical value, super structures with large lattice constants are formed. If the lattice of the overlayer and the substrate are rigid and parallel, the super structure lattice constant gets \(a_{ovl}/|{M}|\), where \(a_{ovl}\) is the 1x1 lattice constant of graphene or h-BN (\(\approx\) 0.25 nm).
+
+Figure 6: Schematic view of a moiré and a dislocation in an overlayer system. a) in a moiré pattern \(\sim(n+1)\)\(sp^{2}\) units fit on \(n\) substrate units, and the directions of the substrate and the adsorbate lattice do not necessarily coincide. b) in a dislocation network \(n+1\)\(sp^{2}\) units coincide with \(n\) substrate units. Here the lock-in energy is large for on-top and weak or repulsive for bridge sites, where the dislocation evolves.
+
+Figure 7: Relief view of a constant current scanning tunneling microscopy image of a corrugated layer boron nitride (nanomesh) (30x30 nm², I_t_=2.5 nA, V_s_=-1 V). This nanostructure with 3.2\(\pm\)0.1 nm periodicity consists of two distinct areas: the wires which are 1.2+0.2 nm broad and the holes with a diameter of 2.0\(\pm\)0.2 nm. The corrugation is 0.07\(\pm\)0.02 nm. From [43].
+
+For rigid \(sp^{2}\) layers, i.e. if there would be no lateral lock-in energy available, we expect flat floating layers, reminiscent to incommensurate moiré patterns without a directional lock-in of the structures (see Figure 6 a)). For the case of h-BN/Pd(111) such a tendency to form moiré type patterns, also without a preferential lock-in to a substrate direction, were found [44]. These h-BN films have the signature of the electronic structure of flat single layers (see Section 2.4). The lock-in energy is the energy that an epitaxial system gains on top of the average adhesion energy if the overlayer locks into preferred bonding sites. It involves the formation of commensurate coincidence lattices between the substrate and the adsorbate layer with dislocations (see Figure 6 b)). The lock-in energy is expected to be proportional to the bond energy shown in Figure 3. For the case of \(h\)-BN/Rh(111) the lattice mismatch (Eqn. 1) is -6.7 %, and \(13\times 13\) BN units coincide with \(12\times 12\) Rh units [45, 46]. With the room temperature lattice constants of h-BN and Rh this leads to a residual compression of the 13 h-BN units by 0.9 %. Figure 7 shows a relief-view of this super structure coined ”h-BN nanomesh” [45]. It has a super cell with a lattice constant of 3.2 nm i.e. (12x12) Rh(111) units and displays the peculiar topography of a mesh with ”wires” and ”holes” or ”pores”. It turned out that this structure is a corrugated single layer of hexagonal boron nitride with two electronically distinct regions that are related to the topography [47, 48]. This structure also has the ability of trapping single molecules in its holes (see Section 2.6.2). The accompanying variation of the local coordination of the substrate and the adsorbate atoms divides the unit cells into regions with different registries. In reminiscence to Figure 5 the notation (B,N)=(top,hcp) refers to the local configuration, where a B atom sits on top of the substrate atom in the first layer and N on top of the hexagonal close packed (hcp) site that is on top of the substrate atom in the second layer. Again 3 regions can be distinguished with atoms in (fcc,top), (top,hcp) and (hcp,fcc) configurations (see Figure 8). A force field theory approach indicated that the (B,N)=(fcc,top) sites correspond to the tightly bound regions of the h-BN layers, i.e. the holes, while the weakly bound or even repelled regions correspond to the wires [47]. The corrugation, i.e. height difference of the h-BN layer from the top of the substrate is, in accordance between experiment and theory about 0.05 nm. This is sufficient to produce a distinct functionality (see Section 2.6.2) [48]. For h-BN/Ru(0001) a structure very similar to that of h-BN/Rh(111) was found [49].
+
+Figure 8: Views of the height modulated graphene (g) and \(h\)-BN nanomesh (BN) on Ru(0001), M and V denote mounds (hcp,fcc) and valleys of the graphene, H and W holes (fcc,top) and wires of the \(h\)-BN nanomesh. The six ball model panels illustrate the three different regions ((fcc,top), (top,hcp) and (hcp,fcc)), which can be distinguished in both systems [50].
+
+For the graphene case the situation is related, though not identical. The difference lies in the fact that the base in the 1x1 unit cell of free-standing graphene consists of two identical carbon atoms C_A_ and C_B_, while that of h-BN does not. C_A_ and C_B_ become only distinguishable by the local coordination to the substrate. In g/Ru the local (fcc,top) and (top,hcp) coordination leads to close contact between the (C_A_,C_B_) atoms and the substrate [51] while (B,N) is strongly interacting only in the (fcc,top) coordination [34]. As a result, twice as many atoms are bound in strongly interacting regions in g/Ru when compared to \(h\)-BN/Ru. In reminiscence to morphological terms the strongly bound regions of g/Ru(0001) are called valley (V) and the weakly bound region with the (C_A_,C_B_) atoms on (hcp,fcc) sites mounds (M). The fact that (top,hcp) leads to strong bonding for graphene but weak bonding for \(h\)-BN gives rise to an inverted topography of the two layers: a connected network of strongly bound regions for graphene (valleys) and a connected network of weakly bound regions for \(h\)-BN (wires). Also for the graphene case ”moiré”-type superstructures were found, where g/Ir(111) is the best studied so far [7].
+
+### Electronic structure I: Work function
+
+#### 2.3.1 Flat layers: Vertical polarization
+
+The work function, i.e. the minimum energy required to remove an electron from a solid, is material dependent. There are excellent reviews on the topic [52, 53]. For our purpose, where we want to discuss the electric fields near the surface, it is sufficient to recall the Helmholtz equation that relates the work function \(\Phi\) of a flat surface with vertical electric dipoles:
+
+\[\Phi=\frac{e}{\epsilon_{0}}N_{a}\cdot p\] (2)
+
+where \(e\) is the elementary charge, \(\epsilon_{0}\) the permittivity and \(N_{a}\) the areal density of the dipoles \(p\). If the dipoles are assigned to the atoms, we get e.g. for Ni(111) with a work function of 5.2 eV and an in plane lattice constant of 0.25 nm a dipole of 0.7 Debye (1 Debye= 3.34 \(\cdot 10^{-30}\) Cm). It has to be said that this is the classical view of the work function and all quantum mechanical effects may be incorporated empirically into Equation 2 in using an effective dipole.
+
+Now the influence of overlayers shall be analyzed. If a single layer of a medium is placed on top of the substrate, the electric fields in the surface dipole polarize the layer i.e. decreases the work function by \(\Delta\Phi_{s}=\frac{e}{\epsilon_{0}}N_{a}\cdot p_{ind}\) by screening. The induced dipole \(p_{ind}\) is proportional to the electric field perpendicular to the surface\(E_{\perp}\): \(p_{ind}=\alpha\cdot E_{\perp}\), where \(\alpha\) is the polarizability. In Figure 9 it is shown how a polarizable medium screens out the electric field and decreases the work function. These considerations apply for dielectrica and metals, if they are not in contact with the substrate. The strong vertical distance dependence of the electric field in the surface dipole layer involves a correlation with the screening induced work function shift \(\Delta\Phi_{s}\). In the case of contact of a metallic overlayer with the Fermi level of the substrate, no vertical dipoles in the sense of Figure 9 are induced, although the work function may change. However, as we know from the Smoluchowski effect [54] a corrugation induces a non-uniform surface charge density and thus lateral electric fields (see next Section).
+
+Figure 9: Schematic view of the effect of a thin insulated, polarizable single layer (SL) on the work function. The layer gets polarized and accordingly reduces the surface dipole. The dashed line is the electrostatic potential without single layer, while the solid line shows the effect of the single layer.
+
+For a dielectric, as it is h-BN, the screening causes, e.g. in the case of h-BN/Ni(111) a work function lowering of 1.6 eV [36], which corresponds to an induced dipole of 0.3 Debye. In this case charge gets displaced, but not transferred from the \(sp^{2}\) layer to the substrate. If the medium is metallic, as it is for most graphene cases, we expect a charge transfer that aligns the chemical potentials of the substrate and the overlayer. A strong interaction also alters the chemical potentials and in turn influences the charge transfer. In any case the charge redistribution changes the work function, i.e. the surface dipole. For weakly bound graphene theory predicts a correlation between the work function of the substrate and the doping level [55].
+
+#### 2.3.2 Corrugated layers: Lateral electric fields, dipole rings
+
+If the surface is flat the lateral electric fields will only vary for ionic components in the two sublattices of the \(sp^{2}\) networks. For the case of h-BN the different electronegativities of the two elements cause a local charge transfer from the boron to the nitrogen atoms. By means of density functional theory calculations it has been found that for a free-standing h-BN sheet about 0.56 electrons are transferred from B to N (B,N)=(0.56,-0.56)\(e^{-}\)[36]. This yields a sizable Madelung contribution to the lattice stability of 2.4 eV per 1x1 unit cell (see Appendix A.2). If the h-BN sits on a metal this Madelung energy is reduced by a factor of 1/2 due to the screening of the ionic charges. For h-BN on Ni(111) the ionicity slightly increases (B,N)=(0.65,-0.59) \(e^{-}\), where the net charge displacement of 0.06 \(e^{-}\) towards the substrate is consistent with the above mentioned work function decrease due to polarization [36]. It is expected that the relatively strong local ionicity of the h-BN surface has an influence on the diffusion of atoms with the size of the BN bond length of 0.15 nm. If the surface is not flat, but corrugated, this causes lateral electric fields on the length scale of the corrugation, which is particularly interesting for trapping atoms or molecules [17]. Lateral fields may occur in dielectric or above metallic overlayers. Figure 10 shows the lateral and vertical electrostatic potential in a corrugated single layer nanostructure. For the case of a dielectric (Figure 10 a)) a different distance of the layer from the surface imposes a different screening and lateral potential variations, also within the dielectric. For the case of a metal (Figure 10 b)) a corrugation causes lateral potential variations, reminiscent to the Smoluchowski effect [54], but not within the layer. Also, it was found for g/Ru(0001) that the metallic case causes smaller lateral potential variations [50].
+
+Figure 10: Contour plot of the electrostatic potential at the surface of a corrugated single layer on a flat metal. The energies continuously increase from the Fermi level \(E_{F}\) towards the vacuum level \(E_{V}\) at \(z=\infty\), with \(E_{F}<E_{1}<E_{2}<E_{3}<E_{4}<E_{V}\). Left for a single layer dielectric, right for a single layer metal.
+
+For dielectrics or insulators the key for the understanding of the lateral electrostatic potential variation came from the \(\sigma\) band splitting (see Section 2.4.2). This splitting, which is in the order of 1 eV, is also reflected in N1s core level x-ray photoelectron spectra [56], where the peak assignment is in line with the \(\sigma\) band assignment [49, 48]. Without influence of the substrate the energy of the \(\sigma\) band, which reflects the in plane \(sp^{2}\) bonds, are referred to the vacuum level. This means that the sum of the work function and the binding energy as referred to the Fermi level, is a constant. Vacuum level alignment arises for physisorbed systems, as e.g. for noble gases [57, 58], or as it was proposed for h-BN films on transition metals [59]. The \(\sigma\) band splitting causes the conceptual problem of aligning the vacuum level and the Fermi level with two different work functions. The _local_ work function, or the electrostatic potential near the surface may, however, be different. This electrostatic potential with respect to the Fermi level can be measured with photoemission of adsorbed Xe (PAX) [60, 57]. Xenon does not bond strongly to the substrate and thus the core level binding energies are a measure for the potential energy difference between the site of the Xe core (Xe has a van der Waals radius of about 0.2 nm) and the Fermi level. Recently, the method of photoemission from adsorbed Xe was successfully applied to explore the electrostatic energy landscape of h-BN/Rh(111) [17]. In accordance with density functional theory calculations it was found that the electrostatic potential at the Xe cores in the holes of the h-BN/Rh(111) nanomesh is 0.3 eV lower than that on the wires. This has implications for the functionality, since these sizable potential gradients polarize molecules, and it may be used as an electrostatic trap for molecules or negative ions. The peculiar electrostatic landscape has been rationalized with dipole rings, where in plane dipoles, sitting on the rim of the corrugations produce the electrostatic potential well. For in plane dipoles that sit on a ring the electrostatic potential energy in the center of the ring becomes
+
+\[\Delta E_{pot}=\frac{e}{4\pi\epsilon_{0}}\frac{P}{R^{2}}\] (3)
+
+Figure 11: Schematic drawing of a dipole ring where in plane dipoles \({\bf{p}}_{i}\) are sitting on a ring with radius \(R\). a) the electrostatic potential energy \(E_{pot}\) and b) the polarization energy, which is proportional to \((\nabla E_{pot})^{2}\). From [43].
+
+where \(e\) is the elementary charge, \(R\) the radius of the hole and \(P=\sum|\textbf{p}_{i}|\) the sum of the absolute values of the dipoles on the ring. For \(R\)=1 nm and \(\Delta E_{pot}\)=0.3 eV, \(P\) gets 10 Debye, which is equivalent to 5.4 water molecules with the hydrogen atoms pointing to the center of the holes. The strong lateral electric fields in the BN nanomesh may be exploited for trapping molecules, or negatively charged particles. It can also act as an array of electrostatic nanolenses for slow charged particles that approach or leave the surface. Figure 11 shows the electrostatic potential in a dipole ring and the square of the related electric fields. The square of the gradient of the electrostatic potential, or the electric field, is expected to be proportional to the polarization induced bond energy \(E_{pol}\):
+
+\[E_{pol}={\bf E}_{\parallel}\cdot{\bf{p}}_{ind}=\alpha\cdot{\bf{E}}_{\parallel}^{2}\] (4)
+
+where \({\bf{E}}_{\parallel}\) is the lateral electric field, \({\bf{p}}_{ind}\) the induced dipole, and \(\alpha\) the polarizability. For the case of h-BN/Rh(111) the origin of the in plane electrostatic fields is not the Smoluchowski effect, where the delocalisation of the electrons at steps forms in plane dipoles [54]. It is due to the contact of differently bonded boron nitride with different local work functions since the screening depends on the vertical displacement of the dielectric from the metal, i.e. on the corrugation (see Section 2.3.1).
+
+The concept of dipole rings is not restricted to the above mentioned polarization of corrugated dielectrics. As it was recently shown for the case of g/Ru(0001), lateral electric fields also occur above corrugated metals, where the dipoles are created due to lateral polarization like in the Smoluchowski effect [50].
+
+### Electronic structure II: Bandstructure, Fermisurfaces
+
+The band structure of \(sp^{2}\) layers on transition metals has the same signature as free-standing single layers. The binding energy difference between the in plane \(\sigma\) bands and the out of plane \(\pi\) bands does not remain constant, when the layers come into contact with a substrate. This is a consequence of the bonding via the \(\pi\) orbitals, while for the \(\sigma\) bands vacuum level alignment is observed.
+
+#### 2.4.1 Flat layers: The generic case
+
+The electronic bandstructure of h-BN on Ni(111) is similar to that of a free-standing layer of h-BN, with \(\sigma\) and \(\pi\) bands that have the generic structure of a \(sp^{2}\) honeycomb lattice (see Appendix). In Figure 12 angle resolved photoemission data from h-BN/Ni(111) along the \(\Gamma K\) azimuth are compared with density functional theory calculations. In the bottom panel the bandstructure calculations are shown for h-BN/Ni(111) and free-standing h-BN. The band width of the \(\sigma\) bands increases less than 1 % by the adsorption of the h-BN. The widths of the spin split \(\pi\) bands increase in the spin average by 4%, where the width is larger for the minority spins. This band structure with \(\sigma\) and \(\pi\) bands is generic for all \(sp^{2}\) systems. Though, the \(\sigma\) and the \(\pi\) band do not shift equally in energy upon adsorption of the \(sp^{2}\) layer. The shift between the \(\sigma\) and the \(\pi\) bands of about 1.3 eV indicates that the interaction between the Ni substrate and the h-BN \(\pi\) and \(\sigma\) bands is not the same. Also other substrates like Pt(111) and Pd(111) cause a similar h-BN band structure, although h-BN/Pt(111) and h-BN/Pd(111) are systems with large negative lattice mismatch [21, 44]. For these ’flat’ cases the bonding to the substrates is rather independent of the positions within the unit cell. In the Section on corrugated layers (2.4.2) we will see that this changes when substrates with mismatch and stronger lock-in energy are used. For Ni(111), Pd(111) and Pt(111) it was found empirically that the \(\sigma\) bands have, within \(\pm\) 100 meV, the same binding energy, if they are referred to the vacuum level [59]. This vacuum alignment confirms that the \(\sigma\) bands do not strongly interact with the underlying metal. The binding energy of the \(\pi\) band, on the other hand, varies, and indicates the \(\pi\) bonding to the substrate.
+
+For graphene the electronic structure is also well understood [21]. The \(\pi\) bands are of particular interest since they are decisive for the issue on whether the overlayer is metallic or not. On substrates the ideal case of free-standing graphene may generally not be realized, since the site selective \(p_{z}\) interaction with the substrate is causing a symmetry breaking between the two carbon sublattices \(C_{A}\) and \(C_{B}\). This opens a gap at the Dirac point. The Dirac point coincides with the \(K\) point of the Brillouin zone of graphene, where the bandstructure is described by cones on which electrons behave quasi relativistic (see Appendix).
+
+Figure 12: Experimental gray scale dispersion plot on the \(\Gamma K\) azimuth from angle resolved He I\(\alpha\) photoemission. The gray scale reproduces the intensity (white maximum). Bottom: Spin resolved theoretical band structure of h-BN/Ni(111) along \(M\Gamma\) and \(\Gamma K\) in the hexagonal reciprocal unit cell (Brillouin zone). b) Spin up, the radius of the circles is proportional to the partial \(p_{x}=p_{y}\) charge density centered on the nitrogen sites (\(\sigma\) bands). Thick yellow lines show the bands of a buckled free-standing h-BN monolayer, which have been aligned the \(h\)-BN/Ni(111) \(\sigma\) band at \(\Gamma\). c) Spin down, the radius of the circles is proportional to the partial \(p_{z}\) charge density on the nitrogen sites (\(\pi\) bands). From [36].
+
+The magnitude of the gap is a measure for the anisotropy of the interaction of the \(C_{A}\) and the \(C_{B}\) carbon atoms with the substrate. The position of the Fermi level with respect to the Dirac energy is decisive on whether we deal with \(p\)-type or \(n\)-type graphene (or h-BN). Since the center of the gap of h-BN lies below the Fermi level, this means that e.g. h-BN/Ni(111) is \(n\)-type. Figure 13 shows the \(\pi\) band electronic structure of \(sp^{2}\) honeycomb lattices near the Fermi energy and around the \(K\) point. If the Fermi energy lies above the Dirac energy \(E_{Dirac}\), which is the energy of the center of the gap, the majority of the charge carriers move like electrons (\(n\)-type). Correspondingly, if the Fermi energy lies below the Dirac point the charge carriers move like holes (\(p\)-type). This \(n\)-type, \(p\)-type picture is analogous to the semiconductors, where the Fermi level takes, depending on its position, the role of the acceptors (\(p\)-type) and the donors (\(n\)-type), respectively. Nagashima et al. first measured the \(\pi\) gaps at \(K\) for the case of one and two layers of graphene on TaC(111), where they found \(n\)-type behavior and a gap of 1.3 eV for the single layer and about 0.3\(\pm\)0.1 eV for the double layer [61].
+
+Figure 13: Schematic diagram showing the \(\pi\) band electronic structure of \(sp^{2}\) honeycomb lattices around the \(K\) point. For free-standing graphene the \(\pi\) and the \(\pi^{*}\) band lie on cones intersecting at the Dirac point. Interaction with a support causes a shift of the Fermi energy, and, if it is anisotropic a gap opens. Here (\(E_{Fermi}>E_{Dirac}\)), the case for \(n\)-type conduction is shown (for details see text).
+
+#### 2.4.2 Corrugated layers: \(\sigma\) band splitting
+
+Figure 14: He \(I_{\alpha}\) normal emission photoemission spectra of h-BN/Ni(111), h-BN/Pd(111) and h-BN/Rh(111). While h-BN/Ni(111) and h-BN/Pd(111) show no sizable \(\sigma\) band splitting, the splitting into \(\sigma_{\alpha}\) and \(\sigma_{\beta}\) for h-BN/Rh(111) is about 1 eV (From [43]).
+
+The super structures as described in Section 2.2.2 are also reflected in the electronic structure. Figure 14 shows valence band photoemission spectra for three different \(h\)-BN single layer systems in normal emission. We would like to draw the attention on the \(\sigma\) bands at about 5 eV binding energy (see Table 1). Figure 14 shows valence band photoemission data for h-BN/Ni(111), h-BN/Pd(111) and h-BN/Rh(111). Hexagonal boron nitride has 3 \(\sigma\) bands and one \(\pi\) band that are occupied. Along \({\Gamma}\), i.e. perpendicular to the \(sp^{2}\) plane, the two low binding energy \(\sigma\) bands are degenerate, while the third is not accessible for He I_α_ radiation. Accordingly the \(\sigma\) bands of h-BN Ni(111) and of h-BN/Pd(111) show one single peak. Those of h-BN/Rh(111) are weak and split in a \(\sigma_{\alpha}\) and a \(\sigma_{\beta}\) contribution [49]. In measuring the spectra away from the \({\Gamma}\) point it is seen that both, the \(\sigma_{\alpha}\) and the \(\sigma_{\beta}\) bands split in two components each, as it is known from angular resolved measurements of h-BN single layers with \(\sigma_{\alpha}\) bands only [59] (see Figure 12).
+
+\begin{table}
+\begin{tabular}{c|c|c|c|c|c}
+\hline
+Substrate & \(E_{B}^{F}\sigma_{\alpha}\) & \(E_{B}^{F}\sigma_{\beta}\) & \(\Phi\) & \(E_{B}^{V}\sigma_{\alpha}\) & Reference \\
+\hline
+Ni(111) & 5.3 & & 3.5 & 8.8 & [62] \\
+Rh(111) & 4.57 & 5.70 & 4.15 & 8.7 & [49] \\
+Pd(111) & 4.61 & & 4.26 & 8.9 & [43] \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Experimental values (in eV) for the photoemission binding energies referred to the Fermi level \(E_{B}^{F}\) of the \(\sigma\) bands for h-BN single layers on three different substrates along \(\Gamma\), and the corresponding work functions \(\Phi\). The binding energies with respect to the vacuum level \(E_{B}^{V}\) are determined by \(E_{B}^{F}+\Phi\).
+
+The \(\sigma\) band splitting indicates two electronically different regions within the h-BN/Rh(111) unit cell. They are related to strongly bound and weakly bound h-BN. Later on, h-BN/Ru(0001) [49] was found to be very similar to h-BN/Rh(111). Theoretical efforts [47] and atomically resolved low temperature STM [48] showed that the h-BN/Rh(111) nanomesh is a corrugated single sheet of h-BN on Rh(111) (see Section 2.2.2). The peculiar structure arises from the site dependence of the interaction with the substrate atoms that causes the corrugation of the h-BN sheet, with a height difference between strongly bound regions and weakly bound regions of about 0.05 nm.
+
+Interestingly, no \(\sigma\) band splitting could be found for the g/Ru(0001) system, although these graphene layers are also strongly corrugated [50]. This is likely related to the inverted topography (see Figure 8) and the metallicity of graphene on ruthenium. The metallicity can be investigated locally by means of scanning tunneling spectroscopy. It has however to be said that tunneling spectroscopy is most sensitive to the density of states at the \(\Gamma\) point but the graphene conduction electrons reside around the \(K\) point. This experimental limitation does not exist in angular resolved photoemission, where on the other hand the sub-nanometer resolution of scanning probes is lost. With photoemission it is possible to determine the average Fermi surface of the \(sp^{2}\) layers. The Fermi surface indicates the doping level, i.e. the number of electrons in the conduction band. In the case of graphene cuts across Dirac cones (see Figure 13) are measured. If the Fermi surface map is compared to another constant energy surface near the Fermi energy, it can be decided whether the Dirac point lies below or above the Fermi energy, i.e. whether we deal with \(n\)- or \(p\)-type graphene. In Figure 15 the Fermi surface map as measured for g/Ru(0001) and h-BN/Ru(0001) are compared. For the case of g/Ru(0001) extra intensity at the \(K\) points indicates Dirac cones. The cross sections of which correspond to 5% of the area of the Brillouin zone, or 0.1 \(e^{-}\). The cross section shown in Figure 15 c) does not resolve a single cone with a bimodal distribution around \(K\), as it is e.g. the case for g/SiC [63], but a single peak from which an average cone diameter at the Fermi energy is determined. The measurement of the bandstructure along \(\Gamma K\) indicates that these 0.1 \(e^{-}\) are donated from the substrate to the graphene. For the case of h-BN/Ru(0001) the Fermi surface shows no peak at \(K\) which is expected for an insulating or semiconducting \(sp^{2}\) layer [43]. All features that can be seen in this Fermi surface map stem from the underlying Ru(0001) substrate.
+
+Figure 15: Fermi surface maps. (a) g/Ru(0001). (b) \(h\)-BN/Ru(0001). The hexagons indicate the surface Brillouin zones of Ru(0001) (red dashed), graphite (green solid) and \(h\)-BN (blue solid). (c) and (d) show the normalized intensities of azimuthal cuts along the dashed yellow sectors in (a) and (b) respectively. From [50].
+
+### Sticking and intercalation
+
+Graphite is well known for its ability to intercalate atoms [64], and there are also reports on intercalation in h-BN [65]. Intercalation is the reversible inclusion of guest atoms or molecules between other host molecules. For the host material graphite or \(h\)-BN intercalation occurs between the honeycomb sheets, where the bonding in the sheet is strong and the bonding between the sheets is relatively weak. Here, we deal with single \(sp^{2}\) layers on transition metals and use the term ”intercalation” also for irreversible intercalation, as it is observed if metal atoms slip below the \(sp^{2}\) layer. For the case of g/Ni(111) it has e.g. been found that Cu, Ag and Au intercalate irreversibly, although they form no graphite intercalation compounds [30].
+
+Intercalation is preceded by sticking (adsorption), and diffusion of the intercalating species. In the following the model case of cobalt on h-BN/Ni(111) is discussed in more detail. Figure 16 shows the growth of cobalt on h-BN/Ni(111). On flat terraces, as shown in Figure 16 a) three different patterns are observed. (i) three dimensional (3D) clusters, whose heights scale with the lateral diameter, (ii) triangular, two dimensional (2D) islands with a constant apparent height and (iii) line patterns [66]. Often these line patterns are found to be connected to 2D islands, and 3D islands tend to nucleate on such lines. A careful analysis relates the lines with domain boundaries, where (B,N)=(fcc,top) and (B,N)=(hcp,top) domains touch [26] (see Section 2.2.1). The 2D islands are irreversibly intercalated Co below the h-BN layer, while the 3D clusters remain on top of the h-BN. This results e.g. in the property that the 3D islands may be removed, cluster by cluster, with a STM manipulation procedure [66].
+
+Figure 16: Cobalt on h-BN/Ni(111). a) Scanning tunneling microscopy image, showing triangular two dimensional (2D) intercalated Co islands, and circular three dimensional (3D) clusters and a defect line connecting to a intercalated island (27x27 nm). b) Schematic side view of the situation, showing the topology of the 3D and the 2D agglomerates. Data from [66].
+
+The sticking coefficient, i.e. the probability that an impinging atom sticks on the surface, is determined by X-ray photoelectron spectroscopy (XPS). While the sample was exposed to a constant flux of about 3.5 monolayers of Co per hour, the measurement of the Co uptake on the sample gives a measure for the sticking probability [66]. Figure 17 a) shows the temperature dependence of the sticking coefficient of cobalt atoms that are evaporated by sublimation onto h-BN/Ni(111). Clearly, the sticking is not unity, as it is commonly assumed for a metal that is evaporated onto a metal. Also, the sticking is strongly temperature dependent. This means that at higher temperature more Co atoms scatter back into the vacuum and that the bond energy of the individual Co atoms must be fairly small. The solid line in Figure 17 a) indicates the result of an extended Kisliuk model [67] for the sticking of Co on h-BN/Ni(111). This model predicts about 30% of the Co atoms not to thermalize on the surface and to directly scatter back, and says that the activation energy for diffusion is about 190 meV smaller than the desorption energy [66].
+
+The intercalation is also thermally activated. In Figure 17 b) the intercalated amount of Co as compared to the total amount of Co on the surface is shown for low Co coverages. The fact that at low substrate temperatures almost no Co slips below the \(h\)-BN, indicates that the intercalation must be thermally activated. The number \(N^{2D}\) of intercalated atoms, may be modeled with a simple ansatz like:
+
+\[N^{2D}=k_{2}\exp{-\frac{E_{A}^{2D}}{k_{B}T}}\] (5)
+
+where \(k_{2}\) is the reaction constant that comprises the sticking and the diffusion of Co to the intercalation site, \(E_{A}^{2D}\) the thermal activation energy for intercalation at this site and \(k_{B}T\) the thermal energy. For the 3D clusters the number \(N^{3D}\) of atoms in the clusters is assumed to be constant:
+
+\[N^{3D}=k_{3}\] (6)
+
+where \(k_{3}\) is the reaction constant that comprises the sticking and the diffusion of Co to the 3D cluster nucleation sites. The backreaction, i.e. the dissolution of atoms from 2D islands or 3D clusters is neglected, since there the Co bond energies are much larger. The solid line in Figure 17 b) shows a fit with \(E_{A}^{2D}=0.24\pm 0.1\rm{eV}\) and a ratio \(\frac{k_{2}}{k_{3}}=\frac{1}{250}\). In \(\frac{k_{2}}{k_{3}}\) the temperature dependence of the sticking and the diffusion cancel. A temperature dependence of the 3D nucleation site density is not included in this kinetic modeling, because the data do not allow the extraction of more than two independent parameters. The small value of \(\frac{k_{3}}{k_{2}}\) is an indication that a Co atom finds an intercalation site more easily than a 3D cluster. This is consistent with the assignment that the defect lines (see Figure 16 a)) act as collectors for the intercalation, since the probability for a diffusing particle to hit a line is much larger than hitting a point like 3D cluster.
+
+Figure 17: Temperature dependence of a) sticking coefficient of Co from a sublimation source at about 1400 K and b) intercalation of Co below h-BN on Ni(111). Data from [66].
+
+### Functionality
+
+The functionality of \(sp^{2}\) single layers bases on a set of attractive properties like a, compared to clean transition metal surfaces, low reactivity and high thermal stability. They also feature a change in electron transport perpendicular and parallel to the surface. In a metal \(sp^{2}\)-layer metal hetero-junction the layer changes the phase matching of the electrons in the two electrodes. This is particularly interesting in view of spintronic applications where in a magnetic hetero-junction the resistance for the two spin components is not the same, and magnetoresistance or spin filtering is expected [68]. If the layers are insulating or dielectric, as it is the case for h-BN they act as atomically sharp tunneling junctions. The corrugated \(sp^{2}\) layers bear, on top of the potential of the flat layers, the possibility of using them as templates for molecular architectures.
+
+#### 2.6.1 Flat layers: Tunneling junctions
+
+Figure 18: \(C_{60}\) on h-BN/Ni(111). a) Scanning tunneling microscopy shows that a monolayer of \(C_{60}\) wets the substrate and forms a regular 4x4 structure. b) The valenceband photoemission spectrum is dominated by molecular orbitals of \(C_{60}\). The intensity and energy position of the orbitals is strongly temperature dependent. Note the shift of the photoemission leading edge of 80 meV, which corresponds to the transfer of about half an electron onto the molecules in going from 150 K to room temperature. From [69].
+
+An insulating single layer on a metal acts as a tunneling junction between the substrate and the adsorbate. It decouples adsorbates from the metal. This decoupling also influences the time scale on which the electronic system equilibrates. The electronic equilibration time scale increases and eventually becomes comparable to that of molecular motion. This opens the doors for physics beyond the Born-Oppenheimer approximation, where it is assumed that the electronic system is always in equilibrium with the given molecular coordinates. Such an example is the C₆₀/h-BN/Ni(111) junction [69]. Figure 18 a) shows 3 substrate terraces separated by the Ni(111) terrace height of 0.2 nm. The whole surface is wetted by a hexagonally closed packed C₆₀ layer with a 4x4 super structure, with one C₆₀ on 4 h-BN/Ni(111) unit cells. The electronic structure of the C₆₀ layer shows a distinct temperature dependence that is in line with the onset of molecular libration. The freezing of the molecular rotation at low temperature is well known from C₆₀ in the bulk of fullerene [70] and C₆₀ at the surface of fullerene [71]. The C₆₀ molecular orbitals as measured with photoemission shift by about 200 meV in going from 100 K to room temperature. The shift is parallel to the work function, which indicates a vacuum level alignment of the molecular orbitals of C₆₀. This shift signals a significant charge transfer from the substrate to the adsorbate. As can be seen in the inset of Figure 18 b) it causes an upshift of the leading edge in photoemission. This upshift translates in an average charge transfer of about 0.5 \(e^{-}\) onto the lowest unoccupied molecular orbital (LUMO) of the C₆₀ cage. The LUMO occupancy therefore changes by about a factor of 7 in going from 100 K to room temperature, which is a molecular switch function. The normal emission photoemission intensity of the highest occupied molecular orbital (HOMO) as a function of the substrate temperature shows the phase transition between 100 K and room temperature. The fact that the phase transition is visible in the normal emission intensity gives a direct hint that the molecular motion that leads to the phase transition must be of rocking type (cartwheel) since azimuthal rotation of the C₆₀ molecules would not alter this intensity. The experiments also show that C₆₀ is indeed weakly bound to the substrate, since it desorbs at temperatures of \(\sim\)500 K, which is only 9% higher than the desorption temperature from bulk C₆₀. If we translate the work function shift in the phase transition into a pyroelectric coefficient \(p_{i}=\partial{P_{S}}/\partial{T}\), where \(P_{S}\) is the polarization we get an extraordinary high value of -2000 \(\mu\)C/m²K, which is larger than that of the best bulk materials.
+
+The correlation of the molecular orientation with the charge on the LUMO can be rationalized with the shape of the LUMO of the C₆₀. The LUMO wave function is localized on the pentagons of the C₆₀ cage and at low temperature C₆₀ does not expose the pentagons towards the h-BN/Ni(111). Therefore the charge transfer is triggered by the onset of rocking motion where the LUMO orbitals get a larger overlap with the Fermi sea of the nickel metal. In turn this will increase electron tunneling to the LUMO. In an adiabatic picture, however, the back tunneling probability is as large as the forth tunneling probability, and the magnitude of the charge transfer effect could not be explained. It was therefore argued that the magnitude of the effect is a hint for non adiabatic processes, i.e. that the back tunneling rate gets lower due to electron self trapping. This self trapping can only be efficient, if the electron tunneling rate is low, i.e. if the electron resides for times that the molecule needs to change its coordinates.
+
+#### 2.6.2 Corrugated layers: Templates
+
+Templates are structures that are able to host objects in a regular way. In nanoscience they play a key role and act as a scaffold or construction lot for supramolecular self assembly that allow massive parallel production processes on the nanometer scale. Therefore the understanding of the template function is of paramount importance. It requires the exploration of the atomic structure that defines the template unit cell geometry, and the electronic structure that imposes the bond energy landscape for the host atoms or molecules. On surfaces the bond energy landscape has a deep valley perpendicular to the surface that is periodically corrugated parallel to the surface. The perpendicular valley is responsible for the adsorption and the parallel corrugation governs surface diffusion. If the surface shall act as a template, and e.g. impose a lateral ordering on a length scale larger than the (1x1) unit cell, it has to rely on reconstruction and the formation of super structures. For the case of the h-BN/Rh(111) the superstructure has a size of 12x12 Rh(111) unit cells on top of which 13x13 h-BN unit cells coincide (see Section 2.2.2). It is a corrugated \(sp^{2}\) layer which displays a particular template function for molecular objects with a size that corresponds with the nanostructure. The super cell divides into different regions, the wires, where the layer is weakly bound to the substrate and the ”holes” or ”pores” where h-BN is tightly bound to the substrate. The holes have a diameter of 2 nm and are separated by the lattice constant of 3.2 nm (see Section 2.2.2).
+
+Figure 19 documents the template function of h-BN/Rh(111) for three different molecules at room temperature. For C₆₀ with a van der Waals diameter of about 1 nm it is found that h-BN is wetted, and 12 C₆₀ molecules sit in one Rh (12x12) unit cell. It can also be seen that the ordering is not absolutely perfect. There are super cells with one C₆₀ missing and such where one extra C₆₀ is ”coralled” in the holes of the h-BN nanomesh. If we take a molecule that has the size of the holes of 2 nm, here naphthalocyanine (Nc) (C₄₈H₂₆N₈), we observe that the molecules self assemble in the holes. Apparently, the trapping potential is larger than the molecule-molecule interaction that would lead to the formation of Nc islands with touching molecules. If we take a molecule with an intermediate size, i.e. copper phthalocyanine (Cu-Pc) (C₃₂H₁₆CuN₈) with a van der Waals diameter of 1.5 nm, it is seen that they also assemble in the nanomesh holes. There is a significant additional feature compared to the Nc case, i.e. Cu-Pc does not sit in the center of the holes but likes to bind at the rims. This observation gives an important hint to the bond energy landscape within this superstructure.
+
+Figure 19: Room temperature scanning tunneling microscopy (STM) images of molecules trapped in h-BN/Rh(111) nanomesh. a) C₆₀: Individual molecules are imaged throughout this region, following closely the topography (15x15 nm). The positions in the hole centers are occupied by either zero or one C₆₀ molecule; at two places, large protrusions may represent additional corralled molecules (From [45]). b) Naphthalocyanine (Nc) (C₄₈H₂₆N₈): Site-selective adsorption (120x120 nm). The inset (19x19 nm) on the top right shows an enlargement. The inset on the right is a schematic representation of the molecule in the h-BN nanomesh. (From [48]). c) Copper phthalocyanine (Cu-Pc) (C₃₂H₁₆CuN₈) molecules trapped at the rim of the holes (18 x 18 nm). At the given tunneling conditions, the mesh wires map dark, the holes map grey and Cu-Pc are imaged as bright objects. The inset shows a magnified model of the trapped Cu-Pc molecule. (From [17]).
+
+Figure 20: Temperature dependence of normal emission valence band photoemission spectra from Xe/h-BN/Rh(111) [17] and Xe/g/Ru(0001) [50]. a) Energy distribution curves extracted for three different temperatures (66, 75, and 82 K) for Xe/h-BN/Rh(111). b) Spectral weight of the Xe on the wires as a function of temperature. c) Spectral weight of the Xe in the holes as a function of temperature. d) Energy distribution curves extracted for three different temperatures (67, 87, and 96 K) for Xe/g/Ru(0001). e) Spectral weight of the Xe on the mounds as a function of temperature. f) Spectral weight of the Xe in the valleys as a function of temperature. The solid lines in b), c), e) and f) are fits obtained from a zero-order desorption model.
+
+In Section 2.3.2 it was shown that corrugated \(sp^{2}\) layer systems have relatively large lateral electric fields due to dipole rings. These fields may polarize molecules and provide an additional bond energy. Figure 11 shows that this bonding scales with \(\alpha\cdot E^{2}_{\parallel}\), where \(E_{\parallel}\) is the lateral electric field, and \(\alpha\) the polarizability of the molecule. \(E_{\parallel}^{2}\) is largest at the rims of the h-BN nanomesh holes or the g/Ru(0001) mounds. This feature in the bond energy landscape is in line with the observation that molecules like Cu-Pc like to sit at the corrugation rims of the \(sp^{2}\) layers. A quantitative measure for the adsorption energy can be obtained from thermal desorption spectroscopy (TDS) which was performed for xenon.
+
+\begin{table}
+\begin{tabular}{|l|r|r|r|r|r|r|}
+\hline
+ & \multicolumn{3}{c|}{\(h\)-BN/Rh(111)} & \multicolumn{3}{c|}{g/Ru(0001)} \\
+\hline
+Phase & \(C^{W}\) & \(C^{H}\) & \(R^{H}\) & \(C^{M}\) & \(C^{V}\) & \(R^{V}\) \\
+\hline
+\(E_{d}\)(meV) & \(181\) & \(184\) & \(208\) & \(222\) & \(222\) & \(234\) \\
+\(N_{1}\) & 25 & 17 & 12 & 13 & 14 & 23 \\ \hline
+\end{tabular}
+\end{table}
+Table 2: Experimentally determined Xe desorption energies \(E_{d}\) and Xe atoms per unit cell at full coverages \(N_{1}\) for \(h\)-BN/Rh(111) [17] and g/Ru(0001) [50]. For all fits an attempt frequency \(\nu\) of \({1.2\times 10^{12}}\) Hz has been used.
+
+Figure 20 shows a comparison of Xe/h-BN/Rh(111) and Xe/g/Ru(0001) TDS, where the surfaces are heated with a constant heating rate \(\beta=dT/dt\), and where the remaining Xe on the surface was monitored with photoemission from adsorbed xenon (PAX). Since the Xe core level energy is sensitive to the local electrostatic potential it also indicates where it is sitting in the \(sp^{2}\) super cell. This allows a TDS experiment where the Xe bond energy on different sites in the super cell is inferred [17, 50].
+
+The desorption data for Xe/h-BN/Rh(111) on the wires and in the holes are shown in Figure 20 b) and c), respectively, and for Xe/g/Ru(0001) on the mounds and the valley in Figure 20 e) and f). The data of the weakly bound \(sp^{2}\) layer regions (wires and mounds) are well described with a zero order desorption model where \(dN\) molecules desorb on the temperature increase \(dT\):
+
+\[-dN=\frac{\nu}{\beta}\cdot\exp(-\frac{E_{d}}{k_{B}T})\cdot dT\] (7)
+
+where \(\nu\) is the attempt frequency in the order of 10¹² Hz, \(\beta\) the heating rate, \(E_{d}\) the desorption energy and \(k_{B}T\) the thermal energy. The data are fitted to the integrals of Equation 7 where the initial coverage \(N_{1}\) is taken from the intensity of the photoemission peaks, the sizes of the Xe atoms and the super cells. From this it is e.g. found that 25 Xe atoms cover the wires in the h-BN/Rh(111) unit cell. The desorption energies of 169, 181, 222 and 249 meV for Xe/Xe [72], Xe_W_/h-BN/Rh(111), Xe_M_/g/Ru(0001) and for Xe/graphite [73] indicate that they are similar, but have a trend to increase in going to more metallic substrates. The initial coverages and desorption energies are summarized in Table 2. For the strongly bound regions (hole and valley) this single desorption energy picture does not hold. For Xe/h-BN/Rh(111) it turned out that 12 Xe atoms in the holes have to be described with a larger bond energy. Correspondingly different phases \(C\) and \(R\) have been introduced, where \(C\) stands for coexistence and \(R\) for ring or rim [17]. Twelve Xe atoms fit on the rim of the h-BN/Rh(111) nanomesh holes and thus a dipole ring induced extra bond energy of 13% was inferred. For g/Ru(0001) the situation is less clear cut, since a two phase fit fits the data less well than those of the h-BN/Rh(111) case. This may be due to the different topography of the two systems (see Figure 8). The fact that the extra bond energy in Xe/g/Ru(0001) is 4% only is rationalized with the lower local work function difference, i.e. the smaller Xe core level energy splitting in Xe/g/Ru(0001) (240 meV) than in Xe/h-BN/Rh(111) (310 meV).
+
+## Appendix A Atomic and Electronic Structure in Real and Reciprocal Space
+
+Graphene and hexagonal boron nitride layers are isoelectronic. Both, the (C,C) and the (B,N) building blocks have 12 electrons and both have a honeycomb network structure with fairly strong bonds of 6.3 and 4.0 eV for C=C and B=N, respectively. The electronic configurations of the constituent atoms are shown in Figure 21.
+
+Figure 21: Boron, carbon and nitrogen are neighbors in the periodic table. Accordingly, h-BN and graphene are isoelectronic and are constituted by 12 electrons per unit cell.
+
+### \(sp^{2}\) hybridization
+
+The strong bonding within the \(sp^{2}\) network foots on the hybridization of the 2s and the 2p valence orbitals. The 2s and the 2p energies in atomic boron, carbon and nitrogen are close in energy (in the order of 10 eV) and have similar spacial extension. If the atoms are assembled into molecules, where the overlap between adjacent bonding orbitals is maximised, the linear combination of the s and p orbitals provides higher overlap than that of two 2p orbitals. Figure 22 shows a \(s\) and a \(p_{x}\) wavefunction and their \(sp_{x}\) hybrid, which is a coherent sum of the wavefunction amplitudes. The phase in the \(p\) wavefunction depends on the direction and produces the lobe along the \(x\) direction, which allows a large overlap with the wave functions of the neighboring atoms.
+
+Figure 22: The coherent sum of a \(|s\rangle\) and a \(|p_{x}\rangle\) wavefunction leads to a directional \(|sp^{2}\rangle\) hybrid. The phases of the wavefunctions are marked with + and -, where negative amplitudes are shaded.
+
+The hybrid orbitals are obtained as a linear combination of atomic orbitals. For the case of carbon \(sp^{3}\), \(sp^{2}\) and \(sp^{1}\) hybrids may be formed, where CH₄ (methene), C₂H₄ (ethene) and C₂H₂ (acetylene) are the simplest hydrocarbons representing these hybrids. The tetrahedral symmetry of \(sp^{3}\) and the planar symmetry of \(sp^{2}\) is also reflected in the two allotropes diamond and graphite, where graphite is the thermodynamically stable phase. For boron nitride, accordingly cubic and hexagonal boron nitrides are found, where here the cubic form is thermodynamically more stable at room temperature and 1 bar pressure.
+
+The \(sp\) hybrid orbitals are obtained in combining \(s\) and \(p\) orbitals. This corresponds to a new base for the assembly of atoms into molecules. In the \(sp^{2}\) hybrid orbitals the index 2 indicates that two \(p\) orbitals are mixed with the \(s\) orbital. The \(2s\) orbital is combined with \(2p_{x}\) and \(2p_{y}\) orbitals, into 3 two fold spin degenerate orbitals that form the \(\sigma\) bonds:
+
+\[|sp^{2}_{1}\rangle={\sqrt{\frac{1}{3}}}|s\rangle+{\sqrt{\frac{2}{3}}}|p_{x}\rangle\] (8)
+
+\[|{sp^{2}_{2}\rangle={\sqrt{\frac{1}{3}}}|s\rangle-{\sqrt{\frac{1}{6}}}|p_{x}\rangle+{\sqrt{\frac{1}{2}}}|p_{y}\rangle}\] (9)
+
+\[|sp^{2}_{3}\rangle={\sqrt{\frac{1}{3}}}|s\rangle-{\sqrt{\frac{1}{6}}}|p_{x}\rangle-{\sqrt{\frac{1}{2}}}|p_{y}\rangle\] (10)
+
+These 3 \(\sigma\) orbitals contain a mixture of \(\frac{1}{3}\) of an \(s\) and \(\frac{2}{3}\) of a \(p\) electron. It can easily be seen that the three orbitals lie in a plane and point in directions separated by angles of 120∘. \(|sp^{2}_{1}\rangle\) points into the direction [\({\sqrt{\frac{2}{3}}},0,0\)], \(|sp^{2}_{2}\rangle\) into the direction [-\(\sqrt{\frac{1}{6}},{\sqrt{\frac{1}{2}}},0\)], and from the scalar product between the two directions, we get the angle of 120∘. In the case of the \(sp^{2}\) hybridization the fourth orbital has pure \(p\) character and forms \(\pi\) bonds.
+
+\[|sp^{2}_{4}\rangle=|p_{z}\rangle\] (11)
+
+\(|sp^{2}_{4}\rangle\) is perpendicular to the \(sp^{2}\)\(\sigma\) bonding plane. For the case of graphene these \(p_{z}\) orbitals on the honeycomb \(sp^{2}\) network are responsible for the spectacular electronic properties of the conduction electrons in the \(\pi\) bands because they are occupied with one electron.
+
+### Electronic band structure
+
+The \(sp^{2}\) hybridization determines the **atomic structure** of both, hexagonal boron nitride and graphene sheets. They form a two dimensional honeycomb structure as shown in Figure 23. The lattice can be described as superposition of two coupled sublattices \(A\) and \(B\) (see Figure 23 a)). In the case of graphene both sublattices are occupied by one carbon atom (\(C_{A},C_{B}\)), while in the case of h-BN one sublattice is occupied by boron atoms and the other sublattice by nitrogen atoms (B,N). The interference of the electrons between these lattices causes the peculiar electronic structure of \(sp^{2}\) layer networks. The 12 electrons in the unit cell are filled into 4 \(1s\) core levels, and into the 16 \(sp^{2}\) hybrids that form 3 \(\sigma\) bonding, one \(\pi\) bonding, one \(\pi^{*}\) antibonding and 3\(\sigma^{*}\) anti bonding bands. In the bonding bands the two adjacent \(sp^{2}\) hybrids are in phase, while in the antibonding case they are not. The atomic orbitals \(sp^{2}_{1}\), \(sp^{2}_{2}\) and \(sp^{2}_{3}\) constitute the in plane \(\sigma\) bands, while the \(sp^{2}_{4}=p_{z}\) orbitals form the \(\pi\) bands. From this it can be seen that for flat layers the \(\sigma\) and the \(\pi\) electrons do not interfere, i.e. may be treated independently. In Figure 23 the real space lattice and the corresponding Brillouin zone in reciprocal space are shown.
+
+Figure 23: Honeycomb structure like that of a single layer h-BN or graphene. a) real space: the \(sp^{2}\) hybridization causes the formation of two coupled sublattices \(A\) and \(B\) with lattice vectors \({\bf a}_{1}\) and \({\bf a}_{2}\) and lattice constant \(a=|{\bf{a}}_{1}|=|{\bf{a}}_{2}|\). b) reciprocal space: Brillouin zone. The high symmetry points \(\Gamma,M\) and \(K\) are marked, where the reciprocal distance \(\Gamma K\) is \(\frac{2\pi}{\sqrt{3}a}\). The distinction of \(K\) and \(K^{\prime}\) is possible if the system is three fold symmetric (trigonal), but not six-fold symmetric (hexagonal).
+
+In \(k\)-space the reciprocal lattice constant is \(\frac{2\pi}{a}\), where \(a\) is the lattice constant of graphene or h-BN, of about 0.25 nm. For the case of graphene, where the two sublattices are indistinguishable, this leads to a gap-less semiconductor with Dirac points at the \(K\) points. In h-BN, the distinguishability of boron and nitrogen leads to an insulator, where the \(\pi\) valence band is mainly constituted by the nitrogen sublattice, and the conduction band by the boron sublattice.
+
+The different electronegativities of boron and nitrogen lead for free-standing h-BN to 0.56 \(e^{-}\) transferred from B to N [36]. This ionicity produces a Madelung energy \(E_{Mad}\):
+
+\[E_{Mad}=\alpha_{Mad}\cdot\frac{1}{4\pi\epsilon_{0}}\cdot\frac{q^{2}}{a}\] (12)
+
+with \(\alpha_{Mad}\)=1.336 for the honeycomb lattice [74], \(q\) the displaced charge (in the above case 0.56 \(e^{-}\)) and \(a\) the lattice constant. It has to be noted that this Madelung energy applies for the free-standing case. If the ionic honeycomb layer sits on top of a metal, the energy in Equation 12 reduces by a factor of 1/2.
+
+#### A.2.1 \(\pi\) bands
+
+It is instructive to recall the basic statements within the framework of the ”tight binding” scheme of Wallace that he developed for the band theory of graphite [18]. Also the description of tight binding calculations of molecules and solids of Saito and Dresselhaus is recommended [75] and the most recent review of Castro Neto et al. [76]. Tight binding means that we approximate the wave functions as superpositions of atomic \(p_{z}\) wave functions on sublattice \(A\) and \(B\), respectively. Furthermore, they are Bloch functions with the periodicity of the lattice. The essential physics lies in the interference between the two sublattices. It is just another beautiful example for quantum physics with two interfering systems. In order to solve the Schrödinger equation with this Ansatz we have to solve the secular equation:
+
+\[\left|\begin{array}[]{cc}H_{AA}-E&H_{AB}\\ H_{BA}&H_{BB}-E\\ \end{array}\right|=0\] (13)
+
+where the matrix elements \(H_{AA}\) and \(H_{BB}\) describe the energy on the sublattices \(A\) and \(B\) and most importantly \(H_{AB}\) the hybridization energy due to interference or hopping of the \(\pi\) electrons between the two lattices.
+
+The solutions for the energies \(E_{-}\) (bonding) and \(E_{+}\) (antibonding) are:
+
+\[E_{\pm}=\frac{1}{2}\left(H_{AA}+H_{BB}\pm{\sqrt{(H_{AA}-H_{BB})^{2}+4|H_{AB}|^{2}}}\right)\] (14)
+
+Here we show the simplest result that explains the essential physics. It is the degeneracy of the bonding \(\pi\) and the antibonding \(\pi^{*}\) band at the \(K\) point of the Brillouin zone, if the two sublattices are indistinguishable. This means that at \(K\) the square root term in Equation 14 has to vanish.
+
+The tight binding ansatz delivers values for \(H_{AA}\), \(H_{BB}\) and \(H_{AB}\):
+
+\[H_{AA}=E_{A}\] (15)
+
+\[H_{BB}=E_{B}\] (16)
+
+where \(E_{A}\) and \(E_{B}\) are the unperturbed energies of the atoms on sublattices \(A\) an \(B\). The square of the interference term \(H_{AB}\) is \(k\)-dependent and gets:
+
+\[\left|H_{AB}\right|^{2}=\gamma_{AB}^{2}\left(1+4\cos(k_{x}a)\cos(k_{y}a/\sqrt{3})+4\cos^{2}(k_{y}a/\sqrt{3})\right)\] (17)
+
+\(k_{x}\) and \(k_{y}\) are coordinates in \(k\)-space pointing along \(x\) and \(y\), respectively, and \(a\) is the lattice constant (see Figure 23). \(\gamma_{AB}\) describes the hybridization between the sublattices and is proportional to the electron hopping rate between two adjacent sites on sublattice \(A\) and sublattice \(B\). Basically it determines the \(\pi\) band width, which turns out to be \(3\gamma_{AB}\). If hopping within the sublattices, e.g. between two adjacent A-sites, is allowed this leads to \(k\)-dependent corrections in \(H_{AA}\) and \(H_{BB}\) and to a symmetry breaking between the \(\pi\) and the \(\pi^{*}\) band [18]. If the phase of the electron wave functions is considered, the hexagonal symmetry is broken and a trigonal symmetry, i.e. two distinct \(K\) points, \(K\) and \(K^{\prime}\) have to be considerd [77, 76]. For equivalent sublattices, i.e. \(H_{AA}\) = \(H_{BB}\) the \(\pi\) bandstructure is given by the hopping between the two sublattices i.e. by \(H_{AB}\) and \(H_{BA}\), respectively. From Equation 17 it is seen that \(H_{AB}\) vanishes at the \(K\) point, i.e. for \({\bf{k}}=(k_{x},k_{y})=(0,\frac{2\pi}{{\sqrt{3}}a})\). The fact that the electrons in the two sublattices do not interfere if they are at the \(K\) point of the Brillouin zone, is a direct consequence of the symmetry of the crystal, and does not change if hopping between non nearest neighbors is included in the model. It should also be mentioned that these results are expected for the electronic structure of any system that forms a honeycomb lattice.
+
+In Figure 24 the generic tight binding band structure of a \(sp^{2}\) hybridized lattice with two sublattices is shown for hopping \(\gamma_{AB}\) between the sublattice A and B, only. Figure 24 a) presents the case of graphene, i.e. equivalent sublattices \(A\) and \(B\). The two \(p_{z}\) electrons from the atoms \(C_{A}\) and \(C_{B}\) fill the \(\pi\) band. The highest energy corresponds to the Fermi energy and lies at the \(K\) point. This means that the Fermi surface of this system consists of points at the \(K\) points. The peculiarity that the resulting Fermi surface encloses no volume leads to the notion that graphene is a gapless semiconductor. The band structure in the vicinity of the \(K\) point is most interesting. The dispersion \(E(k)\) of the electrons is linear, i.e. \(\left(\frac{\partial E}{\partial k}\right)_{K}=v_{F}=const.\). The linear dispersion resembles to that of massless photons, or relativistic particles with \(E\gg m_{o}c^{2}\), where \(m_{o}\) is the rest mass. It constitutes so called Dirac cones at the \(K\) point of the Brillouin zone (see Figure 24) a)). From Equation 17 we get with \(\frac{\partial{E}}{\partial{k}}\) at the \(K\) point the Fermi velocity \(v_{F}=\frac{a\cdot\gamma_{AB}}{\hbar}\) , which is for a \(\pi\) bandwidth of 6 eV or hopping rate of \(\frac{1}{10\rm{fs}}\) about \(c/300\). In brackets a seeming contradiction has to be clarified: The second derivative of \(E(k)\) at the \(K\) point is zero. With the relation for the second derivative of \(\frac{\partial^{2}E}{\partial k^{2}}=\frac{\hbar^{2}}{m^{*}}\), the effective mass \(m^{*}\) of the electrons and the holes is infinite. i.e. electrons and holes at the \(K\) point may not be accelerated, as it is the case for photons. However, since the Fermi velocity is not zero, electrons have not to be accelerated in order to be transported.
+
+In Figure 24 b) the result for a lattice with two inequivalent sublattices that corresponds to the case of h-BN are shown. For the sake of simplicity, the same hybridization \(\gamma_{AB}\) has been chosen. The symmetry between the \(A\) and the \(B\) lattice is broken, if the energy of the unperturbed atoms \(E_{A}\) and \(E_{B}\) are not the same (see Equation 14), which is obviously the case for boron and nitrogen. The band structure is similar, but at the \(K\) point a gap with the magnitude of \(|E_{A}-E_{B}|\) opens, and no Dirac physics is expected.
+
+Figure 24: Tight binding \(\pi\) and \(\pi^{*}\) band structures of honeycomb lattices as shown in Figure 23 along \(\Gamma KM\Gamma\). The hopping rate \(\gamma_{AB}/\hbar\) is kept constant for both cases. a) Case for two undistinguishable sublattices \(A\) and \(B\) with \(E_{A}\)=\(E_{B}\). Note the emergence of a gapless semiconductor: If each sublattice contributes one electron, the Fermi surface is constituted by the Dirac point at \(K\). b) Case for two distinguishable sublattices \(A\) and \(B\) with \(E_{A}>E_{B}\). Note: At \(K\) a gap with the magnitude \(E_{g}=E_{A}-E_{B}\) opens, and with an even number of \(\pi\) electrons the system is an insulator.
+
+#### A.2.2 \(\sigma\) bands
+
+The \(\sigma\) bands form the strong bonds between the atoms in the \(x-y\) plane. For flat layers they are orthogonal to the \(\pi\) bands and can be treated independently. The case is more involved than that of the \(\pi\) bands since here the 3 atomic orbitals (\(s\), \(p_{x}\) and \(p_{y}\)) on the two sublattices give rise to 6 bands. Also the overlap between the different atomic orbitals gets larger and the \(s\)-\(p\) mixing is a function of the \(k\) vector. With the tight binding ansatz similar to Equations 14-17 3 \(\sigma\) bands were derived [75]. Essentially the secular equation is now the determinant of a \(6\times 6\) matrix leading to 6 bands. At \(\Gamma\) the lowest lying band, \(\sigma_{0}\), has \(s\) character and the two remaining \(\sigma\) bands, \(\sigma_{1}\) and \(\sigma_{2}\) are degenerate and mainly \(p_{x}\) and \(p_{y}\) derived.
+
+It is interesting to note that also for the \(\sigma\) bands the bandstructure forms cones, if the base of the honeycomb lattice is homonuclear (graphene). They are reminiscent to the Dirac cone, where the \(\pi\) and the \(\pi^{*}\) band touch. At \(K\) and a binding energy of about 13 eV the \(\sigma_{0}\) and the \(\sigma_{1}\) band touch. For heteronuclear bases in the honeycomb (h-BN) also a gap opens, as it is observed for the \(\pi\) and \(\pi^{*}\) band. Of course, the conical band touching is less important for graphene and h-BN, since both involved \(\sigma\) bands remain fully occupied. The measurement of this gap, nevertheless, would open a way to distinguish the \(\sigma\) and the \(\pi\) bonding to the substrate.
+
+Figure 25: Density functional theory (DFT) bandstructure calculation along \(\Gamma KM\Gamma\) for graphene and h-BN single layers. The eigenvalues for discrete \(k\)-points are shown as dots. The solid lines connect the eigenvalues in the Brillouin zone where the band assignment from the tight binding ansatz with a 2s and 2p basis set holds: Three \(\sigma\) bands \(\sigma_{0}\), \(\sigma_{1}\) and \(\sigma_{2}\) and the two \(\pi\) bands \(\pi\) and \(\pi^{*}\) are shown. The three \(\sigma^{*}\) antibonding bands are more difficult to resolve since they are also mixing in the DFT calculation with 3s and 3p contributions. a) Graphene. Note the Dirac cone at \(K\) at the Fermi level \(E_{F}\), where the \(\pi\) and the \(\pi^{*}\) bands touch. For the \(\sigma_{0}\) and the \(\sigma_{1}\) band a similar conical band touching occurs at about -13 eV. b) h-BN. Note the band narrowing and the opening of a gap at the \(K\) point for the \(\pi\)-\(\pi^{*}\) and the \(\sigma_{0}\)-\(\sigma_{1}\) band cones. Calculations by courtesy of Peter Blaha.
+
+Figure 25 shows a state of the art Density Functional Theory (DFT) bandstructure calculation for a single layer graphene and h-BN, respectively. These calculations consider a much larger basis set than the 2s and the 3 2p orbitals as it is done in the tight binding picture. Accordingly more bands are found. The dots in Figure 25 are the energy eigenvalues for given \(k\) vectors. The three \(\sigma\), the \(\pi\) and the \(\pi^{*}\) band reproduce the tight binding result [75]. At higher energies also 3s and 3p orbitals contribute eigenvalues and the identification of the antibonding bands becomes difficult.
+
+**Acknowledgements**
+
+Most material presented in this Chapter was obtained with Jürg Osterwalder and thanks to the empathy and work of our students Wilhelm Auwärter, Matthias Muntwiler, Martina Corso and Thomas Brugger. It is also a big pleasure to acknowledge Peter Blaha, who started in an early stage of our endeavor to contribute significantly with theory to the understanding of \(sp^{2}\) single layers.
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+

arxiv_ground_truth/0905.2570.md

@@ -0,0 +1,346 @@
+# Reverberation Mapping of the Optical Continua of 57 MACHO Quasars
+
+Justin Lovegrove ¹
+²
+
+Footnote 1: affiliation: School of Physics and Astronomy, University of Southampton, Southampton, UK SO171BJ
+
+Footnote 2: affiliation: Harvard-Smithsonian Centre for Astrophysics, Cambridge, Massachusetts, USA 02144
+
+jl805@soton.ac.uk
+
+###### Contents
+
+1. 1 Abstract
+2. 2 Introduction
+3. 3 Theoretical Background
+4. 4 Observational Details
+5. 5 Theoretical Method
+6. 6 Results
+7. 7 Conclusions
+8. 8 Future Work
+9. 9 Acknowledgements
+
+List of Figures
+1. 1 Degeneracy in \(\theta\) and \(\epsilon\)
+2. 2 Manually-produced plots
+3. 3 Comparison of quasar data to white noise
+4. 4 Comparison of quasar and star data
+5. 5 Fit of data to model
+6. 6 Interpretations of \(\epsilon\)
+7. 7 Relative luminosities and radii
+
+## 1 Abstract
+
+Autocorrelation analyses of the optical continua of 57 of the 59 MACHO quasars reveal structure at proper time lags of \(544\pm 5.2\) days with a standard deviation of 77 light days. Interpreted in the context of reverberation from elliptical outflow winds as proposed by Elvis (2000) [1], this implies an approximate characteristic size scale for winds in the MACHO quasars of \(544\pm 5.2\) light days. The internal structure variable of these reflecting outflow surfaces is found to be \(11.87^{o}\pm 0.40^{o}\) with a standard deviation of \(2.03^{o}\).
+
+## 2 Introduction
+
+Brightness fluctuations of the UV-optical continuum emission of quasars were recognised shortly after the initial discovery of the objects in the 1960s [2]. Although several programmes were undertaken to monitor these fluctuations, little is yet known about their nature or origin. A large number of these have focused on comparison of the optical variability with that in other wavebands and less on long-timescale, high temporal resolution optical monitoring. Many studies have searched for oscillations on the \(\sim\) day timescale in an attempt to constrain the inner structure size (eg [3]). This report, however, is concerned with variability on the year timescale to evidence global quasar structure.
+
+In a model proposed by Elvis [1] to unite the various spectroscopic features associated with different “types” of quasars and AGN (eg broad absorption lines, X-ray/UV warm absorbers, broad blue-shifted emission lines), the object’s outer accretion disc has a pair of bi-conical extended narrow emission line regions in the form of outflowing ionised winds. Absorption and emission lines and the so-called warm absorbers result from orientation effects in observing these outflowing winds. Supporting evidence for this is provided by a correlation between polarisation and broad absorption lines found by [4]. Outflowing accretion disc winds are widely considered to be a strong candidate for the cause of feedback (for a discussion of the currently understood properties of feedback see [5]). Several models have been developed [6, 7] to simulate these winds. [6] discusses different launch mechanisms for the winds - specifically the balance between magnetic forces and radiation pressure - but finds no preference for one or the other, while [7] discusses the effect of rotation and finds that a rotating wind has a larger thermal energy flux and lower mass flux, making a strong case for these winds as the source of feedback. The outflow described by [1] is now usually identified with the observationally-invoked “dusty torus” around AGN [8].
+
+[9] demonstrated for the MACHO quasars that there is no detectable lag time between the V and R variability in quasars, which can be interpreted as demonstrating that all of the optical continuum variability originates in the same region of the quasar.
+
+[10] observed the gravitationally lensed quasar Q0957+561 to measure the time delay between the two images and measure microlensing effects. In doing so, they found a series of autocorrelation subpeaks initially attributed to either microlensing or accretion disc structure. These results were then re-interpreted by [11] as Elvis’ outflowing winds at a distance of \(2\times 10^{17}cm\) from the quasar’s central compact object. A model applied by [12] to the quasar Q2237+0305, to simulate microlensing, found that the optimal solution for the system was one with a central bright source and an extended structure with double the total luminosity of the central source, though the outer structure has a lower surface brightness as the luminosity is emanating from a larger source, later determined by [13] to lie at \(8.4\times 10^{17}cm\).
+
+[13] continued on to argue that since magnetic fields can cause both jets and outflows, they therefore must be the dominant effect in AGN. [14] however pointed out that the magnetic field required to power the observed Elvis outflows is too great to be due to the accretion disc alone. They therefore argue that all quasars and AGN have an intrinsically magnetic central compact object, which they refer to as a MECO, as proposed by [15], based upon solutions of the Einstein-Maxwell equations by [16]. One compelling aspect to this argument is that it predicts a power-law relationship between Elvis outflow radius and luminosity, which was found in work by Kaspi et al [17] and updated by Bentz et al [18], if one assumes the source of quasar broad emission lines to be outflow winds powered by magnetic fields. The [17] and [18] results were in fact empirically derived for AGN of \(Z<0.3\) and [17] postulates that there may be some evolution of this relation with luminosity (and indeed one might expect some time-evolution of quasar properties which may further modify this scaling relation) so generalising these results to quasars may yet prove a fallacy. The radius of the broad line region was found to scale initially by [17] as \(R_{blr}\propto L^{0.67}\), while [18] found \(R_{blr}\propto L^{0.52}\).
+
+Another strength of the MECO argument is that while [19] found quasar properties to be uncorrelated using the current standard black hole models, [14] and [20] found a homogeneous population of quasars using the [16] model. [21] used microlensing observations of 10 quadruply-lensed quasars, 9 of which were of known redshift including Q2237+0305, to demonstrate that standard thin accretion disc models, such as the widely-accepted Shakura-Sunyaev (S-S) disc [22], underestimate the optical continuum emission region thickness by a factor of between 3 and 30, finding an average calculated thickness of \(3.6\times 10^{15}cm\), while observed values average \(5.3\times 10^{16}cm\). [23] found a radius of the broad line region for the Seyfert galaxy NGC 5548 of just under 13 light days when the average of several spectral line reverberations were taken, corresponding to \(R_{blr}=3.3\times 10^{16}cm\). When the scaling of [17] and [18] is taken into account, the [23] result is comparable to the [13] and [11] results (assuming \(\frac{L_{quasar}}{L_{seyfert}}\sim 10^{4}\), then [18] would predict a quasar \(R_{blr}\) of approximately \(3\times 10^{18}\)). Also, given that black hole radius scales linearly with mass, as does the predicted radius of the inner edge of the accretion disc, a linear mass-\(R_{blr}\) relationship might also be expected. Given calculated Seyfert galaxy black hole masses of order \(10^{8}M_{o}\) and average quasar masses of order \(10^{9}M_{o}\), this would scale the Seyfert galaxy \(R_{blr}\) up to \(3.3\times 10^{17}cm\). While these relations are not self-consistent, either of them may be found consistent with the existing quasar structure sizes. [24] also found structure on size scales of \(10^{16}cm\) from microlensing of SDSS J1004+4112 which would then scale to \(10^{18}cm\). These studies combined strongly evidence the presence of the Elvis outflow at a radial distance of approximately \(10^{18}cm\) from the central source in quasars which may be detected by their reverberation of the optical continuum of the central quasar source.
+
+The [12] result is also in direct conflict with the S-S accretion disc model, which has been applied in several unsuccessful attempts to describe microlensing observations of Q2237. First a simulation by [25] used the S-S disc to model the microlensing observations but predicted a large microlensing event that was later observed not to have occurred. [26] then attempted to apply the S-S disc in a new simulation but another failed prediction of large-amplitude microlensing resulted. Another attempt to simulate the Q2237 light curve by [27] produced the same large-amplitude microlensing events. These events are an inherent property of the S-S disc model where all of the luminosity emanates from the accretion disc, hence causing it all to be lensed simultaneously. Only by separating the luminosity into multiple regions, eg two regions, one inner and one outer, as in [12], can these erroneous large-amplitude microlensing events be avoided.
+
+Previous attempts have been made to identify structure on the year timescale, including structure function analysis by Trevese et al [28] and by Hawkins [29, 30]. [28] found strong anticorrelation on the \(\sim 5\) year timescale but no finer structure - this is unsurprising as their results were an average of the results for multiple quasars taken at low temporal resolution, whereas the size scales of the Elvis outflow winds should be dependant on various quasar properties which differ depending on the launch mechanism and also should be noticed on smaller timescales than their observations were sensitive to. [29] also found variations on the \(\sim 5\)-year timescale again with poor temporal resolution but then put forth the argument that the variation was found to be redshift-independant and therefore was most likely caused by gravitational microlensing. However [31] demonstrated that microlensing occurs on much shorter timescales and at much lower luminosity amplitudes than these long-term variations. [30] used structure function analysis to infer size scales for quasar accretion discs but again encountered the problem of too infrequent observations. In this paper and [32] it was argued that Fourier power specra are of more use in the study of quasars, which were then used in [32] to interpret quasar variability. However, since reverberation is not expected to be periodic, Fourier techniques are not suited to its detection. Hawkins’ observations of long-timescale variability were recognised in [33] as a separate phenomenon to the reverberation expected from the Elvis model; this long-term variability remains as-yet unexplained. Hawkins’ work proposed this variation to be indicative of a timescale for accretion disc phenomena, while others explained it simply as red noise.
+
+[34] demonstrated that there is a correlation between the optical and x-ray variability in some but not all AGN, arguing that x-ray reprocessing in the accretion disc is a viable source of the observed variability, combined with viscous processes in the disc which would cause an inherent mass-timescale dependance, as in the S-S disc the temperature at a given radius is proportional to mass\({}^{-\frac{1}{4}}\). In this case a red-noise power spectrum would describe all quasar variability. For the purposes of this report however, the S-S disc model is regarded as being disproven by the [12] and [21] results in favour of the Elvis outflow model and so the temperature-mass relation is disregarded. A later investigation [35] demonstrated that the correlation between X-ray and optical variability cannot be explained by simple reprocessing in an optically thick disc model with a corona around the central object. This lends further support to the assumption that the [12] model of quasars is indeed viable.
+
+[36] demonstrated from Fourier power spectra that shorter-duration brightness events in quasars statistically have lower amplitudes but again their temporal resolution was too low to identify reverberation on the expected timescales. Also as previously discussed, non-periodic events are extremely difficult to detect with Fourier techniques. All of these works are biased by the long-term variability recognised by [33] as problematic in QSO variability study. A survey by [37] proposed that quasars could be identified by their variability on this timescale but the discovery by [33] that long-timescale variation is not quite a universal property of quasars somewhat complicates this possibility. The spread of variability amplitudes is also demonstrated in [38] for 44 quasars observed at the Wise Observatory. The [38] observations are also of low temporal resolution and uncorrected for long-term variability thus preventing the detection of reverberation patterns.
+
+This project therefore adopts the [1] and [12] model of quasars consisting of a central compact, luminous source with an accretion disc and outflowing winds of ionised material with double the intrinsic luminosity of the central source. The aim is to search for said winds via reverberation mapping. Past investigations such as [39] have attempted reverberation mapping of quasars but have had large gaps in their data, primarily due to the fact that telescope time is usually allocated only for a few days at a time but also due to seasonal dropouts. Observations on long timescales, lacking seasonal dropouts and with frequent observations are required for this purpose, to which end the MACHO programme data have been selected. Past results would predict an average radius of the wind region of order \(10^{18}cm\).
+
+The assumption will be made that all structure on timescales in the region of \(c\Delta t=10^{18}cm\) is due to reverberation and that the reverberation process is instantaneous. The aim is simply to verify whether this simplified version of the model is consistent with observation, not to compare the model to other models.
+
+## 3 Theoretical Background
+
+Reverberation mapping is a technique whereby structure size scales are inferred in an astrophysical object by measuring delay times from strong brightness peaks to subsequent, lower-amplitude peaks. The subsequent peaks are then assumed to be reflections (or absorptions and re-emissions) of the initial brightness feature by some external structure (eg the dusty torus in AGN, accretion flows in compact objects). Reverberation mapping may be used to study simple continuum reflection or sources of specific absorption/emission features or even sources in entirely different wavebands, by finding the lag time between a brightness peak in the continuum and a subpeak in the emission line or waveband of interest. This technique has already been successfully applied by, among many others, [23] and [11].
+
+When looking for continuum reverberation one usually makes use of the autocorrelation function, which gives the mean amplitude, in a given light curve, at a time \(t+\Delta t\) relative to the amplitude at time \(t\). The amplitude of the autocorrelation is the product of the probability of a later brightening event at dt with the relative amplitude of that event, rendering it impossible to distinguish by autocorrelation alone, eg a 50% probability of a 50% brightening from a 100% probability of a 25% brightening, since both will produce the same mean brightness profile. The mathematical formulation of this function is:
+
+\[AC(\Delta t)=\frac{1}{N_{obs}}\cdot\sum_{t}\frac{I(t+\Delta t)\cdot I(t)}{\sigma^{2}}\] (1)
+
+where I(t) represents the intensity at a time t relative to the mean intensity, such that for a dimming event I(t) is negative. Hence \(AC(\Delta t)\) becomes negative if a brightening event at t is followed by a dimming event at \(t+\Delta t\) or if a dimming event at t is followed by a brightening event at \(t+\Delta t\). \(\sigma\) is the standard deviation of I - in rigorous mathematics the \(\sigma^{2}\) term is in fact the product of the standard deviations for I(t) and I(\(t+\Delta t\)) but since they are idential in this case, \(\sigma^{2}\) may be used. The nature of the autocorrelation calculation also has a tendancy to introduce predicted brightness peaks unrelated to the phenomena of interest. Given autocorrelation peaks at lag times \(t_{1}\) and \(t_{2}\), a third peak will also be created at lag \(t_{2}-t_{1}\), of amplitude \(\frac{A(t_{2})}{A(t_{1})}\), which is then divided by the number of data points in the brightness record.
+
+## 4 Observational Details
+
+The MACHO survey operated from 1992 to 1999, observing the Magellanic Clouds for gravitational lensing events by Massive Compact Halo Objects - compact objects in the galactic halo, one of the primary dark matter candidates. The programme was undertaken from the Mount Stromlo Observatory in Australia, where the Magellanic Clouds are circumpolar, giving brightness records free from seasonal dropouts. Information on the equipment used in the programme can be found at http://wwwmacho.anu.edu.au/ or in [40]. 59 quasars were within the field of view of this programme and as a result highly sampled light curves for all of these objects were obtained. These quasars were observed over the duration of the programme in both V and R filters. The brightness records for the 59 quasars in both V and R filters are freely available at http://www.astro.yale.edu/mgeha/MACHO/ while the entire MACHO survey data are available from http://wwwmacho.anu.edu.au/Data/MachoData.html.
+
+## 5 Theoretical Method
+
+Throughout this study the following assumptions were made:
+
+1. 1.That all autocorrelation structure on the several hundred day timescale was due to reverberation from the Elvis outflows, which are also the source of the broad emission lines in quasars. We denote the distance to this region as \(R_{blr}\) - the “radius of the broad line region”.
+2. 2.That the timescale for absorption and re-emission of photons was negligable compared to the light travel time from the central source to the outflow surface.
+3. 3.That the internal structure variable was less than \(45^{o}\) for all objects as found by [11] and [13]. Note that an internal structure variable of \(\epsilon\) and inclination angle of \(\theta\) cannot be distinguished by autocorrelation alone from an internal structure variable of \(90^{o}-\epsilon\) with an inclination angle of \(90^{o}-\theta\), as demonstrated in Fig. 1. The assumed low \(\epsilon\) is further justified by the fact that the [1] model predicts the winds to be projected at an angle of \(30^{o}\) from the accretion disc plane, contraining \(\epsilon\leq 30^{o}\) as the reverberating surfaces must lie at lower inclinations if they are in fact part of the outflow structure.
+4. 4.That extinction was negligable for all objects. Reliable extinction maps of the Magellanic Clouds are not available and information about the quasars’ host galaxies is also unavailable so extinction calculations are not possible. This assumption should be reasonable as any extinction would have a noticeable impact on the colour of the quasar (given the accepted relation \(3.2\cdot E(B-V)=A_{V}\) where \(E(B-V)\) is the colour excess and \(A_{V}\) is the V absorption).
+
+First a series of predictions were made for the relative luminosities of the 59 quasars in the sample using their redshifts (presented in [41]) and apparent magnitudes, combined with the mean quasar SED presented in [42]. Two estimates were produced for these, one for the V data and one for the R data. This calculation was of interest as it would predict the expected ratios of \(R_{blr}\) between the objects in our sample via the [17] and [18] relations. These relations are so far only applied to nearby AGN and so this investigation was carried out to test their universality. Firstly, the apparent magnitudes were converted to flux units via the equation
+
+\[m=-2.5\cdot\log{f}\] (2)
+
+where m is the apparent magnitude and f the flux. Then the redshifts of the objects were used to calculate their distances using the tool at
+http://astronomy.swin.edu.au/ elenc/Calculators/redshift.php with \(H_{0}=71Mpc/km/s\). Using this distance and the relation
+
+\[f=\frac{L}{4\pi d^{2}}\] (3)
+
+where L is the luminosity, f the flux and d the distance, the luminosity of each source in the observed waveband was calculated. It was recognised, however, that simply taking the ratio between these luminosities was not a fair representation of the ratio of their absolute luminosities as cosmological redshift would cause the observed region of the quasar spectrum to shift with distance and the luminosity varies over the range of this spectrum. Hence the mean quasar SED of [42] was used to convert the observed luminosities of the objects to expected luminosities at a common frequency - in this case 50 GHz. These values were then divided by the minimum calculated luminosity so that their relative values are presented in Table 1. For this purpose it was assumed that the spectral shape of the quasar is independant of bolometric luminosity.
+
+Next a calculation was made to predict the dependance of the reverberation pattern of a quasar on its orientation to the observer’s line of sight. This was produced using the geometric equations initially presented in [11] but reformulated to become more general. [11] discusses ”case 1” and ”case 2” quasars with different orientations - ”case 1” being where the nearest two outflow surfaces lie on the near side of the accretion disc plane and ”case 2” being where the nearest two surfaces are on the near side of the rotation axis. These equations become generalised by recognising that the two cases are in fact degenerate and from reverberation alone one cannot distinguish a \(13^{o}\) internal structure variable in ”case 1” from a \(77^{o}\) internal structure variable in ”case 2”. This is also demonstrated in Fig. 1. For the purposes of Fig. 1, the [11] interpretation of \(\epsilon\) is adopted but later it will be demonstrated that in fact there are several possible meanings of \(\epsilon\), though this degeneracy is the same for all interpretations.
+
+(a)\(\epsilon\) **and \(\theta\) as in Case I of [11]**
+
+(b)\(\theta^{1}=\epsilon\) **and \(\epsilon^{1}=\theta\)**
+
+(c)\(\theta^{1}=90^{o}-\theta\) **and \(\epsilon^{1}=90^{o}-\epsilon\)**
+
+Figure 1: Three different quasar orientations and internal structure variables can give the same series of reverberation lags
+
+The generalisation is then
+
+\[t_{1}=\frac{R_{blr}}{c}\cdot(1-\cos(\theta-\epsilon))\] (4)
+
+\[t_{2}=\frac{R_{blr}}{c}\cdot(1-\cos(\theta+\epsilon))\] (5)
+
+\[t_{3}=\frac{R_{blr}}{c}\cdot(1+\cos(\theta+\epsilon))\] (6)
+
+\[t_{4}=\frac{R_{blr}}{c}\cdot(1+\cos(\theta-\epsilon))\] (7)
+
+This prediction could later be compared to the MACHO observations. A value of \(\epsilon\) was not inserted until the project had produced results, so that the mean calculated value of \(\epsilon\) could be inserted into the simulation for comparison with the results.
+
+In the data analysis it was quickly noticed that one of the 59 MACHO quasars - MACHO 42.860.123 - was very poorly observed by the survey, totalling only 50 observations over the programme’s seven-year lifetime. The V and R data for the remaining objects were processed in IDL to remove any 10-\(\sigma\) data points – 5-\(\sigma\) would seem a more appropriate value at first glance, but inspection of the data showed relevant brightness peaks between 5- and 10-\(\sigma\). At the same time all null entries (nights with no observation) were also removed. The data were then interpolated to give a uniformly-spaced brightness record over time, with the number of bins to interpolate over equal to the number of observations. The case was made for using a fixed number of bins for all objects, eg 1000, but for some data records as few as 250 observations were made and so such an interpolation would serve only to reinforce any remaining spurious points. By the employed method, a small amount of smoothing of the data was introduced. The data then had their timescales corrected for cosmological redshift to ensure that all plots produced and any structure inferred were on proper timescales.
+
+In the beginning of the project, to understand the behaviour of the data and appreciate the complexities and difficulties of the investigation, a list was compiled of the most highly-sampled (\(>600\) observations) quasars, yielding 30 objects. Each of these objects had their light curves and autocorrelations examined and a key feature presented itself immediately; the data showed in some cases a long-timescale (\(\sim 1000\) proper days), large-amplitude (\(\sim 0.8\) magnitude) variation that dominated the initial autocorrelation function. Since this variation was seen only in some quasars (and is on a longer timescale than the predicted reverberation), this signal was removed by applying a 300-day running boxcar smooth algorithm over the data, before subtracting this smoothed data (which would now be low-frequency variation dominated) from the actual brightness record. The timescale for smoothing was determined by examination of the brightness profiles of the objects, which showed deviations from this long-timescale signature on timescales below 300 days. Autocorrelation analysis of the uncorrected data also found the first autocorrelation minimum to lie before 300 days. Further, if the central brightness pulse has a duration beyond 300 days it will be extremely difficult (if even possible) to resolve the delay to the first reverberation peak, which is expected to be at a maximum lag time of several hundred days. The new autocorrelation calculations for the long-term variability-corrected data showed from visual inspection the autocorrelation patterns expected from reverberation. However, as is described in section 2, the presence of autocorrelation peaks alone do not necessarily constitute a detection of reverberation.
+
+To determine the reality of the autocorrelation peaks, a routine was written to identify the ten largest-amplitude brightening and fading events in each low-pass filtered brightness record and produce the mean brightness profiles following them - a mean brightening profile and a mean dimming profile. Inspection of these brightness profiles demonstrated that any spurious autocorrelation peaks were a small effect, easily removed by ignoring any peaks within one hundred days of a larger-amplitude peak. The justification for this is simple - if two reverberations occur within one hundred days of each other (which has a low probability of occurence when one considers the required quasar orientation), resolving them within the brightness record or autocorrelation will become difficult, particularly for the most poorly-sampled, low-redshift objects with a time resolution of \(\sim 10\) days and given that the average half-width of a central brightness peak was found to be approximately 70 days. There is also the consideration of the shape of the reverberation signal from each elliptical surface, which has not yet been calculated but may have differing asymmetry for each of the four surfaces.
+
+Finally it was noticed that the brightening and dimming profiles do not perfectly match and in some cases seem to show structure of opposing sign on the same timescales - if a brightening event was usually followed by another brightening event at a time lag \(t\), a dimming event was sometimes also followed by a brightening event at a lag of \(t\). The fact that this effect is not seen in every object makes it difficult to interpret or even identify as a real effect. The result of this is that the mean brightness profile does not always perfectly match the autocorrelation, but usually demonstrates the same approximate shape. These arguments are not quantifiable and so are discarded from future investigation, which will focus on the results of an automated analysis. At this point it was also noted that the highest-redshift quasar in the survey, MACHO 208.15799.1085 at a redshift of 2.77, contained only \(\sim 700\) proper days’ worth of observations. It was felt that this would not be sufficient time to observe a significant number of reverberation events, which were observed to occur on timescales of order \(\sim 500\) proper days, and so the object was discarded from further analysis. Fig. 2 gives one example of the manually-produced light curves, its autocorrelation and the effect of long-term variability.
+
+Figure 2: **Plots produced in manual inspection of the data for one quasar.** Top left is the raw data, top right its autocorrelation. On the bottom left is the data remaining after subtraction of long-term variability and bottom right is the corrected data’s autocorrelation function.
+
+In the automatically-processed, long-term variability-corrected data, reverberation patterns were recognised in the autocorrelation by an IDL routine designed to smooth the autocorrelation on a timescale of 50 days and identify all positive peaks with lags less than 1000 days. If two or more peaks were identified within 100 days of each other, the more prominent peak was returned. As a result, we estimate the accuracy of any reverberation signatures to be approximately 50 days. Therefore a 50-day smooth will reduce the number of peaks for the program to sort through while not detracting from the results, simultaneously removing any double-peaks caused by dips in autocorrelation caused by unrelated phenomena. The 50-day resolution simply reflects the fact that the occurence of two peaks within one hundred days can not be identified with two separate reverberations.
+
+This technique was also applied to a ’Brownian noise’ or simple red noise simulation with a spectral slope of -2 for comparison. This was generated using a random number generator in the following way. First an array (referred to as R) of N random numbers was produced and the mean value subtracted from each number in said array. A new, blank array (S) of length N+1 was created, with values then appended as follows:
+
+\[\mbox{if }R[i]>0\mbox{ then }S[i+1]=S[i]+1\] (8)
+
+\[\mbox{if }R[i]<0\mbox{ then }S[i+1]=S[i]-1\] (9)
+
+The seeds for the random number generator were taken as the raw data for each quasar in each filter, so that 114 different simulations were produced to aid in later comparison with results. [43] outline a method for producing more rigorous red noise simulations of different spectral slopes but to allow more time to focus on the observations and their interpretation, only this simplified calculation was performed.
+
+Basic white noise simulations were also produced but produced none of the predicted autocorrelation structure and so were only followed through the early data processing stages. An example white noise autocorrelation is given in Fig. 3 with a quasar autocorrelation curve overplotted. Fig. 4 shows the light curve and autocorrelation function for a star observed by the MACHO programme compared to one of the quasars to demonstrate the difference in autocorrelation function and demonstrate that the observed structure is a feature of the quasars themselves and not induced by observational effects.
+
+Figure 3: **Quasar (+) and white noise (.) autocorrelation functions**
+
+Figure 4: **Top: Quasar (+) and star data (.)** demonstrating that the long-term variability observed in the quasars is not an observational effect. **Bottom: Quasar (bold) and star autocorrelation functions** after 300-day boxcar smooths have been applied, showing that the autocorrelation structure observed in quasars is intrinsic to the quasars themselves
+
+## 6 Results
+
+Once the data were processed and autocorrelation peaks identified, the positions of the peaks were then used to calculate the radial distance from central source to outflow region (\(R_{blr}\)), the inclination angle of the observer’s line of sight from the accretion disc plane (\(\theta\)) and a factor termed the ’internal structure variable’ which is the angle from the accretion disc plane to the reverberating outflow regions (\(\epsilon\)). These are calculated from equations (2-5). These can then produce
+
+\[(4)+(5)+(6)+(7)=t_{1}+t_{2}+t_{3}+t_{4}=4\cdot\frac{R_{blr}}{c}\] (10)
+
+or \(R_{blr}=c\cdot\langle t\rangle\)
+
+\[(6)-(5)=t_{3}-t_{2}=2\cdot\frac{R_{blr}}{c}\cdot\cos(\theta+\epsilon)\] (11)
+
+or \(\theta+\epsilon=\arccos(\frac{t_{3}-t_{2}}{2\cdot\langle t\rangle})\)
+
+\[(7)-(4)=t_{4}-t_{1}=2\cdot\frac{R_{blr}}{c}\cdot\cos(\theta-\epsilon)\] (12)
+
+or \(\theta-\epsilon=\arccos(\frac{t_{4}-t_{1}}{2\cdot\langle t\rangle})\)
+
+From (11) and (12) we then obtain
+
+\[\theta=\frac{\arccos(\frac{t_{3}-t_{2}}{2\cdot\langle t\rangle})+\arccos(\frac{t_{4}-t_{1}}{2\cdot\langle t\rangle})}{2}\] (13)
+
+\[\epsilon=\frac{\arccos(\frac{t_{3}-t_{2}}{2\cdot\langle t\rangle})-\arccos(\frac{t_{4}-t_{1}}{2\cdot\langle t\rangle})}{2}\] (14)
+
+Consideration of the geometry involved also shows that three special cases must be considered. First is the case where \(\theta=90^{o}-\epsilon\). In this case, peaks two and three are observed at the same time and so only three distinct brightness pulses are seen. In analysing the data, the simplifying assumption was made that all 3-peak autocorrelation signatures were caused by this effect. Another case to be considered is when \(\theta=90^{o}\), in which cases pulses 1 and 2 are indistinguishable, as are pulses 3 and 4. Again, the assumption was made that all two-peak reverberation patterns were caused by this orientation effect. Notice that in these cases the equation for \(R_{blr}\) is not modified, but the equations for \(\epsilon\) in the 3-peak and 2-peak cases become
+
+\[\epsilon=\frac{\arcsin(\frac{t_{3}-t_{1}}{2\cdot\langle t\rangle})}{2}\] (15)
+
+\[\epsilon=\frac{\arcsin(\frac{t_{2}-t_{1}}{2\cdot\langle t\rangle})}{2}\] (16)
+
+respectively. Finally there is the case of zero inclination angle, where peaks one and two arrive at the same time as the initial brightening event, followed by peaks three and four which then arrive simultaneously. In this situation no information about the internal structure variable can be extracted. Thankfully, no such events were observed. The results calculated for the 57 studied quasars, from equations (10) and (13-16), are presented in Table 1. The redshifts presented in Table 1 are as given in [41]. Additionally a column has been included showing \(R_{blr}\) in multiples of the minimum measured \(R_{blr}\) for comparison with the relative luminosities of the quasars.
+
+This yields an average \(R_{blr}\) of 544 light days with an RMS of 74 light days and an average \(\epsilon\) of \(11.87^{o}\) with an RMS of \(2.03^{o}\). A plot of \(t_{1}\), \(t_{2}\), \(t_{3}\), \(t_{4}\) as a function of \(\theta\) for the 57 objects was then compared to the predictions of a simulation using the calculated average \(\epsilon\) from these observations, for the data from both filters. This entire process was then repeated for the red noise simulations, of which 114 were produced such that the resulting plots could be directly compared. The simulations produced by the R data were treated as R data while the simulations produced by V data were treated as V data, to ensure the simulations were treated as similarly to the actual data as possible. The process was then repeated for 10 stars in the same field as the example quasar that was studied in detail, MACHO 13.5962.237. Notice that while the quasar data show trends very close to the model, the star data exhibit much stronger deviations (see Fig. 5).
+
+\begin{table}
+\begin{tabular}{l c c c c c c c c c r}
+\hline \hline
+\multicolumn{1}{c}{MACHO ID} & z & n(V) & V & n(R) & R & \(R_{blr}\) & \(\theta\) & \(\epsilon\) & \(R_{rel}\) & \multicolumn{1}{c}{\(L_{rel}\)} \\
+ & & & & & & (ld) & (_o_) & (_o_) & & \\
+\hline
+2.5873.82 & 0.46 & 959 & 17.44 & 1028 & 17.00 & 608 & 71 & 9.5 & 1.67 & 37 \\
+5.4892.1971 & 1.58 & 958 & 18.46 & 938 & 18.12 & 560 & 70 & 12.0 & 1.53 & 54 \\
+6.6572.268 & 1.81 & 988 & 18.33 & 1011 & 18.08 & 578 & 71 & 12.5 & 1.58 & 79 \\
+9.4641.568 & 1.18 & 973 & 19.20 & 950 & 18.90 & 697 & 72 & 11.0 & 1.91 & 24 \\
+9.4882.332 & 0.32 & 995 & 18.85 & 966 & 18.51 & 589 & 64 & 12.0 & 1.61 & 6 \\
+9.5239.505 & 1.30 & 968 & 19.19 & 1007 & 18.81 & 579 & 64 & 12.0 & 1.59 & 28 \\
+9.5484.258 & 2.32 & 990 & 18.61 & 396 & 18.30 & 481 & 83 & 10.5 & 1.32 & 78 \\
+11.8988.1350 & 0.33 & 969 & 19.55 & 978 & 19.23 & 541 & 67 & 14.0 & 1.48 & 3 \\
+13.5717.178 & 1.66 & 915 & 18.57 & 509 & 18.20 & 572 & 59 & 13.0 & 1.57 & 62 \\
+13.6805.324 & 1.72 & 952 & 19.02 & 931 & 18.70 & 594 & 85 & 9.5 & 1.63 & 41 \\
+13.6808.521 & 1.64 & 928 & 19.04 & 397 & 18.74 & 510 & 81 & 11.0 & 1.40 & 38 \\
+17.2227.488 & 0.28 & 445 & 18.89 & 439 & 18.58 & 608 & 82 & 8.0 & 1.67 & 4 \\
+17.3197.1182 & 0.90 & 431 & 18.91 & 187 & 18.59 & 567 & 73 & 16.0 & 1.55 & 24 \\
+20.4678.600 & 2.22 & 348 & 20.11 & 356 & 19.87 & 439 & 68 & 14.0 & 1.20 & 18 \\
+22.4990.462 & 1.56 & 542 & 19.94 & 519 & 19.50 & 556 & 64 & 12.5 & 1.52 & 17 \\
+22.5595.1333 & 1.15 & 568 & 18.60 & 239 & 18.30 & 565 & 73 & 9.0 & 1.54 & 41 \\
+25.3469.117 & 0.38 & 373 & 18.09 & 363 & 17.82 & 558 & 65 & 12.0 & 1.53 & 15 \\
+25.3712.72 & 2.17 & 369 & 18.62 & 365 & 18.30 & 517 & 72 & 12.0 & 1.42 & 73 \\
+30.11301.499 & 0.46 & 297 & 19.46 & 279 & 19.08 & 546 & 68 & 12.5 & 1.50 & 6 \\
+37.5584.159 & 0.50 & 264 & 19.48 & 258 & 18.81 & 562 & 70 & 12.5 & 1.54 & 7 \\
+48.2620.2719 & 0.26 & 363 & 19.06 & 352 & 18.73 & 605 & 65 & 13.0 & 1.66 & 3 \\
+52.4565.356 & 2.29 & 257 & 19.17 & 255 & 18.96 & 447 & 85 & 8.0 & 1.22 & 44 \\
+53.3360.344 & 1.86 & 260 & 19.30 & 251 & 19.05 & 496 & 67 & 10.0 & 1.36 & 33 \\
+53.3970.140 & 2.04 & 272 & 18.51 & 105 & 18.24 & 404 & 69 & 13.5 & 1.11 & 75 \\
+58.5903.69 & 2.24 & 249 & 18.24 & 322 & 17.97 & 491 & 80 & 8.5 & 1.35 & 104 \\
+58.6272.729 & 1.53 & 327 & 20.01 & 129 & 19.61 & 518 & 70 & 12.5 & 1.42 & 15 \\
+59.6398.185 & 1.64 & 279 & 19.37 & 291 & 19.01 & 539 & 83 & 9.5 & 1.48 & 29 \\
+61.8072.358 & 1.65 & 383 & 19.33 & 219 & 19.05 & 471 & 67 & 16.0 & 1.29 & 29 \\
+61.8199.302 & 1.79 & 389 & 18.94 & 361 & 18.68 & 475 & 69 & 14.0 & 1.30 & 40 \\
+63.6643.393 & 0.47 & 243 & 19.71 & 243 & 19.29 & 536 & 67 & 11.0 & 1.47 & 5 \\
+63.7365.151 & 0.65 & 250 & 18.74 & 243 & 18.40 & 625 & 83 & 10.5 & 1.71 & 19 \\
+64.8088.215 & 1.95 & 255 & 18.98 & 240 & 18.73 & 464 & 67 & 17.5 & 1.27 & 46 \\
+64.8092.454 & 2.03 & 242 & 20.14 & 238 & 19.94 & 485 & 73 & 10.5 & 1.33 & 16 \\
+68.10972.36 & 1.01 & 267 & 16.63 & 245 & 16.34 & 670 & 84 & 10.0 & 1.84 & 216 \\
+75.13376.66 & 1.07 & 241 & 18.63 & 220 & 18.37 & 561 & 67 & 10.0 & 1.54 & 36 \\
+77.7551.3853 & 0.85 & 1328 & 19.84 & 1421 & 19.61 & 489 & 69 & 14.0 & 1.34 & 9 \\
+78.5855.788 & 0.63 & 1457 & 18.64 & 723 & 18.42 & 491 & 77 & 12.0 & 1.35 & 19 \\
+206.16653.987 & 1.05 & 741 & 19.56 & 581 & 19.28 & 486 & 69 & 14.5 & 1.33 & 15 \\
+206.17052.388 & 2.15 & 803 & 18.91 & 781 & 18.68 & 365 & 37 & 13.5 & 1.00 & 53 \\
+207.16310.1050 & 1.47 & 841 & 19.17 & 885 & 18.85 & 530 & 90 & 8.5 & 1.45 & 31 \\
+207.16316.446 & 0.56 & 809 & 18.63 & 880 & 18.44 & 590 & 66 & 12.5 & 1.62 & 16 \\
+208.15920.619 & 0.91 & 836 & 19.34 & 759 & 19.17 & 621 & 70 & 13.0 & 1.70 & 15 \\
+208.16034.100 & 0.49 & 875 & 18.10 & 259 & 17.81 & 742 & 82 & 9.0 & 2.03 & 23 \\
+211.16703.311 & 2.18 & 733 & 18.91 & 760 & 18.56 & 374 & 90 & 6.5 & 1.02 & 57 \\
+211.16765.212 & 2.13 & 791 & 18.16 & 232 & 17.87 & 447 & 76 & 13.0 & 1.22 & 108 \\
+1.4418.1930 & 0.53 & 960 & 19.61 & 340 & 19.42 & 577 & 64 & 11.5 & 1.58 & 6 \\
+1.4537.1642 & 0.61 & 1107 & 19.31 & 367 & 19.15 & 635 & 79 & 9.0 & 1.74 & 9 \\
+5.4643.149 & 0.17 & 936 & 17.48 & 943 & 17.15 & 699 & 80 & 9.0 & 1.92 & 8 \\
+6.7059.207 & 0.15 & 977 & 17.88 & 392 & 17.41 & 613 & 83 & 11.5 & 1.68 & 4 \\
+13.5962.237 & 0.17 & 879 & 18.95 & 899 & 18.47 & 524 & 66 & 12.5 & 1.44 & 2 \\
+14.8249.74 & 0.22 & 861 & 18.90 & 444 & 18.60 & 579 & 69 & 13.5 & 1.59 & 3 \\
+28.11400.609 & 0.44 & 313 & 19.61 & 321 & 19.31 & 504 & 70 & 13.5 & 1.38 & 5 \\
+53.3725.29 & 0.06 & 266 & 17.64 & 249 & 17.20 & 553 & 69 & 14.0 & 1.52 & 1 \\
+68.10968.235 & 0.39 & 243 & 19.92 & 261 & 19.40 & 566 & 83 & 12.0 & 1.55 & 3 \\
+69.12549.21 & 0.14 & 253 & 16.92 & 244 & 16.50 & 497 & 68 & 12.5 & 1.36 & 9 \\
+70.11469.82 & 0.08 & 243 & 18.25 & 241 & 17.59 & 544 & 79 & 10.5 & 1.49 & 1 \\
+82.8403.551 & 0.15 & 836 & 18.89 & 857 & 18.55 & 556 & 56 & 12.0 & 1.52 & 1 \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Properties of the MACHO quasars. z is the quasar’s redshift, n(V) and n(R) the number of V and R band observations respectively, V and R respectively are the average V and R magnitudes, \(R_{blr}\) the calculated radial distance from the central source to the reverberating regions, \(\theta\) the calculated inclination angle and \(\epsilon\) the calculated internal structure variable. \(R_{rel}\) is the calculated \(R_{blr}\) in units of the smallest \(R_{blr}\) found and \(L_{rel}\) is the luminosity relative again to the minimum calculated luminosity. The last five columns are each an average of the values calculated separately from the V and R data.
+
+(a)Quasar R (left) and V (right) data
+
+(b)Star data fit to model
+
+Figure 5: **Autocorrelation peak lags normalised to calculated \(R_{blr}\) vs calculated inclination angle.** The solid lines represent the model’s predictions.
+
+The red-noise simulation process described in section 5 was repeated ten times (so that in total 1140 individual red noise simulations were created), so that a statistic on the expected bulk properties of red noise sources could be gathered. It was found that on average, \(18.5\%\) of pure red-noise sources had no autocorrelation peaks detected by the program used to survey the MACHO quasars, with an RMS of \(\pm 6.3\%\). Since all 57 studied quasars show relevant autocorrelation structure, it may be concluded that the observed structure is a \(9\sigma\) departure from the red-noise distribution, indicating a greater than \(99\%\) certainty that the effect is not caused by red-noise of this type.
+
+With an error of \(\pm 50\) days in the determination of each lag time, an error of \(\pm 35\) days is present in the calculation of \(t_{2}-t_{1}\), \(t_{3}-t_{2}\), etc. This also yields an error of \(\pm 25\) days in the calculated radius of the outflow region. If the error in \(\cos(\theta\pm\epsilon)\) is small, ie \(\sigma(R_{blr})\ll R_{blr}\), the small angle approximation may be applied in calculating the error in radians, before converting into degrees. Using the average value of \(R_{blr}=544ld\) and therefore an error in \(\cos(\theta\pm\epsilon)\) of \(\pm 0.0643\), the error in \(\theta\pm\epsilon\) is found to be \(\pm 3.69^{o}\) - sufficiently low to justify the small angle approximation. The errors in \(\theta\) and \(\epsilon\) are therefore both equal to \(\pm 2.61^{o}\). The errors in the calculated mean values for \(R_{blr}\) and \(\epsilon\) are then \(\pm 3.3\) light days and \(\pm 0.35^{o}\) respectively. The 100-day resolution limit for reverberation peaks also restricted the value of \(\theta\) to greater than \(50^{o}\) - at lower inclinations the program was unable to resolve neighbouring peaks as they would lie within 100 days of each other for a quasar with \(R_{blr}\sim 544\) light days. For this reason, the calculated inclination angles are inherently uncertain. The contribution to the RMS of \(\epsilon\) is surprisingly small for the two- and three-peak objects.
+
+The availability of data in two filters enables an additional statistic to be calculated - the agreement of the R and V calculated values. For one quasar, MACHO 206.17057.388, one autocorrelation peak was found in the V data while three were found in the R. Inspection of the data themselves showed that the R data was much more complete and so the V result for this object was discarded. The mean deviation of the R or V data from their average was found to be 44 days, giving an error on the mean due to the R/V agreement of \(\pm 4.1\) days. The mean deviation of the calculated \(\theta\) due to the R/V agreement is \(7.1^{o}\) and that for \(\epsilon\) is \(2.04^{o}\), yielding an error in \(\epsilon\) due to the difference in R and V autocorrelation functions of \(\pm 0.19^{o}\).
+
+The RMS deviation of the observed lag times from those predicted by a model with internal structure variable \(\epsilon=11.87^{o}\) is found to be \(4.29^{o}\), which is acceptable given a systematic error of \(\pm 2.61^{o}\) and an RMS in the calculated \(\epsilon\) of \(2.03^{o}\). Combining the two errors on each of the mean \(R_{blr}\) and \(\epsilon\) yields total errors on their means of \(\pm 5.26\) light days and \(0.40^{o}\) respectively.
+
+The interpretation of \(\epsilon\) is not entirely clear - it is presented in [11] as the angle made by the luminous regions of the outflowing winds from the accretion disc. However, as is demonstrated in Fig. 6a, \(\epsilon\) may in fact represent the projection angle of the shadow of the accretion disc onto the outflowing winds if the central luminosity source occupies a region with a radius lower than the disc thickness, thus giving by geometry the ratio of the inner accretion disc radius to the disc thickness. If the central source is extended above and below the accretion disc, however, then \(\epsilon\) may represent, as shown in Fig. 6b, the ratio of disc thickness to \(R_{blr}\). There may also be a combination of effects at work.
+
+(a)Interpretation of \(\epsilon\) for a central source much smaller than the thickness of the accretion disc
+
+(b)Interpretation of \(\epsilon\) if the central source extends above and below the accretion disc
+
+Figure 6: **Interpretations of \(\epsilon\)**
+
+An accurate way of discriminating between these two interpretations may be found by analysing the mean reverberation profiles from the four outflow surfaces and producing a relation between the timing of the central reverberation peak and that of the onset of reverberation by a specific outflow surface. The two figures above demonstrate how the timing of reverberation onset is interpreted in the two cases of compact vs extended central source but to discriminate, models must be produced of the expected reverberation profiles for these two cases. Results from [44] would predict the former, as indeed would [31] and [12] and so figure 6a would seem favourable, though analysis of the reverberation profiles will enable a more concrete determination of the meaning of \(\epsilon\). Models may then be produced of the expected brightness patterns for differing outflow geometries and compared to the observed light curves to determine the curvature and projection angles of the outflows.
+
+The calculated luminosities may now be compared to the calculated \(R_{blr}\) for this sample to determine whether the [17] or [18] relations hold for quasars. Fig. 7 demonstrates that there is absolutely no correlation found for this group of objects. This is not entirely unsurprising as even for nearby AGN both [17] and [18] had enormous scatters in their results, though admittedly not as large as those given here. One important thing to note is that given a predicted factor 200 range in luminosity, one might take [18] as a lower limit on the expected spread, giving a predicted range of \(R_{blr}\) of approximately a factor 15. The sensitivity range of this project in \(R_{blr}\) is a function of the inclination angle of the quasar but in theory the \(R_{blr}\) detectable at zero inclination would be 500 light days with a minimum detectable \(R_{blr}\) at this inclination of 50 light days. This factor 10 possible spread of course obtains for any inclination angle giving possibly an infinite spread of \(R_{blr}\). For the calculated RMS of 77 light days, the standard deviation is 10.2 light days, 2% of the calculated mean. The probability of NOT finding a factor 15 range of \(R_{blr}\) if it exists is therefore less than 1%. Therefore it can be stated to high confidence that this result demonstrates that the [17] and [18] results do not hold for quasars. The failure of [17] for this data set is evident from the fact that it predicts an even larger spread in \(R_{blr}\) than [18] which is therefore even less statistically probable from our data. It is possible that this is a failure of the SED of [42], that there are some as-yet unrecognised absorption effects affecting the observations or that the assumption of negligable absorption/re-emission times is incorrect but the more likely explanation would seem to be that no correlation between \(R_{blr}\) and luminosity exists for quasars.
+
+Figure 7: **Relative luminosities and broad line radii.** Absolutely no correlation is seen between \(R_{rel}\) (the relative \(R_{blr}\)) and \(L_{rel}\) (the relative luminosity), demonstrating that the [17] and [18] results do not hold for quasars. The error bars are so small that they are contained within the data points.
+
+A calculated average \(R_{blr}\) of 544 light days corresponds to \(1.4\times 10^{18}cm\), compared to the predicted average of \(\sim 10^{18}cm\). The value found here is in remarkable agreement with those previously calculated, made even more surprising by the fact that so few results were available upon which to base a prediction. Many quasar models predict a large variation in quasar properties, see for example [19], so we conclude that quasars and perhaps AGN in general are an incredibly homogeneous population.
+
+## 7 Conclusions
+
+Brightness records of 57 quasars taken from the MACHO survey in R and V colour filters have been analysed to show the presence of autocorrelation structure consistent with biconic outflowing winds at an average radius of \(544\pm 5.2\) light days with an RMS of 74 light days. An internal structure variable of \(11.87\pm 0.40^{o}\) was found, with an RMS of \(2.9^{o}\). The accuracy of the program designed to determine the timing of the reverberation peaks limited its temporal resolution to 100 days, resulting in the quoted systematic errors in the mean values calculated. With longer-timescale, more regularly sampled data this temporal resolution can be improved - this may also be achieved with more sophisticated computational techniques combined with brightness models not available in this project.
+
+The correlations between radius of the broad emission line region and luminosity found by [17] and [18] for nearby AGN do not appear to hold for quasars. This may be indicative of some time or luminosity evolution of the function as no redshift-independant correlation is found in this data set. If there is some relation it is more likely to be time-evolving since any luminosity dependance would most likely be noticeable in Fig. 7, which it is not.
+
+The presence of reverberation in 57 of the 57 quasars analysed implies that the outflowing wind is a universal structure in quasars - a verifiable result since this structure may be identified in regularly-sampled quasars in other surveys. While it is acknowledged that red noise may yet be responsible for the brightness fluctuations observed, the results are so close to the initial model’s prediction that noise seems an unlikely explanation, especially given the corroborating evidence for the theory [11, 12, 21, 23, 24].
+
+The continuum variability of quasars, though well observed, is still not well understood. The results of this study would suggest that an understanding of these fluctuations can only be found by recognising that several physical processes are at work, of which reverberation is of secondary importance in many cases. It does however appear to be universally present in quasars and possibly in all AGN. Given that quasars are defined observationally by the presence of broad, blue-shifted emission lines, of which outflowing winds are the proposed source, this result is strong support for the [1] model.
+
+## 8 Future Work
+
+Several phenomena identified in the MACHO quasar light curves remain as yet unexplained.
+
+1. 1.What is the source of the long-term variability of quasars? Is it a random noise process or is there some underlying physical interpretation? It has been suggested that perhaps a relativistic orbiting source of thermal emission near the inner accretion disc edge may be the source of such fluctuation. Modelling of the expected emission from such a source must be undertaken before such a hypothesis can be tested.
+2. 2.Why is it that the brightness profile following a dimming event sometimes agrees perfectly with the brightening profile while at other times it is in perfect disagreement? Again, is this a real physical process? Work by [45] on stratified wind models presents a situation where the central object brightening could increase the power of an inner wind, increasing its optical depth and thus shielding outer winds. This would result in negative reverberation. Further investigation may demonstrate a dependance of this effect on whether the central variation is a brightening or fading.
+3. 3.What is the mean profile of each reverberation peak? This profile may yield information about the geometry of the outflowing wind, thus enabling constraints to be placed on the physical processes that originate them.
+4. 4.Can quasars be identified by reverberation alone? Or perhaps by the long-term variability they exhibit? With surveys such as Pan-STARRS and LSST on the horizon, there is growing interest in devloping a purely photometric method of identifying quasars.
+5. 5.LSST and Pan-STARRS will also produce light curves for thousands of quasars which can then be analysed in bulk to produce a higher statistical accuracy for the long-term variability properties of quasars. It is evident that the sampling rate will only be sufficient for reverberation mapping to be performed with LSST and not Pan-STARRS.
+6. 6.Is there a time- or luminosity-evolving relation between \(R_{blr}\) and luminosity? Comparison of \(R_{blr}\), luminosity and redshift may yet shed light on this question.
+
+## 9 Acknowledgements
+
+I would like to thank Rudy Schild for proposing and supervising this project, Pavlos Protopapas for his instruction in IDL programming, Tsevi Mazeh for his advice on the properties and interpretation of the autocorrelation function and Phil Uttley for discussion and advice on stochastic noise in quasars. This paper utilizes public domain data originally obtained by the MACHO Project, whose work was performed under the joint auspices of the U.S. Department of Energy, National Nuclear Security Administration by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, and the Mount Stromlo and Siding Spring Observatory, part of the Australian National University.
+
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+

arxiv_ground_truth/0905.3502.md

@@ -0,0 +1,369 @@
+# Extrinsic noise passing through a Michaelis-Menten reaction: A universal response of a genetic switch
+
+Anna Ochab-Marcinek
+
+ochab@ifka.ichf.edu.pl
+Department of Soft Condensed Matter, Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland
+
+###### Abstract
+
+The study of biochemical pathways usually focuses on a small section of a protein interactions network. Two distinct sources contribute to the noise in such a system: intrinsic noise, inherent in the studied reactions, and extrinsic noise generated in other parts of the network or in the environment. We study the effect of extrinsic noise entering the system through a nonlinear uptake reaction which acts as a nonlinear filter. Varying input noise intensity varies the mean of the noise after the passage through the filter, which changes the stability properties of the system. The steady-state displacement due to small noise is independent on the kinetics of the system but it only depends on the nonlinearity of the input function.
+
+For monotonically increasing and concave input functions such as the Michaelis-Menten uptake rate, we give a simple argument based on the small-noise expansion, which enables qualitative predictions of the steady-state displacement only by inspection of experimental data: when weak and rapid noise enters the system through a Michaelis-Menten reaction, then the graph of the system’s steady states vs. the mean of the input signal always shifts to the right as noise intensity increases.
+
+We test the predictions on two models of _lac_ operon, where TMG/lactose uptake is driven by a Michaelis-Menten enzymatic process. We show that as a consequence of the steady state displacement due to fluctuations in extracellular TMG/lactose concentration the _lac_ switch responds in an asymmetric manner: as noise intensity increases, switching off lactose metabolism becomes easier and switching it on becomes more difficult.
+
+keywords: Genetic switches and networks , noise in biological systems , nonlinear input function 
+Figure 1: Steady-state displacement due to noise passing through a nonlinear uptake reaction. Extrinsic noise \(P_{t}\) enters the system through the uptake reaction which acts as a nonlinear filter \(h()\): for example, when the input signal has a Gaussian distribution, the output signal would have a skewed distribution. For small noise, varying the intensity of the input noise, without changing its mean \(\bar{P}\), varies the mean of the output noise \(\langle h(P_{t})\rangle\). This causes a shift \(\Delta(\bar{P})\) of the steady states of the system. The displacement direction depends on the shape of the filtering function \(h(P_{t})\). The small-noise expansion method enables qualitative prediction of the displacement direction, by inspection of the following experimental data (here, an example sketch): the graph of the system’s steady states vs. \(\bar{P}\) and the graph of the uptake rate vs. \(\bar{P}\). Real experimental data can be found e.g. in (Ozbudak et al., 2004).
+
+## 1 Introduction
+
+When studying biochemical pathways, we usually focus on a very small section extracted from a large network of protein interactions. Two distinct sources contribute to the noise in the studied system: Intrinsic noise due to the randomness of molecular collisions which result in chemical reactions of interest, and extrinsic noise, generated in other parts of the network or in the external environment (Shibata, 2005).
+
+Computational methods developed based on the Gillespie algorithm (Gillespie, 1976, 2007) enormously increased the popularity of theoretical studies of intrinsic stochasticity in biochemical reactions (see the most cited papers: Thattai and Oudenaarden (2001); Rao et al. (2002); Ozbudak et al. (2002)). On the other hand, growing attention is being paid to the study of extrinsic noise (Swain et al., 2002; Raj and van Oudenaarden, 2008; Shahrezaei and Swain, 2008; Shahrezaei et al., 2008; Lei et al., 2009), in particular to the problem of discrimination between the effects of intrinsic and extrinsic noise components in studied systems (Elowitz et al., 2002; Raser and O’Shea, 2004; Paulsson, 2004; Pedraza and Oudenaarden, 2005; Newman et al., 2006). Experiments on _lac_ gene in _E. coli_(Elowitz et al., 2002) and _PHO5_ and _GAL1_ genes in _Saccharomyces cerevisiae_(Raser and O’Shea, 2004) have demonstrated that the contribution of extrinsic noise to gene expression may be significant.
+
+Noise propagates across a gene network by passing from one subsystem to the other through reactions which link particular sections of the network, or which link the network with the external environment. In this paper, we study the effect of extrinsic fluctuations entering the studied system through a nonlinear uptake reaction which acts as a nonlinear filter. In this way, one of the reaction rates in the system depends in a nonlinear way on a fluctuating extrinsic variable. Shahrezaei et al. (2008) simulated (with a Gillespie algorithm modified by inclusion of a randomly varying reaction rate) a simple general model of gene expression where one of the reaction rates was an exponential function of a slowly fluctuating Gaussian process. The resulting reaction rate had an asymmetric, log-normal distribution. This particular shape of the nonlinear filter was chosen to conform the experimentally measured distributions of gene expression rates (Rosenfeld et al., 2005). Shahrezaei et al. (2008) found that the presence of this kind of noise shifts the mean concentration of the reaction product, however they did not explain that effect with analytical calculations.
+
+While Shahrezaei et al. (2008) studied numerically the effect of extrinsic fluctuations whose average lifetime was longer than the shortest timescale of the system (thus interfering the relaxation dynamics of the system’s variables), we focus on the analytical study of the effect of weak and rapid extrinsic noise, whose instantaneous fluctuations are too fast to directly interfere the system’s dynamics. The assumption of weakness and rapidity of the noise allows to use the small-noise expansion method (Gardiner, 1983). A measure for the rapidity of the random fluctuations is the correlation time (Horsthemke and Lefever, 1984). When the characteristic time scale of the noise (the correlation time) is much faster than the time scale of the reactions within the system of interest, then the noisy input signal contributes to the kinetics of the system as an average signal only (Horsthemke and Lefever, 1984). One could expect that, in such a case, varying the input noise intensity does not influence the behavior of the system. However, the non-linear filtering function can transform the output noise distribution in such a way that its mean varies as the input noise intensity is varied (see Fig. 1). The system may therefore respond to extrinsic noise by shifting its steady states with respect to the deterministic ones (Ochab-Marcinek, 2008; Rocco, 2009; Gerstung et al., 2009).
+
+A particularly interesting case is a multistable system, for example a bistable switch (Laurent and Kellershohn, 1999; Ferrell, 2002; Wilhelm, 2009), a chemical reaction system where two distinct steady states are possible under the same external conditions. In the presence of a noisy input signal, the bistable region can move to a concentration range where otherwise only one steady state was present, or bistability can disappear for concentrations at which the system was initially bistable (Ochab-Marcinek, 2008). Thus, as noise intensity increases, the existing steady states of the bistable system can be stabilized or destabilized, which results in the increase or decrease of the escape time from one steady state to the other (Ochab-Marcinek, 2008). The switching time in a bistable genetic switch has been recently investigated by Cheng et al. (2008), as the measure for the robustness of the cellular memory to intrinsic noise. The simplicity of the system (one-dimensional equation of kinetics) allowed them for analytical calculation of the switching time as the mean first-passage time derived from the Fokker-Planck equation (Gardiner, 1983). At a fixed noise level, Cheng et al. (2008) studied the dependence of the switching time on the transcription rate. On the other hand, Lei et al. (2009) simulated a similar system with a transcription delay. Extrinsic noise was there applied in the same log-normal form as in the above-mentioned paper by Shahrezaei et al. (2008) and the noise was strong enough to allow for bi-directional switching. Lei et al. (2009) showed that the switching frequency decreases as the delay in the negative feedback loop increases.
+
+The study of stochastic control of metabolic pathways from the perspective of the small-noise expansion is relatively novel. It was initially presented in (Ochab-Marcinek, 2008) where we used the small-noise expansion to calculate the noise-induced steady-state displacement for an arbitrary filtering function and tested it on the reduced Yildirim-Mackey model (Yildirim and Mackey, 2003) of the _lac_ operon. Using a numerical simulation of that model, we showed that the induction time of the operon increases and the uninduction time decreases as a consequence of the noise-induced displacement. Recently, (Rocco, 2009) studied fluctuations in enzyme activity in Michaelis-Menten kinetics, expressing the noise-induced steady-state displacement as the Stratonovich drift. On the other hand, (Gerstung et al., 2009) used the small-noise expansion method to study the steady-state displacement of concentrations of cis-regulatory promoters due to intrinsic fluctuations in transcription factor concentration combined with extrinsic fluctuations in the transcription factor synthesis rate. The non-linear input function (the filter) was the probability of promoter occupation in the form of \(h(n)=n/(K+n)\), as a function of the fluctuating number \(n\) of free transcription factors.
+
+While in (Gerstung et al., 2009) the above form of the filtering function arised from the stationary solution of Master equation for the probability of finding \(n\) free transcription factors, in my present work the filtering function of the same form is the uptake rate of a certain substance P (the input signal), which enters the system through a catalytic Michaelis-Menten reaction (see Fig. 1). In a reaction that follows Michaelis-Menten kinetics, the reactant combines reversibly with an enzyme to form a complex. The complex then dissociates into the product and the free enzyme. If the total enzyme concentration does not change over time, and the concentration of the substrate-bound enzyme changes much more slowly than those of the product and substrate (the quasi-steady-state assumption), then the intermediate enzymatic steps can be reduced in the equations of chemical kinetics. The rate of product formation (P uptake) is then proportional to \(h(P)=P/(K_{M}+P)\), where \(P\) is the concentration of P (Atkins, 1998).
+
+We show that, when the P concentration \(P_{t}\) fluctuates with a Gaussian distribution, then \(h(P_{t})\) has an asymmetric distribution, similarly as in Shahrezaei et al. (2008) and Lei et al. (2009). A simple calculus argument, based on the small-noise expansion method, allows to show that the noise-induced steady-state displacement depends solely on the shape of the nonlinear input function (Gerstung et al., 2009). Consequently, we argue that the Michaelis-Menten type uptake function (monotonically increasing and concave) always generates the same type of the steady-state displacement. We show that this argument enables qualitative predictions of the steady-state displacement only by inspection of experimental data, i.e. a) the graph of the uptake function vs. the mean of the input signal and b) the graph of system’s steady states vs. the mean of the input signal.
+
+We show how, as a consequence of the steady-state displacement, the induction/uninduction time of the _lac_ operon changes depending on the intensity of TMG/lactose fluctuations. To illustrate the universality of the phenomenon, we compare the results for the Ozbudak model of _lac_ operon (Ozbudak et al., 2004), where the extracellular TMG uptake rate is an experimentally fitted function \(\sim P^{0.6}\) (increasing and concave), with the results for the reduced Yildirim-Mackey model (Ochab-Marcinek, 2008; Yildirim and Mackey, 2003), where the extracellular lactose uptake rate is given by the Michaelis-Menten formula \(P/(K_{M}+P)\). For both models we show how the Gaussian distribution of the input signal generates the skewed distribution of the Michaelis-Menten uptake rate. We analyze the validity range of the small-noise expansion in both cases. We compare the noise-induced steady state displacement and the noise-induced changes in the induction/uniduction time for the Ozbudak model with the previously published results for the reduced Yildirim-Mackey model (Ochab-Marcinek, 2008).
+
+## 2 Theory
+
+### Small-noise expansion method
+
+The input concentration of a substance P is modeled by the stochastic process \(P_{t}\) with the mean \(\bar{P}\) and the variance \(\sigma^{2}\). The passage through the uptake reaction generates the output noise \(h(P_{t})\).
+
+The equations of kinetics of the studied system are:
+
+\[\dot{\vec{X}}=\vec{F}(\vec{X},h(P_{t})),\] (1)
+
+where \(\vec{X}\) is the vector of concentrations of substrates, and the system depends on \(P_{t}\) through the function \(h(P_{t})\) only.
+
+Assume that the noise is weak and slower than the time scale of the filter response but faster than the characteristic time scale of the system (1). Then the system experiences the output noise (after the passage through the filter) as its mean, \(\langle h(P_{t})\rangle\)(Horsthemke and Lefever, 1984). If the mean value of the output noise differs from the deterministic value of \(h(\bar{P})\), then the steady states of
+
+\[\dot{\vec{X}}=\vec{F}(\vec{X},\langle h(P_{t})\rangle)\] (2)
+
+are shifted with respect to steady states of the deterministic system
+
+\[\dot{\vec{X}}=\vec{F}(\vec{X},h(\bar{P})).\] (3)
+
+It is possible to approximately find the displacement without the knowledge of the equations of kinetics, just by transformation of the \(X\) vs. \(\bar{P}\) graph. While the deterministic system has stationary states \(\vec{X}^{*}\), the stochastic system has quasi-steady states \(\langle\vec{X}\rangle\), around which its trajectories fluctuate (assuming that a noise-induced transition between multiple steady states is very unlikely). In the stochastic system, the steady states differ from the deterministic ones. This difference can be graphically shown as the shift of the graph of the steady states \(\vec{X}^{*}\) vs. \(\bar{P}\) along the \(\bar{P}\) axis ¹ (see Fig. 1) by a function \(\Delta(\bar{P})\), such that
+
+Footnote 1: There are two reasons to consider the horizontal shift (with respect to \(\bar{P}\)) and not the vertical one (with respect to \(\langle\vec{X}\rangle\)): a) The input signal \(P_{t}\) is the primary source of noise so it is easier to perform the calculations based on the dependence of the steady states on the variations in \(P_{t}\) than on the variations of \(\vec{X}\), which anyway depend those of \(P_{t}\). b) We will analyze the noise-induced shift in bistable systems, which can have two steady states for one value of \(\bar{P}\), so the vertical shift would be ambiguous.
+
+\[\langle\vec{X}(P_{t})\rangle=\vec{X}^{*}(\bar{P}+\Delta(\bar{P})).\] (4)
+
+The system depends on \(P_{t}\) only in the function of the output process \(h(P_{t})\), so
+
+\[\langle\ \vec{X}(h(P_{t}))\ \rangle=\vec{X}^{*}(\ h(\bar{P}+\Delta(\bar{P}))\ ).\] (5)
+
+As assumed above, the system only experiences the mean of the output process (Fig. 1):
+
+\[\langle\ \vec{X}(h(P_{t}))\ \rangle=\vec{X}^{*}(\ \langle h(P_{t})\rangle\ ),\] (6)
+
+And therefore, from the combination of Eqs. (5) and (6):
+
+\[\langle h(P_{t})\rangle=h(\bar{P}+\Delta(\bar{P})).\] (7)
+
+In other words, when the input process \(P_{t}\) is stochastic (fluctuating around the mean \(\bar{P}\)), then the mean of the output process \(h(P_{t})\) is equal to the deterministic output under the shift \(\Delta(\bar{P})\) of the input flux.
+
+Assume that \(\Delta(\bar{P})\) depends on \(\bar{P}\) so weakly that it can be treated as a constant \(\Delta\). Expand the output process \(h(\bar{P}+\Delta)\) around the mean of the input process:
+
+\[\langle h(P_{t})\rangle\approx h(\bar{P}+\Delta)=h(\bar{P})+h^{\prime}(\bar{P})\Delta+...\] (8)
+
+(As shown in the Results section and in (Ochab-Marcinek, 2008), this approximation works well for the example systems under study. In case when \(\Delta(\bar{P})\) depends steeply on \(\bar{P}\), the approximation may break down.)
+
+On the other hand, expansion of \(\langle h(P_{t})\rangle\) around \(\bar{P}\) with respect to a fluctuation \(\delta P\) of the input process (\(P_{t}=\bar{P}+\delta P\) at a given time \(t\)) yields:
+
+\[\langle h(P_{t})\rangle=\langle h(\bar{P}+\delta P)\rangle=h(\bar{P})+h^{\prime}(\bar{P})\langle\delta P\rangle+\frac{1}{2}h^{\prime\prime}(\bar{P})\langle\delta P^{2}\rangle+...\] (9)
+
+But the mean deviation from the mean of the input process \(\langle\delta P\rangle=0\) and the variance of the input process \(\langle\delta P^{2}\rangle=\sigma^{2}\). Combination of Eqs. (8) and (9) yields the approximate formula for the noise-induced shift of steady states with respect to deterministic steady states:
+
+\[\Delta(\bar{P})=\frac{h^{\prime\prime}(\bar{P})}{2h^{\prime}(\bar{P})}\sigma^{2}\] (10)
+
+### Michaelis-Menten uptake rate as the non-linear filter
+
+In the case of a system where the filtering function is generated by Michaelis-Menten kinetics (Atkins, 1998), the small-noise expansion method is valid when the noise is slower than the intermediate enzymatic reactions, which had been reduced into \(h(\bar{P})\). Then, the enzymatic reactions are not perturbed by the fluctuations in P concentration, and the concentrations of the reactants can reach their steady state while the P concentration is approximately constant.
+
+The noise correction \(\Delta(\bar{P})\) to the Michaelis-Menten rate \(h(\bar{P})=\bar{P}/(K_{M}+\bar{P})\) can be viewed as a correction to the Michaelis constant \(K_{M}\). The displacement \(\bar{P}+\Delta(\bar{P})=\bar{P}(1+\Delta(\bar{P})/\bar{P})\) is equivalent to the correction
+
+\[K_{M,noise}(\bar{P})=\frac{K_{M}}{1+\frac{\Delta(\bar{P})}{\bar{P}}}=\frac{K_{M}}{1-\frac{\sigma^{2}}{(K+\bar{P})\bar{P}}}.\] (11)
+
+When the input signal \(P_{t}\) has a Gaussian distribution with the mean \(\bar{P}\) and variance \(\sigma^{2}\), the output signal \(h(P_{t})=P_{t}/(K+P_{t})\) (Fig. 2) has an asymmetric probability density function (Fig. 3 b):
+
+\[p(h)=\frac{K}{\sqrt{2\pi\sigma^{2}}(1-h)^{2}}\exp\left[-\frac{1}{2\sigma^{2}}\left(\frac{Kh}{1-h}-\bar{P}\right)^{2}\right]\] (12)
+
+(See the Appendix A for the details of the calculation.) The mean \(\langle h(P_{t})\rangle=\int hp(h)dh/\mathcal{N}\) of (12) exists if \(0\geq h\geq 1\), which corresponds to \(0\geq\bar{P}\geq+\infty\). The normalization constant \(\mathcal{N}=(1/2)(\mathrm{erf}[\bar{P}\sqrt{2}/(2\sigma)]+1)\), and the accurate values of \(\langle h(P_{t})\rangle\) can be computed numerically.
+
+Figure 2: The TMG uptake rate (17) for the Ozbudak model (solid line) and the lactose uptake rate (19) for the Yildirim-Mackey model (dashed line). The mathematical formulas for the uptake rates are different in both models, but as they describe the same uptake mechanism, their graphs have a similar Michaelis-Menten shape with \(h^{\prime}>0\) and \(h^{\prime\prime}<0\), which generates the same type of the steady-state shift due to noise in extracellular TMG/lactose concentration (see the Results section).
+
+Figure 3: Probability density functions for the TMG/lactose uptake rates when the fluctuating concentrations of the extracellular TMG/lactose have a Gaussian distribution with the variance \(\sigma^{2}\). Vertical lines indicate the mean, which is slightly shifted with respect to the deterministic uptake rate (solid line). a) Ozbudak model. b) Yildirim-Mackey model.
+
+## 3 Models
+
+The _lac_ operon is one of the most extensively studied examples of a bistable genetic switch. In a certain range of extracellular TMG/lactose concentration, the gene transcription can be in one of two discrete states, either fully induced or uninduced. Bistability is generated by the positive feedback loop, where high lactose/TMG concentration in the cell causes weak repression of the _lac_ gene transcription and, in consequence, more permease is produced which pumps more lactose/TMG into the cell. According to experimental data (Huber et al., 1980; Wright et al., 1981; Page and West, 1984; Lolkema et al., 1991; Ozbudak et al., 2004) the rate of the TMG/lactose uptake into the cell is of Michaelis-Menten type. Extracellular lactose concentration experienced by _E. coli_ living in its natural environment may fluctuate, due to high mobility of the bacterium (see e.g. DiLuzio et al. (2005)), granularity of the intestinal content and motions of intestinal villi.
+
+### Ozbudak model
+
+The model (Ozbudak et al., 2004) is based on the experiment where a gene coding for a fluorescent protein was incorporated under the control of the _lac_ promoter in E. coli bacteria, in order to indicate the transcription activity of the _lac_ gene. The equations of chemical kinetics for this system are the following:
+
+\[\frac{R}{R_{T}}=\frac{1}{1+(x/x_{0})^{2}}\] (13)
+
+\[\tau_{y}\frac{dy}{dt}=\alpha\frac{1}{1+R/R_{0}}-y\] (14)
+
+\[\tau_{x}\frac{dx}{dt}=\beta_{G}h(T)y-x\] (15)
+
+\(x\) denotes the intracellular TMG concentration, \(y\) is the concentration of permease in green fluorescence units, \(R_{T}\) is the total concentration of the repressor, and \(R\) is the concentration of active repressor. The active fraction of the repressor is a function of the TMG concentration \(x\), with half-saturation concentration \(x_{0}\). \(\alpha\) is the maximum rate of generation of permease, \(R_{0}\) is the half-saturation of the repressor. Permease is depleted in a time scale \(\tau_{y}\), due to a combination of degradation and dilution. TMG enters the cell at the rate \(h(T)\) (Fig. 2) proportional to \(y\) and to the glucose uptake rate \(\beta_{G}\), and is depleted with time constant \(\tau_{x}\).
+
+Steady states of \(y\) (in \(x_{0}\) units, further on called the ’scaled units’) are given by:
+
+\[y=\alpha\frac{1+(\beta_{G}h(T)y)^{2}}{1+\frac{R_{T}}{R_{0}}+(\beta_{G}h(T)y)^{2}}\] (16)
+
+\(T\) is measured in \(\mathrm{\mu M}\). In a certain range of TMG concentrations, the cubic equation (16) has three roots (two stable fixed points separated by one unstable fixed point), which very well reproduces the experimentally observed switch-like behavior of _lac_ operon. In this study, we assume that the extracellular glucose concentration is zero. Then, the system is bistable for \(3.39<T<24.40\ \mathrm{\mu M}\). We have chosen the value of \(\tau_{x}=1\ \mathrm{min}\) significantly larger than the time scale of the noise (20), and at the same time much smaller than \(\tau_{y}=216\ \mathrm{min}\), which conforms the experimental results (Mettetal et al., 2006) reporting \(\tau_{x}\) less than the measurement resolution of \(35\ \mathrm{min}\).
+
+The TMG uptake rate function
+
+\[h(T)=1.23\times 10^{-3}T^{0.6},\] (17)
+
+as well as the values of the parameters (Table 1), have been fitted from the experimental data (Ozbudak et al., 2004). Since the formula (17) is the result of fitting, it does not have the classical Michaelis-Menten form \(T/(K+T)\), but its graph has the characteristic Michaelis-Menten shape, increasing and concave.
+
+### Yildirim-Mackey model
+
+The Yildirim-Mackey model (Ochab-Marcinek, 2008; Yildirim and Mackey, 2003) consists of three equations of chemical kinetics for mRNA (\(M\)), allolactose (\(A\)) and lactose (\(L\)) concentrations in the _E. coli_ cell:
+
+\[\begin{array}[]{lll}\frac{dM}{dt}&=&\alpha_{M}\ \frac{1+K_{1}\ A^{2}}{1+K_{2}R_{tot}+K_{1}\ A^{2}}+\Gamma_{0}-\tilde{\gamma}_{M}\ M\\ &&\\ \frac{dA}{dt}&=&k_{B}\ M\left(\alpha_{A}\ \frac{L}{K_{L}+L}-\beta_{A}\ \frac{A}{K_{A}+A}\right)-\tilde{\gamma}_{A}\ A\\ &&\\ \frac{dL}{dt}&=&k_{P}\ M\left(\alpha_{L}\ h(L_{e})-\beta_{L}\ \frac{L}{K_{L1}+L}\right)-\alpha_{A}\ k_{B}\ M\ \frac{L}{K_{L}+L}-\tilde{\gamma}_{L}\ L\\ \end{array}\] (18)
+
+\(\alpha\) and \(\beta\) denote the gain and loss rates for the reactions. \(K_{1}\) is the equilibrium constant for the repressor-allolactose reaction. \(K_{2}\) is the equilibrium constant for the operator-repressor reaction, and \(R_{tot}\) is the total amount of the repressor. The \(\tilde{\gamma}=\gamma+\mu\) are the coefficients for the terms representing decay of species due to chemical degradation (\(\gamma\)) and dilution (\(\mu\)). Even if allolactose is totally absent, on occasion repressor will transiently not be bound to the operator and RNA polymerase will initiate transcription. Although the mRNA production rate \(dM/dt\) would be then nonzero (a leakage transcription), it is necessary to add an empirical constant \(\Gamma_{0}\) to the model to obtain a leakage rate that agrees with experimental values (Yildirim and Mackey, 2003). The allolactose gain and loss rates and the lactose loss rate depend on the \(\beta\)-galactosidase concentration (an anzyme breaking down lactose into allolactose), which is proportional (\(k_{B}\) factor) to the mRNA concentration. Similarly, the lactose gain and loss rates depend on the permease concentration, proportional (\(k_{P}\) factor) to the mRNA concentration. See Table 2 for the values of parameters.
+
+The system is bistable for \(27.7\ \mathrm{\mu M}\ <L_{e}<61.8\ \mathrm{\mu M}\). The lactose uptake rate function
+
+\[h(L_{e})=\frac{L_{e}}{K_{L_{e}}+L_{e}}\] (19)
+
+has the Michaelis-Menten form, to conform the data of Lolkema et al. (1991); Huber et al. (1980); Page and West (1984); Wright et al. (1981).
+
+Figure 4: Comparison of the steady-state displacements calculated using the small-noise expansion method for the Ozbudak model (a) and the Yildirim-Mackey model (b). Rectangles indicate bistable regions of both models. In the Ozbudak model, the noise-induced shift of steady states is weaker than in the Yildirim-Mackey model and it depends more strongly on the extracellular sugar concentration. The displacement is the strongest on the left boundary of the bistable region and the weakest on its right boundary.
+
+### Fluctuations
+
+The fluctuations in TMG/lactose concentration have been modeled by Ornstein-Uhlenbeck noise. For the small-noise expansion to be valid, the correlation time of the noise
+
+\[\tau_{OU}=1.2\ \mathrm{s}\] (20)
+
+has been chosen significantly larger than the time scale of the enzymatic TMG/lactose uptake by permease (\(0.1\ \mathrm{s}\), according to Wright et al. (1986)), but smaller than the fastest time scale of the system: \(\tau_{\mathrm{sys}}=\tau_{x}=1\ \mathrm{min}\) for the Ozbudak model and \(\tau_{\mathrm{sys}}\approx 10^{-1}\ \mathrm{min}\) for Yildirim-Mackey model (Yildirim and Mackey, 2003). Taking into account the bacterial motility (mean velocity of _E. coli_ is \(\sim 30\mu\mathrm{m/s}\), (DiLuzio et al., 2005)), the granularity of the intestinal content and motions of intestinal villi, one can suppose that the fluctuation rapidity assumed here is realistic (Ochab-Marcinek, 2008).
+
+The details of the numerical simulation of the stochastic process are presented in the Appendix B.
+
+## 4 Results
+
+### Asymmetric distribution of the Michaelis-Menten uptake rate
+
+For the Mackey model, the probability density function of the Michaelis-Menten uptake rate \(h\) is given by the formula (12). For the Ozbudak Model, the probability density function of the uptake rate is described by a different formula (see the Appendix A for the details of the calculation):
+
+\[p(h)=\frac{5}{3}\frac{h^{2/3}}{b^{5/3}\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{1}{2\sigma^{2}}\left(\left(\frac{h}{b}\right)^{\frac{5}{3}}-\bar{T}\right)^{2}\right]\] (21)
+
+with \(b=1.23\times 10^{-3}\), because \(h(T)\) is given by the fitted function (17) which, however, has a similar shape to the Michaelis-Menten function. Consequently, the probability density functions for both models also have similar asymmetric shapes, and for small noise their mean values decrease as the noise intensity increases (Fig. 3). For larger noise, the mean values begin to increase (compare Fig. 7).
+
+### Calculation of the steady-state displacement using small-noise expansion
+
+Figure 5: Comparison of the steady-state displacements (23, 24) calculated using the small-noise expansion method with values obtained from the simulation of the Ozbudak model (a), and corresponding results for the Yildirim-Mackey model (Ochab-Marcinek, 2008) (b). The standard deviations of the fluctuations were \(\sigma=1.5\ \mathrm{\mu M}\) for the Ozbudak model and \(\sigma=10\ \mathrm{\mu M}\) for the Yildirim-Mackey model (Yildirim and Mackey, 2003). The response of both models to fluctuations in extracellular TMG/lactose concentration passing through a Michaelis-Menten-type uptake reaction is very similar: The steady states shift to the right. The displacement is larger for lower TMG/lactose concentrations and smaller for higher concentrations.
+
+The steady-state displacement (10) does not depend on the kinetics of the system. It only depends on the input noise intensity and on the shape of the uptake function: its monotonicity and concavity. Therefore, only knowing the graph of the steady states vs. \(\bar{P}\) , one can predict the changes in its stability due to noise passing through the uptake reaction.
+
+For a monotonically increasing and concave uptake function, such as Michealis-Menten (Fig. 2), \(h^{\prime}>0\) and \(h^{\prime\prime}<0\), so the steady states always shift to the right (in the direction of higher concentrations of P):
+
+\[\Delta(\bar{P})<0\] (22)
+
+The formulas for TMG/lactose uptake rates (17), (19) are different in both models, but as they describe the same uptake mechanism, their graphs have a very similar Michaelis-Menten shape (Fig. 2) with \(h^{\prime}>0\) and \(h^{\prime\prime}<0\), which causes a steady-state shift to the right. In the Ozbudak model,
+
+\[\Delta(\bar{T})=-\frac{0.2}{\bar{T}}\sigma^{2},\] (23)
+
+while in the Yildirim-Mackey model (Ochab-Marcinek, 2008),
+
+\[\Delta(\bar{L}_{e})=-\frac{1}{(K_{L_{e}}+\bar{L}_{e})}\sigma^{2}.\] (24)
+
+Within the bistable regions, the steady-state displacement due to noise in the Ozbudak model is smaller than in the Yildirim-Mackey model (Figs. 4, 5). A common feature of both models is a larger shift for low extracellular TMG/lactose concentrations and a smaller shift for high concentrations.
+
+Figure 6: The values of the steady-state displacement, calculated using the small-noise expansion method, compared with the values obtained from simulations. a) the Ozbudak model, b) the Yildirim-Mackey model. The noise intensities were same as in Fig. 5.
+
+Figure 7: The range of validity of the small-noise expansion: Comparison of the steady-state displacement calculated using different methods (simulation, accurate calculation, small-noise expansion) for constant values \(\bar{T}\) or \(\bar{L}_{e}\) as noise intensity and time scale of the noise are changed. Top panels: Varying noise intensities \(\sigma^{2}\). a) Ozbudak model, \(\bar{T}=3.6\ \mu\mathrm{M}\), \(\tau_{OU}=1.2\ \mathrm{s}=0.02\ \mathrm{min}\). b) Mackey model, \(\bar{L}_{e}=30\ \mu\mathrm{M}\), \(\tau_{OU}=1.2\ \mathrm{s}=0.02\ \mathrm{min}\). Bottom panels: Varying noise time scales \(\tau_{OU}\). c) Ozbudak model, \(\bar{T}=3.6\ \mu\mathrm{M}\), \(\sigma^{2}=1\ \mu{M}^{2}\). d) Mackey model, \(\bar{L}_{e}=30\ \mu\mathrm{M}\), \(\sigma^{2}=50\ \mu{M}^{2}\) (the accurate value overlaps with the approximated one).
+
+In the Ozbudak model as well as in the Yildirim-Mackey model, the results obtained analytically by small-noise expansion (23, 24) were in a very good agreement with the simulation results (Figs. 5, 6). The behavior of the Ozbudak model was very similar to the behavior of the Yildirim-Mackey model: In both cases the graph of the steady states vs. the mean extracellular TMG/lactose concentration shifted to the right. The shift was larger for lower TMG/lactose concentrations and smaller for higher concentrations.
+
+### Range of validity of the small-noise expansion
+
+For both models (within the given choice of parameters), we compared the values of \(\Delta\) calculated using different methods for chosen constant values \(\bar{T}\) or \(\bar{L}_{e}\) as noise intensity and time scale of the noise were changed (Fig. 7). The values of \(\Delta\) were: a) obtained from the simulation, b) calculated accurately (the mean \(\langle h(P_{t})\rangle\), computed using the distributions (12) or (21), was substituted into the equations of kinetics instead of \(h(P_{t})\)), and c) calculated using the small-noise expansion. The results are consistent with those expected: The expansion (8) is valid when \(\Delta\ll\bar{P}\), and indeed, in both models the approximation breaks down when \(\Delta/\bar{T}\) or \(\Delta/\bar{L}_{e}\) are greater than the order of \(10^{-2}\). Moreover, the time scale of noise should be less than the time scale of the system, and for the Yildirim-Mackey model the approximation is valid for \(\tau_{OU}<0.1\ \mathrm{min}\) while the fastest characteristic time for the left bifurcation point was \(\tau_{sys}=0.4\ \mathrm{min}\)(Ochab-Marcinek, 2008). For the Ozbudak model, the approximation breaks down at \(\tau_{OU}\approx 0.1\mathrm{min}\), while \(\tau_{sys}=1\mathrm{min}\). The results of the accurate calculation differ slightly from the small-noise approximation because of the Taylor expansion cut-off. There is also a systematic difference, increasing with noise intensity, between the simulation results and those calculated accurately. This difference is due to the reflecting boundary conditions used in the simulations, which add a contribution from the trajectories reflected at \(P_{t}=0\).
+
+Figure 8: Increasing noise intensity inhibits induction and accelerates uninduction in the studied models of the _lac_ operon. Mean induction/uninduction time was measured in simulations of the Ozbudak model (a) with different noise intensities and different mean TMG concentrations \(\bar{T}\): \(24.5\ \mathrm{\mu M}\) (A), \(24.6\ \mathrm{\mu M}\) (B), \(24.7\ \mathrm{\mu M}\) (C), \(3.41\ \mathrm{\mu M}\) (D), \(3.40\ \mathrm{\mu M}\) (E), \(3.39\ \mathrm{\mu M}\) (F). These results are compared with the results for the Yildirim-Mackey model (b) (Ochab-Marcinek, 2008), where the mean extracellular lactose concentrations \(\bar{L}_{e}\) were: \(62\ \mathrm{\mu M}\) (A), \(63\ \mathrm{\mu M}\) (B), \(65\ \mathrm{\mu M}\) (C), \(27.9\ \mathrm{\mu M}\) (D), \(27.8\ \mathrm{\mu M}\) (E), \(27.7\ \mathrm{\mu M}\) (F).
+
+Figure 9: Increasing noise intensity inhibits induction and accelerates uninduction in the Ozbudak model of the _lac_ operon: Example trajectories for different initial states of the system: a) Induced, left bifurcation point (\(\bar{T}=3.39\ \mu\)M). b) Uninduced, right bifurcation point (\(\bar{T}=24.4\ \mu\)M). c) Uninduced, initial concentrations same as the coordinates of the right bifurcation point, but the TMG concentration is slightly beyond the bistability region (\(\bar{T}=24.5\ \mu\)M).
+
+### Induction/uninduction time
+
+We compared the mean induction/uninduction times for both models, obtained from the simulations with different noise intensities and different TMG concentrations (Fig. 8). Both models are robust to fluctuations in extracellular TMG/lactose concentration. Noise-induced switching from the induced to uninduced state due to noise driving the system out of the steady state (Horsthemke and Lefever, 1984) was possible only for concentrations close to the boundaries of the bistable region.
+
+Therefore, to study the uninduction we performed simulations in which the system started within the bistable region, in the induced state very close to the left bifurcation point (Tab. 3). When the noise intensity is zero, the system remains in the steady state and the uninduction time is infinite. But as the noise intensity increases, the bistable region shifts in the direction of larger TMG concentrations, effectively destabilizing the induced state and thus decreasing the mean uninduction time (Fig. 9a).
+
+On the other hand, within the bistable region the noise-driven switching from the uninduced steady state to the induced one was impossible in the range of the noise intensities for which the small-noise expansion method is valid (Fig. 9b). To observe the transitions to the induced state, we had to set the initial conditions outside the bistable region (Tab. 3). The unstable initial positions become closer to the bistable region, and thus become more stable, as this region shifts in the direction of larger TMG concentrations. The trajectories spend more time in the vicinity of the starting point before they switch to the induced state, so the mean induction time increases as noise intensity increases (Fig. 9c). This effect is same as in the Yildirim-Mackey model (Fig. 8 b, Ochab-Marcinek (2008)).
+
+## 5 Discussion
+
+In spite of the differences between the kinetics of the Ozbudak and Yildirim-Mackey models, and differently defined TMG/lactose uptake rate functions (the noise filters), the behavior of both models is qualitatively the same. The steady-state shift due to small noise and the consequent changes in the induction/uninduction times depend on the characteristics (monotonicity, concavity) of the filtering function and not on the details of the kinetics of the system.
+
+Gaussian fluctuations of the extrinsic input signal, which enter the system through the Michaelis-Menten uptake reaction, generate an asymmetric distribution of its rate. This effect is similar as in Shahrezaei et al. (2008) and Lei et al. (2009), where Gaussian noise enters the system through an exponential function. (Note that, however, in Shahrezaei et al. (2008) the noise was too slow (slower than the fastest time-scale of the system) and in Lei et al. (2009) the noise was too strong for the small noise-expansion to be valid.) While in the latter cases the rate distribution is log-normal, the distribution the Michaelis-Menten uptake rate has an opposite skewness. Due to the asymmetry of the distribution, the mean uptake rate varies as the input noise intensity is varied. This gives rise to the shift of the steady-state concentrations of the studied reaction system.
+
+The small-noise expansion turns out to be a useful tool for prediction of the steady-state displacement due to weak and rapid noise. Within the validity range of the method, the distribution of the input noise is not important (it can be non-Gaussian as well): It is only the shape of the uptake function that determines the direction and size of the steady-state displacement for a given input noise intensity. The simplicity of this approximate calculation allows for qualitative predictions, based on experimental data only, of the response of the studied system to extrinsic noise passing the uptake reaction of a given type. Assuming that we can only measure the intensity of the input noise, all information needed to predict the steady-state displacement are: a) Experimental the graph of the system’s steady states vs. the mean input signal and b) Experimental graph of the uptake rate function vs. the mean input signal. Even if this data is not precise or already includes noise (i.e. is shifted due to noise), it enables qualitative predictions of the steady-state displacement when noise intensity increases or decreases. The shape (monotonicity and concavity) of the uptake rate function indicates the direction of the noise-induced shift of the stationary states, whose approximate positions are known from the experimental graph. In particular, when the uptake rate is of Michaelis-Menten type (monotonically increasing and concave), the graph of the system’s steady states vs. mean input signal always shifts to the right. In case of a bistable genetic switch, one can then predict that, as a consequence of such a shift, the extrinsic noise can selectively facilitate or inhibit induction and uninduction.
+
+In principle, it would be possible to measure the shift experimentally on the level of the Michaelis-Menten reaction rate \(h(P)=P/(K_{M}+P)\). The shift could be detected as the noise correction (11) to the Michaelis constant \(K_{M}\), which could be read out from the experimental plot of \(h(P)\). However, in the studied models of the _lac_ operon the noise would be probably too small for experimental detection of the shift of the reaction rate plot. But the computer simulations suggest that it would be easier to experimentally detect the shift by measurement of the switching times of the _lac_ switch.
+
+The measurements of induction/uninduction times in the simulations of two example _lac_ operon models demonstrate that the bistability of the lactose utilization mechanism is robust to fluctuations in extracellular TMG/lactose concentration. However, even as small a noise as used in this study (to fulfill the conditions of validity of the small-noise expansion) can have a significant effect on the induction/uninduction time. The values by which it increases or decreases depend on the choice of model parameters and initial conditions. The effect is the strongest when the initial conditions are close to the bifurcation points (boundaries of the bistable region). For example in the Ozbudak model, when the initial state of the system is the left bifurcation point (induced state), then the uninduction time changes from infinity (at zero noise) to \(\sim\)100 hours (when the standard deviation \(\sigma\) of the TMG fluctuations is 1.5 \(\mu\)M). On the other hand, when the initial state is the right bifurcation point (uninduced state), then the noise-driven induction is impossible. When the initial state is close to the right bifurcation point, but out of the bistable region, the induction time is increased by noise. For example, when the mean TMG concentration is 24.5 \(\mu\)M, then at zero noise the induction time is \(\sim\)190 hours, but at the noise of \(\sigma=1.7\mu\)M the induction time is \(\sim\)250 hours. These effects are qualitatively same as in the Yildirim-Mackey model (Ochab-Marcinek, 2008).
+
+Thus, the fluctuations in extracellular TMG/lactose concentration facilitate the switching off of the TMG/lactose metabolism, but at the same time they prevent the metabolism from switching on. This effect will be valid for any model of _lac_ operon, provided that the TMG/lactose uptake rate is of Michaelis-Menten type. This suggests that in the presence of noise the possibility of random switching on the metabolism of TMG/lactose is more strongly protected than the possibility of random switching it off. One can speculate whether this protection against accidental induction of metabolism is connected with preventing an unnecessary energetic effort. To answer this question, one should analyze the lactose utilization system in _E. coli_ from the energetic point of view.
+
+In other systems, a different direction of the noise-induced steady state displacement is possible. For example, when extrinsic noise enters the system through the exponential function (monotinically increasing and convex), as in Shahrezaei et al. (2008) and Lei et al. (2009), then one can predict that for weak and rapid noise the graph of the steady states vs. mean noise intensity will shift in the opposite direction than that of the systems with the Michaelis-Menten uptake. On the other hand, if the uptake rate is a Hill function \(P^{n}/(K+P^{n})\) then the formula (10) for the steady-state displacement can change its sign for different concentrations of \(P\) and, under certain conditions, bistability may emerge due to noise in place of a graded response. Similar effects are possible for systems with non-monotonic input functions generated by incoherent feed-forward loops (Kaplan et al., 2008; Kim et al., 2008).
+
+The analysis presented applies to weak and rapid noise from one dominating external source. But even in the systems where the contribution of other noises is present, the method may be of use to interpret the experimental measurements in terms of the discrimination between the effects of noises which originate from different sources.
+
+## 6 Acknowledgements
+
+The project was operated within the Foundation for Polish Science TEAM Program co-financed by the EU European Regional Development Fund: TEAM/2008-2/2. We would like to thank Dr. Paweł F. Góra (Jagiellonian University, Kraków, Poland) for valuable discussions and Edward Davis (The University of York, UK) for the help in improving the language of this article.
+
+## Appendix A Transformation of probability density functions
+
+When \(h(P)\) is an either monotonically increasing or decreasing function of a random variable \(P\), and \(P\) is given by the probability density function \(q(P)\), then the probability density function of \(h\) is given by the formula:
+
+\[p(h)=q(P(h))\left|\frac{dP(h)}{dh}\right|,\] (25)
+
+where \(P(h)\) is the inverse function of \(h(P)\). The formula is obtained by the change of variables in the integration (see e.g. Miller and Miller (2004)):
+
+\[\mathrm{Prob}(h(P_{1})<h<h(P_{2}))=\mathrm{Prob}(P_{1}<P<P_{2})=\] (26)
+\[=\int_{P_{1}}^{P_{2}}q(P)dP=\int_{h(P_{1})}^{h(P_{2})}q(P(h))\left|\frac{dP(h)}{dh}\right|dh=\int_{h(P_{1})}^{h(P_{2})}p(h)dh\]
+
+(The absolute value is needed when \(h(P)\) is decreasing.)
+
+## Appendix B Simulation details
+
+The fluctuations in the extracellular TMG/lactose concentration are modelled by the Ornstein-Uhlenbeck (OU) process (Gardiner, 1983) with reflecting boundary conditions in \(P_{t}=0\). \(\xi(t)\) is a Gaussian white noise of intensity \(\gamma\) and autocorrelation \(\langle\xi(t)\xi(s)\rangle=\delta(t-s)\):
+
+\[\frac{dP_{t}}{dt}=-\theta(P_{t}-\bar{P})+\gamma\xi(t),\ \ P_{t}\geq 0.\] (27)
+
+The correlation time of the noise \(\tau_{OU}=1/\theta\). The variance of the OU process is \(\sigma^{2}=\gamma^{2}/2\theta\). We assume a small noise intensity and \(\bar{P}\) sufficiently far from the reflecting boundary, so that the contribution of reflected ’tail’ of the Gaussian distribution can be neglected and we can use formulas for the unbounded OU noise for the mean and variance. The noise intensity is varied in the simulations by varying the value of \(\gamma\). . Numerical integration of the equations of kinetics has been done using the Euler scheme (Press et al., 1993; Mannella, 2002). The timestep \(\delta t=2\cdot 10^{-3}\ \mathrm{min}\) has been chosen significantly smaller than the time scales of the studied systems.
+
+To estimate the mean induction/uninduction time in the simulations, the time was measured until trajectories \(\vec{X}(t)\) got into a close neighborhood of the other deterministic stationary state (of a radius \(D=\sqrt{\sum_{i}\delta X_{i}^{2}}=0.1\) [scaled units] for the Ozbudak model and \(D=5\ \mathrm{\mu M}\) for the Yildirim-Mackey model (Ochab-Marcinek, 2008)). The initial points of the trajectories \(\vec{X}\) were the deterministic steady states within the bistable region, or points outside the bistable region, but close to its boundaries (see Tables 3, 4). The number of simulation runs for calculating the mean induction/uninduction time was \(N=100\) for the Ozbudak model and \(N=1000\) for the Yildirim-Mackey (Yildirim and Mackey, 2003) model.
+
+## Appendix C Tables
+
+\begin{table}
+\begin{tabular}{|l|l|}
+\hline
+\(\rho=1+\frac{R_{T}}{R_{0}}\) & 167.1 _a_ \\
+\(\alpha\) & 100.5 _a_ \\
+\(\beta_{G}(G=0)\) & 100 _a_ \\
+\(\tau_{y}\) & 216 min _b_ \\
+\(\tau_{x}\) & 1 min _c_ \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Parameters of the Ozbudak model: _a_) (Ozbudak et al., 2004), _b_) (Mettetal et al., 2006), _c_) chosen for this study within the range of \(0..35\) min reported by (Mettetal et al., 2006).
+
+\begin{table}
+\begin{tabular}{|l|l||l|l|}
+\hline
+\(\Gamma_{0}\) & \(7.25\times 10^{-7}\ \mathrm{mM/min}\) & \(\mu\) & \(0.0226\ \mathrm{min}^{-1}\) \\
+\(\alpha_{A}\) & \(1.76\times 10^{4}\ \mathrm{min}^{-1}\) & \(\tau_{B}\) & \(2.0\ \mathrm{min}\) \\
+\(\alpha_{B}\) & \(1.66\times 10^{-2}\ \mathrm{min}^{-1}\) & \(\tau_{M}\) & \(0.1\ \mathrm{min}\) \\
+\(\alpha_{L}\) & \(2.88\times 10^{3}\ \mathrm{min}^{-1}\) & \(\tau_{P}\) & \(0.83\ \mathrm{min}\) \\
+\(\alpha_{M}\) & \(9.97\times 10^{-4}\ \mathrm{mM/min}\) & \(K\) & \(7.2\times 10^{3}\) \\
+\(\alpha_{P}\) & \(10.0\ \mathrm{min}^{-1}\) & \(K_{1}\) & \(2.52\times 10^{4}\ \mathrm{mM}^{-2}\) \\
+\(\beta_{A}\) & \(2.15\times 10^{4}\ \mathrm{min}^{-}1\) & \(K_{A}\) & \(1.95\ \mathrm{mM}\) \\
+\(\beta_{L}\) & \(2.65\times 10^{3}\ \mathrm{min}^{-1}\) & \(K_{L}\) & \(0.97\ \mathrm{mM}\) \\
+\(\gamma_{A}\) & \(0.52\ \mathrm{min}^{-1}\) & \(K_{L_{e}}\) & \(0.26\ \mathrm{mM}\) \\
+\(\gamma_{B}\) & \(8.33\times 10^{-4}\ \mathrm{min}^{-1}\) & \(K_{L_{1}}\) & \(1.81\ \mathrm{mM}\) \\
+\(\gamma_{L}\) & \(0.0\ \mathrm{min}^{-1}\) & \(k_{B}\) & \(0.677\) \\
+\(\gamma_{M}\) & \(0.411\ \mathrm{min}^{-1}\) & \(k_{P}\) & \(13.94\) \\
+\(\gamma_{P}\) & \(0.65\ \mathrm{min}^{-1}\) & & \\ \hline
+\end{tabular}
+\end{table}
+Table 2: Parameters of the Yildirim-Mackey model (Yildirim and Mackey, 2003).
+
+\begin{table}
+\begin{tabular}{|l|l|l|l|l|}
+\hline
+\(\bar{T}[\mathrm{\mu M}]\) & \(x_{i}[\mathrm{scaled\ units}]\) & \(y_{i}[\mathrm{scaled\ units}]\) & \(x_{f}[\mathrm{scaled\ units}]\) & \(y_{f}[\mathrm{scaled\ units}]\) \\
+\hline
+A: 24.5 _a_ & 1.012 & 1.210 & 82.23 & 98.09 \\
+B: 24.6 _a_ & 1.012 & 1.210 & 82.44 & 98.10 \\
+C: 24.7 _a_ & 1.012 & 1.210 & 82.65 & 98.12 \\
+D: 3.39 & 13.228 & 51.700 & 0.1577 & 0.6163 \\
+E: 3.40 & 13.707 & 53.475 & 0.1580 & 0.6164 \\
+F: 3.41 & 14.012 & 54.569 & 0.1583 & 0.6164 \\ \hline
+\end{tabular}
+\end{table}
+Table 3: Initial (\(i\) subscript) and final (\(f\) subscript) concentrations of intracellular TMG (\(x\)) and permease (\(y\)) for the simulation measurements of mean switching time in the Ozbudak model. Trajectories A, B, C started from outside the bistable region, but very close to its boundaries. Initial points for these trajectories are the coordinates of the right (_a_) bifurcation point.)
+
+\begin{table}
+\begin{tabular}{|l|l|l|l|l|l|l|}
+\hline
+\(\bar{L}_{e}[\mathrm{\mu M}]\) & \(A_{i}[\mathrm{\mu M}]\) & \(L_{i}[\mathrm{\mu M}]\) & \(M_{i}[10^{-2}\mathrm{\mu M}]\) & \(A_{f}[\mathrm{\mu M}]\) & \(L_{f}[\mathrm{\mu M}]\) & \(M_{f}[10^{-2}\mathrm{\mu M}]\) \\
+\hline
+\(A:62.0^{a}\) & \(14.1\) & \(224\) & \(0.360\) & \(393\) & \(285\) & \(80.8\) \\
+\(B:63.0^{a}\) & \(14.1\) & \(224\) & \(0.360\) & \(399\) & \(289\) & \(82.5\) \\
+\(C:65.0^{a}\) & \(14.1\) & \(224\) & \(0.360\) & \(412\) & \(298\) & \(85.9\) \\
+\(D:27.9\) & \(109.0\) & \(129\) & \(9.406\) & \(4.00\) & \(94.6\) & \(0.212\) \\
+\(E:27.8\) & \(104.0\) & \(129\) & \(8.600\) & \(4.00\) & \(94.0\) & \(0.212\) \\
+\(F:27.7\) & \(96.9\) & \(128\) & \(7.521\) & \(3.98\) & \(93.7\) & \(0.212\) \\ \hline
+\end{tabular}
+\end{table}
+Table 4: Initial (\(i\) subscript) and final (\(f\) subscript) concentrations of allolactose (\(A\)), lactose (\(L\)), and mRNA (\(M\)) for the simulation measurements of mean switching time in the Yildirim-Mackey model (Yildirim and Mackey, 2003). _a_) Trajectories starting from outside the bistable region, but very close to its boundaries. Initial points for these trajectories are the coordinates of the right bifurcation point.
+
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+* Ozbudak et al. (2004) Ozbudak, E. M., Thattai, M., Lim, H., Shraiman, B., van Oudenaarden, A., 2004. Multistability in the lactose utilization network of escherichia coli. Nature 427, 737–740.
+* Page and West (1984) Page, M., West, I., 1984. The transient kinetics of uptake of galactosides into escherichia coli. biochem. Biochem. J. 223, 723–731.
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+* Wright et al. (1981) Wright, J. K., Riede, I., Overath, P., 1981. Lactose carrier protein of escherichia coli: interaction with galactosides and protons. Biochemistry 20, 6404–6415.
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+

arxiv_ground_truth/0906.4764.md

@@ -0,0 +1,316 @@
+# Cooperative Game Theoretic Bid Optimizer for Sponsored Search Auctions
+
+Sriram Somanchi
+
+Electronic Commerce Laboratory
+Computer Science and Automation
+Indian Institute Science, Bangalore
+Email: somanchi@csa.iisc.ernet.in
+
+Chaitanya Nittala
+
+Electronic Commerce Laboratory
+Computer Science and Automation
+Indian Institute Science, Bangalore
+Email: chytu@csa.iisc.ernet.in
+
+Narahari Yadati
+
+Electronic Commerce Laboratory
+Computer Science and Automation
+Indian Institute Science, Bangalore
+Email: hari@csa.iisc.ernet.in
+
+###### Abstract
+
+In this paper, we propose a bid optimizer for sponsored keyword search auctions which leads to better retention of advertisers by yielding attractive utilities to the advertisers without decreasing the revenue to the search engine. The bid optimizer is positioned as a key value added tool the search engine provides to the advertisers. The proposed bid optimizer algorithm transforms the reported values of the advertisers for a keyword into a correlated bid profile using many ideas from cooperative game theory. The algorithm is based on a characteristic form game involving the search engine and the advertisers. Ideas from Nash bargaining theory are used in formulating the characteristic form game to provide for a fair share of surplus among the players involved. The algorithm then computes the nucleolus of the characteristic form game since we find that the nucleolus is an apt way of allocating the gains of cooperation among the search engine and the advertisers. The algorithm next transforms the nucleolus into a correlated bid profile using a linear programming formulation. This bid profile is input to a standard generalized second price mechanism (GSP) for determining the allocation of sponsored slots and the prices to be be paid by the winners. The correlated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of the underlying game. Through detailed simulation experiments, we show that the proposed bid optimizer retains more customers than a plain GSP mechanism and also yields better long-run utilities to the search engine and the advertisers.
+
+Index Terms: Bid Optimizer, Sponsored Search, Cooperative Game Theory, Nash bargaining, Nucleolus.
+
+## I Introduction
+
+Sponsored search auctions have been studied extensively in the recent years due to the advent of targeted advertising and its role in generating large revenues. With a huge competition in providing the sponsored search links, the search engines face an imminent problem which can be called as the _retention problem_. If an advertiser (or alternatively bidder) does not get satisfied because of not getting the right number of clicks or the anticipated payoff, he could drop out of the auction and try sponsored links at a different search engine.
+
+### _Motivation: Retention of Advertisers in Sponsored Search Auctions_
+
+Our motivation to study the retention problem is driven by the compulsions faced by both the search engine and the advertisers.
+
+From the advertisers’ perspective, choosing their maximum-willingness-to-pay such that they get an attractive slot subject to their budget constraints is a challenging problem. The search engines can use various mechanisms for the sponsored search auction as described in [6, 11] but the most popular mechanism is the generalized second price (GSP) auction since it is simple and yields better revenue to the search engine. In the most simple version of GSP, where there are \(k\) slots and \(n\) advertisers (for simplicity assume \(k\leq n\)), the allocation and payment rule are as follows. The allocation rule is that \(n\) advertisers are ranked in descending order based on their bids, with ties broken appropriately, and top \(k\) advertisers’ advertisements are displayed. The payment rule is that every advertiser needs to pay bid amount of the advertiser who is just below his slot and last advertiser is charged the highest bid that has not won any slot. If the non-truthful GSP auction [13] is used by the search engine, the bidders will have an incentive to shade their bids. The bidders would not want to use complicated and computationally intensive bidding strategies as the bidding process is done many times (typically thousands of times) in a day. These advertisers generally build their own software agents or employ third party software agents, which adjust and readjust the bid values on behalf of these advertisers. The bidders typically specify their maximum willingness-to-pay for their keywords for any given day. Hence each keyword has a specific set of bidders bidding on it for the whole day. This scenario constitutes a repeated game between all the bidders bidding for that keyword. In this game, bidders who cannot plan their budget effectively may experience less utilities and thus may drop out of the system.
+
+We now turn to the search engine’s perspective of the retention problem. When the bidders try to know each others’ valuations by submitting and resubmitting bids, they may find a set of strategy profiles which may yield all of them better payoffs. This may lead to collusion among the bidders. Folk theorems [4] suggest that players may be able to increase individual profits by colluding thereby decreasing the search engine’s revenue. Even though the bidders in the keyword auctions are competitors, this collusion against the search engine could be stable. Vorobeychik and Reeves [10] studied this phenomenon and illustrated a particular collusive strategy which is better for all the bidders (hence worst for the search engine) and can be sustainable over a range of settings. Feng and Zhang [3] showed that dynamic price competition between competing advertisers can lead to collusion among them. However, in this dynamic scenario, when the discounted payoffs of the bidders under the collusive strategy are considered, the stability of collusion depends inversely on the number of bidders [4]. That is, the lower the number of bidders in the system, the higher is the stability of the collusion. This motivates us to study the bidder retention problem for the search engine.
+
+Also, due to exponential growth in the space of online advertising and intense competition among the search engine companies, the switching cost for the advertisers to change from one search engine to another is almost zero [6]. Hence, it is imperative for the search engine companies to retain their advertisers to safeguard their market share. Driven by this, the search engine companies have introduced many value added tools, such as bid optimizer, to maximize the bang-per-buck for the bidders. In what follows, we describe the bid optimizer’s role in solving the retention problem.
+
+### _Bid Optimizers_
+
+A _bid optimizer_ is a software agent provided by the search engine in order to assist the advertisers. The bidders are required to provide to the bid optimizer a target budget for the day and a maximum willingness-to-pay. Bid optimizers, currently provided by the search engines, promise to maximize the revenue of advertisers by adjusting the bid amount in each round of the auction based on the projected keyword traffic and remaining budget.
+
+It can be seen that the decisions made by the bid optimizer are crucial to both the search engine and the set of advertisers, who choose to use the bid optimizer. Hence, the objective of a typical bid optimizer is to strike a balance between reduction in revenue of the search engine company versus increase the retention of advertisers. This objective is achieved by providing enhanced utilities to the advertisers, thus ensuring retention of customers, thereby sustaining high levels of revenue to the search engine company in the long run. Designing such intelligent bid optimizers is the subject of this paper.
+
+There are some problems involved in designing bid optimizers.
+
+1. 1.For the search engine, maximizing its short-term revenue (that is, its payoff in a one-shot game) seems to be a viable option. But here, the lower valuation bidders are denied slots due to allocative efficiency concerns. For the bidders, as shown by Cary _et al_[1], where all the high valuation bidders use a particular greedy strategy, it has been proved that none of the bidders except the top \(k\) bidders get the slots after a certain number of rounds of the auction. The above phenomenon can permanently drive away low valuation bidders from the search engine.
+2. 2.Dropping out of the search engine to get better utilities in another search engine is a possible option for the bidders. The low valuation bidders drop out after not getting slots for a certain period of time. The higher valuation bidders can observe this trend and shade their maximum-willingness-to-pay or collude to get better utilities. This may result in the search engine losing revenue. This is a threat to the search engine from the bidders. However, if a large number of bidders remain in the system, the collusion is not stable. The intuition for this is that, high valuation bidders cannot reduce their bids sharply, since they will have the fear of undercutting the lower valuation bidders present in the system and thus losing out on their slots.
+
+Hence, we propose that retaining more number of bidders solves all the problems discussed above. The dependence of the search engine and the bidders on each other for mutual benefit motivates us to use a cooperative approach in general. The above threat model naturally directs us towards using a Nash bargaining model in particular. Our solution can be seen as associating the bid optimizer to a keyword rather than bidders as done by the existing bid optimizers. The overall model of the bid optimizer is depicted in Figure 1.
+
+### _Contributions and Outline of the Paper_
+
+Figure 1: Proposed Bid Optimizer
+
+In this paper, we propose a bid optimizer that uses many ideas from cooperative game theory. The bid optimizer is shown in Figure 1.
+
+* •The inputs to the bid optimizer are the willingness-to-pay values (or valuations) of the bidders.
+* •The output of the bid optimizer is a correlated bid profile, which, when input to a standard GSP auction mechanism, yields utilities to the search engine and the advertisers satisfying the goals set forth in the paper.
+* •The bid optimizer first formulates a characteristic form game involving the search engine and the advertisers. The value for each coalition is defined based on a novel Nash Bargaining formulation with the search engine as one player and a virtual player aggregating all advertisers in that coalition as the other player. The idea of using Nash Bargaining is to ensure a fair share for the search engine and the advertisers.
+* •The nucleolus of the above characteristic form game is selected as the utility profile for the search engine and the advertisers. The choice of nucleolus is based on key considerations such as, bidder retention, stability, and efficiency.
+* •The utility profile represented by the nucleolus is mapped to a correlated bid profile that satisfies individual rationality, retention, stability and efficiency. A linear programming based algorithm is suggested for this purpose.
+
+We carry out experiments to demonstrate the viability and efficacy of the proposed bid optimizer. We show, using a credible bidder drop out model, that the proposed bid optimizer has excellent bidder retention properties and also yields higher long-run revenues to the search engine, when compared to the plain GSP mechanism.
+
+The outline of the paper is as follows. Section II-A1 presents the details of the bid optimizer and introduces the model. In Section II-A3, we present a bid optimization algorithm which uses the Nash Bargaining approach for ensuring the retention of bidders in the system. We then map this fair share for the aggregated bidder to a correlated bid profile in Section II-C. We analyze the properties of our method in Section II-D. We present our experimental results in Section III and conclude the paper in Section IV.
+
+## II Our Approach to Bid Optimization
+
+In this section, we present our algorithm for bid optimization. The algorithm can be divided into three phases as shown in Figure 1: (1) Characteristic form game definition using Nash bargaining, (2) Computing the utility vectors for the players and (3) Inverse mapping of the utility vector into a correlated bid profile. These are discussed in the following sections. The notation in the remainder of the paper is presented in Table I.
+
+### _Characteristic Form Game_
+
+#### II-A1 The Model
+
+The sponsored search auction scenario we consider has \(n\) bidders competing for \(k\) slots of a keyword. We assume that the probability that a bidder \(i\) gets clicked on the \(j\)th slot (or the click-through rate \(CTR_{ij}\)) is independent of the bidder \(i\),that is, \(CTR_{ij}=\beta_{j}\) and we also assume that \(\beta_{1}\geq\beta_{2}\geq\ldots\geq\beta_{k}\). Each bidder \(i\) specifies his maximum willingness-to-pay \(\overline{s}_{i}\) to the bid optimizer. The bid optimizer takes as input all the \(\overline{s}_{i}\)’s of the bidders and suggests them a correlated bid profile. This bid optimization algorithm needs to be invoked only when the number of bidders in the system or their willingness-to-pay change. We also assume that the bid of the player \(i\) could be any real number in \([0,\overline{s}_{i}]\).
+
+Given the above model, we define a bargaining problem ¹ between the search engine and the aggregated bidder and analyze its properties which will help us in formulating a characteristic form game.
+
+Footnote 1: Refer Appendix for the definition of Nash bargaining problem
+
+\begin{table}
+\begin{tabular}{|l|l|}
+\hline
+**Notation** & **Explanation** \\
+\hline \hline
+\(A\) & Auctioneer \\
+\hline
+\(B\) & Aggregated bidder \\
+\hline
+\(n\) & Total number of players \\
+\hline
+\(k\) & Total number of slots. We assume \(k<n\) \\
+\hline
+\(N\) & Set of bidders \(\{1,2,\ldots,n\}\) \\
+\hline
+\(K\) & Set of slots \(\{1,2,\ldots,k\}\) \\
+\hline
+\(\overline{s}_{i}\) & Maximum willingness-to-pay of advertiser \(i\) \\
+\hline
+\(S_{i}\) & Strategy set of bidder \(i,\:\:[0,\overline{s}_{i}]\) \\
+\hline
+\(S\) & Set of all bid profiles \(S_{1}\times S_{2}\times\ldots\times S_{n}\) \\
+\hline
+\(s\) & Bid profile \((s_{1},s_{2},\ldots,s_{n})\in S\) \\
+\hline
+\(u_{i}(s)\) & Utility of bidder \(i\) on bid profile \(s\) \\
+\hline
+\(U_{A}(s)\) & Utility of the auctioneer in the \\
+\hline
+ & Nash bargaining formulation for bid profile \(s\) \\
+\hline
+\(U_{B}(s)\) & Utility of the aggregated bidder in the \\
+\hline
+ & Nash bargaining formulation for bid profile \(s\) \\
+\hline
+\(\overline{U}_{A}\) & \(\max_{s\in S}U_{A}(s)\) \\
+\hline
+\(\overline{U}_{B}\) & \(\max_{s\in S}U_{B}(s)\) \\
+\hline
+\(\beta_{j}\) & Click through rate of any bidder in the \(j^{th}\) slot \\ \hline
+\end{tabular}
+
+\end{table}
+TABLE I: Notation
+
+#### II-A2 Characterization of the Nash Bargaining Solution
+
+The motivation for a cooperative approach is the dependence of the search engine and bidders on each other for their mutual benefit. Given this, the motivation behind choosing a bargaining approach is that the amount of short-term loss (or in other words, the investment of the search engine) for the auctioneer should be chosen based on the bidders present in the system. The Nash bargaining approach provides a framework for this amount to be chosen by the search engine by considering all the bidders as one aggregate agent whose bargaining power depends on all the maximum willingness-to-pay of all the bidders present in the system.
+
+The utility of the aggregated bidder is the sum of the utilities of all the bidders over all possible allocations of slots (outcomes). Now, the bargaining utility space becomes the two dimensional Cartesian space which consists of the utility of auctioneer on one axis and the aggregate bidder’s utility on the other axis. Hence a bargaining solution on this space provides a good compromise for the search engine from its maximum possible revenue and thus gives the required investment of the search engine.
+
+The bargaining space is defined in two dimensional Cartesian space, with utility of auctioneer \(U_{A}(s)\) along the \(x-\)axis and the utility of aggregated bidder \(U_{B}(s)=\sum_{i=1}^{n}u_{i}(s)\) along the \(y-\)axis. Let \(\overline{U}_{A}\) and \(\overline{U}_{B}\) be the maximum possible utilities of the auctioneer and the aggregated bidder respectively. It can be clearly seen that the value \(\overline{U}_{A}\) is attained for the bid profile \(s=(\overline{s}_{1},\overline{s}_{2},\ldots,\overline{s}_{n})\) for which the corresponding \(U_{B}(\overline{s})=\sum_{i=1}^{n}\left(\sum_{j=1}^{k}\beta_{j}y_{ij}(\overline{s})\right)(\overline{s}_{i}-\overline{s}_{i+1})=U_{B}^{{}^{\prime}}\) (say). Similarly, the bid profile \(s=(0,\ldots,0)\) yields the utility pair \((0,\overline{U}_{B})\). Since it is theoretically possible that all the bidders can collude and bid \((0,0,\ldots,0)\), we choose the point \((0,0)\) in this Nash bargaining space as the disagreement point. Ramakrishnan _et. al_ studied this problem in [8] and characterized the solution \((U_{A}^{*},U_{B}^{*})\) to this Nash bargaining(NBS) as
+
+\[(U_{A}^{*},U_{B}^{*})=(\overline{U}_{A},U_{B}^{{}^{\prime}})\:\:\:if\:\overline{U}_{A}\leq\frac{\overline{U}_{B}}{2}\]
+
+\[\hskip 42.67912pt=\left(\frac{\overline{U}_{B}}{2},\frac{\overline{U}_{B}}{2}\right)\:\:\:otherwise\]
+
+#### II-A3 Definition of the Characteristic Form Game
+
+We use the above model to define Nash bargaining solution \(NBS(N)=U_{A}^{*}+U_{B}^{*}\) where \(N\) is the set of bidders participating in the auction.
+
+Let \(N=\{1,2,\ldots,n\}\) be the set of all bidders and let \(0\) represent the search engine. The characteristic form game \(\nu:2^{N\cup\{0\}}\rightarrow\Re\) for each coalition \(C\subseteq N\cup\{0\}\) is now defined as
+
+\[\nu(C)=NBS(C)\:\:\:if\:0\in C\]
+
+\[\hskip 7.11317pt=0\hskip 7.11317ptotherwise\hskip 9.67383pt\]
+
+where \(NBS(C)\) is defined as above. If the search engine is not a part of the coalition, its worth is zero since the players cannot gain anything without the search engine displaying their ads. Otherwise, we associate the sum of utilities in the corresponding Nash bargaining bid profile for that coalition with the search engine as the worth of each coalition. This characteristic function \(\nu\) defines the bargaining power of each coalition with the search engine.
+
+### _Computing a Utility Vector for the Players: Use of Nucleolus_
+
+Since there is an aggregation of the bidders’ revenue taking place in the NBS, we map the utility of the aggregated bidder in the Nash bargaining solution to a correlated bid profile. The NBS gives an aggregate amount of investment the search engine has to make on all the bidders. This investment increases the utility of the aggregated bidder. This utility has to be distributed to the bidders in a way that our goal of retention is reached. Ideally, we would like the allocation to have the following properties.
+
+* •The bidders must not have incentive for not participating in the bid optimizer (individual rationality-IR).
+* •It must retain as many bidders as possible(retention).
+* •The bidders must not have the incentive to shade their maximum willingness to pay (incentive compatibility).
+* •It should be stable both in the one-shot game of GSP and in the cooperative analysis (stability).
+* •It should divide the entire worth of the grand coalition among all the bidders (efficiency).
+
+There are several solution concepts in cooperative game theory that one could employ here,for example, the core, the Shapley value, the nucleolus, etc. We believe the nucleolus is clearly the best choice that satisfies a majority of the above properties. Since nucleolus is defined as the unique utility vector which makes the unhappiest coalition as less unhappy as possible [9], and given that the nucleolus is always in a non-empty core, it is the utility vector that retains the most number of bidders if the core is empty and is the most stable one retaining all the bidders if the core is non-empty. We compute the nucleolus by solving a series of linear programs [4, 5] and obtain the utility vector \((x_{1},x_{2},\ldots,x_{n})\) for the \(n\) players and the search engine (\(x_{0}\)).
+
+### _Mapping the Utility Vector to a Correlated Bid Profile_
+
+#### II-C1 Obtaining a locally envy-free bid profile for each valid coalition
+
+To satisfy the stability criterion in the non-cooperative sense, and ensure truthful participation of all the bidders in the proposed bid optimizer, we aim to find out locally envy-free bids for each of the \(n\choose k\) possible sets of winning bidders. For finding these bids, consider a subset of \((k+1)\) bidders and allocate slots to the bidders in this subset in the sorted order of their willingness-to-pay values to satisfy the requirement for the locally envy-free equilibrium. Now, the bids can be calculated as follows. The \((k+1)^{th}\) bidder bids the reserve price (assumed to be \(0\) here without loss of generality). The bid of the \(k^{th}\) bidder (who pays \(\overline{s}_{k+1}\)) is now calculated by solving for \(b_{k}\) in \(\beta_{k}(\overline{s}_{k}-b_{k+1})=\beta_{k-1}(\overline{s}_{k}-b_{k})\) to satisfy the envy-freeness. Once we obtain \(b_{k}\), we proceed recursively by replacing the \(b_{k+1}\) by \(b_{k}\) and \(k\) by \((k-1)\) in the above equation to get \(b_{k-1}\) and so on till we get the bids of all the \(k\) players. Note that the bid of the first player does not have a role here as long as it is greater than the next highest bid. Thus we obtain _a_ set of bids which are in locally envy-free equilibrium.
+
+#### II-C2 Obtaining a correlated bid profile
+
+The solution given by the nucleolus provides a utility for each bidder. This cannot be used directly in the GSP auction of the search engine. Towards this end, we map the nucleolus to a correlated bid profile which defines the required rotation among the bidders for occupying the slots. This correlated bid profile is what is finally suggested by the bid optimizer, which retains the maximum number of advertisers without hurting the search engine.
+
+The characterization of a correlated bid profile corresponds to assigning the probabilities associated with each of the bid profiles associated with the bidders. There exist several algorithms in general, for finding the correlated bid profile. But we would like to exploit the structure of the problem and obtain a simpler solution without going into the complex details about modifying the ellipsoid algorithm as done in most of the work in this area. See [7] for example. Any correlated strategy we consider here has a subset of size \(k\) bidders bidding their corresponding LEF (locally envy-free) bids (obtained in the previous section) and all other bidders bidding the reserve price. Considering only these \(n\choose k\) strategy profiles corresponding to each subset of size \(k\) bidders winning the slots would suffice since they exhaust all the possible outcomes of the underlying GSP auction.
+
+The probability distribution which yields the utilities suggested by the nucleolus to the players is any distribution which satisfies the constraints that it is a probability distribution, it is individually rational for each player and it must yield the payoffs suggested by the nucleolus to the bidders subject to their budget constraints. This can be obtained by solving a linear program as follows.
+
+\[\min\hskip 14.22636pt\sum_{i\in N\cup\{0\}}z_{i}+\sum_{i\in N}\overline{s}_{i}\left(x_{i}-\left(\sum_{\begin{subarray}{c}C\subseteq N,\:\mid C\mid=k\\ i\in C,\:j=y_{iC}\end{subarray}}p_{C}\beta_{j}(\overline{s}_{i}-b_{j})\right)\right)\hskip 159.3356pt\]
+
+subject to
+
+\[\forall i\in N\:\:\:\:\:\:z_{i} \geq \left(\sum_{\begin{subarray}{c}C\subseteq N,\:\mid C\mid=k\\ i\in C,\:j=y_{iC}\end{subarray}}p_{C}\beta_{j}(\overline{s}_{i}-b_{j})\right)-x_{i}\]
+\[\forall i\in N\:\:\:\:\:\:z_{0} \geq \left(\sum_{\begin{subarray}{c}C\subseteq N,\:\mid C\mid=k\end{subarray}}p_{C}\sum_{\begin{subarray}{c}i\in C,\:j=y_{iC}\end{subarray}}\beta_{j}b_{j+1}\right)-x_{0}\]
+\[\forall i\in N\:\:\:\:\:\:z_{0} \geq x_{0}-\left(\sum_{\begin{subarray}{c}C\subseteq N,\:\mid C\mid=k\end{subarray}}p_{C}\sum_{\begin{subarray}{c}i\in C,\:j=y_{iC}\end{subarray}}\beta_{j}b_{j+1}\right)\]
+\[\forall i\in N\:\:\:\:\:\:z_{i} \geq x_{i}-\left(\sum_{\begin{subarray}{c}C\subseteq N,\:\mid C\mid=k\\ i\in C,\:j=y_{iC}\end{subarray}}p_{C}\beta_{j}(\overline{s}_{i}-b_{j})\right)\]
+\[p_{C}\beta_{j}(\overline{s}_{i}-b_{j}) \geq 0\:\:\:\forall C\subseteq N\:\:\forall i\in C\:\:\forall j\in K\]
+\[\sum_{C\subset N}p_{C} = 1\]
+\[\forall C\subset N\:\:\:\:\:\:p_{C} \geq 0\]
+
+where \(y_{iC}\) denotes the slot that player \(i\) wins in a locally envy-free allocation if only the set \(C\) of players were to win all the slots.
+
+The linear program maps the utility vector suggested by the nucleolus into a correlated bid profile. The objective function minimizes the difference between the utility suggested by nucleolus and the expected utility in the correlated bid profile for each player. The minimization of difference leads to two constraints for each player. This is because for any two variables \(x\) and \(y\),
+
+\[\min\mid x-y\mid\]
+
+is the same as
+
+\[\min z\]
+
+ subject to
+
+\[z\geq x-y\]
+
+\[z\geq y-x\]
+
+In the minimization, the higher valuation bidders are given a preference over the lower valuation bidders. This is done by weighting each player’s difference from the nucleolus in the objective function by their valuation. This is a heuristic to ensure that the error in the inverse mapping of the utility vector to a correlated bid profile is biased towards the higher valuation bidders so that they voluntarily participate in the bid optimizer. Since the only problem to Individual rationality is when the higher valuation bidders shade their willingness-to-pay, this weighing gives the incentive for them to reveal their true valuations.
+
+In the objective function, we minimize the _difference_ (this is done by the first \(4\) constraints of the linear program) between the utility vector and the obtained expected utility in the above linear program since the restriction of the bid profiles to the set of locally envy-free equilibria may not have a feasible correlated bid profile. The minimization is done in such a way that the higher valuation bidders obtain relatively higher utility (due to the weights given to the difference in the objective function) than the lower valuation bidders in case the optimal value of the objective function is non-zero. This is a heuristic to ensure that the error in the inverse mapping of the utility vector to a correlated bid profile is biased towards the higher valuation bidders so they voluntarily participate in the bid optimizer.
+
+### _Properties of the Proposed Solution_
+
+The properties of the proposed solution are as follows:
+
+* •The proposed solution has the bidders participating voluntarily in the bid optimizer for the following reasons. (i) The auctioneer is benefited since he has a guaranteed revenue of at least what is suggested by the nucleolus. (ii) The high valuation bidders are benefited since they are offered the same slots at a relatively lower price. Also, since the nucleolus tries to retain the grand coalition intact, it will be individually rational for the high valuation bidders to participate in the bid optimizer rather than to deviate and bid higher. (iii) The lower valuation bidders are benefited because they get more slots and hence more clicks and their campaign is more effective. Thus, the utility of every player increases and the individual rationality (IR) condition is satisfied.
+* •The bids suggested are in a locally envy-free equilibrium of the game and also are in the core since the nucleolus is always in core if the core is non-empty. This indicates that the proposed solution is strategically stable. In other words, no one can profitably deviate unilaterally from the solution proposed by the bid optimizer.
+* •Retention and efficient division of the worth of the grand coalition are guaranteed by the nucleolus since it is the allocation which tries to retain the grand coalition.
+* •Truthfulness is difficult to satisfy, given that the GSP mechanism is non-truthful. But note that the lower valuation bidders have no incentive to shade their bids. If they do so, they may lose their slots or run into negative utilities. Hence there is a problem only when the higher valuation bidders do not participate in the bid optimizer or they understate their willingness-to-pay. The higher valuation bidders cannot understate their valuations by a large amount since they have a threat of losing their slots to lower valuation bidders who are retained in the system. Also, the higher valuation bidders are given more benefits to participate in the bid optimizer and it is individually rational for them to participate in the bid optimizer.
+
+Hence this solution satisfies all the properties which were mentioned in Section II-B.
+
+## III Experimental Results
+
+This section presents simulation based experimental results to explore the effectiveness of the approach presented in the paper. First we start with a model for the drop outs of bidders.
+
+### _Bidder drop out model_
+
+The bidders drop out if they do not get enough slots (or alternatively clicks) consistently over a period of time. The conditional probability that a bidder drops out given that he did not get a slot in a round may vary from bidder to bidder. Also, the positions of slots occupied by the bidders in the previous few rounds of auction could play an important role in the dropping out of a bidder.
+
+To model this behavior of the bidder dropping out based on the outcomes of the previous auctions and giving more importance to recent outcomes, we propose a discounted weighting of the outcomes of the previous auctions to compute the probability that a bidder will continue in the next round of the auction. This model also captures the myopic human behavior that the bidder’s choice is dependent on only the recent outcomes. That is, the history the bidder looks into, before taking a decision to continue or not for the next round is limited. The amount of history however depends on the bidder in the form of his discounting factor. Let \(x_{-i}\in\{0,1\}\) denote the outcome of the \(i^{th}\) previous round. A “\(1\)“ denotes that the bidder received a click in the \(i^{th}\) previous auction with a zero indicating otherwise. We propose that the probability that the bidder will participate in the next round is given by \(\frac{\sum_{i=1}^{\infty}\gamma^{i}x_{-i}}{\sum_{i=1}^{\infty}\gamma^{i}}=(1-\gamma)\sum_{i=1}^{\infty}\gamma^{i}x_{-i}\) where \(\gamma\) is the discount factor of the bidder. To see how the myopic nature of the bidders is captured, suppose that the bidder’s discount factor \(\gamma=0.95\). The discount factor for the \(101^{st}\) round will be \(0.006\) which is negligible. Hence the bidder’s decision is dependent on at most \(100\) previous auctions. Thus, the discount factor decides the nature of the bidder.
+
+### _Experimental Setup_
+
+Given a fixed set of CTRs, the valuations of the bidders are chosen close enough to each other, to analyse our model in competitive environment. The retention problem, is fundamental in competitive environment, as search engine needs to retain the bidders and allow them compete in further auctions. The results indicate that the proposed bid optimizer not only retains a higher number of bidders than the normal GSP but also yields better cumulative revenue to the search engine in the long run.
+
+Figure 2: Cumulative revenue of the search engine
+
+### _Cumulative Revenue of the Search Engine_
+
+First we consider the cumulative revenue of the search engine for comparing the non-cooperative bidding and using cooperative bid optimizer. We consider \(10\) advertisers with \(5\) slots to be allocated.We run successive auctions and find the cumulative revenue of the search engine after each auction using the two approaches. We run this experiment until the change in the average cumulative revenue after each auction becomes acceptably small. Figure 2 shows a comparison of the cumulative revenue of the search engine under the proposed approach with that of a standard GSP auction.
+
+It can be seen in Figure 2 that though initially the non-cooperative approach (GSP) yields more revenue, after a few runs, the cooperative bid optimizer, with all the solution vectors, starts outperforming. Initially the GSP outcome is better, as the advertisers are bidding their maximum willingness to pay, and hence the search engine gets high levels of revenue. However, as the utilities of the advertisers are less in the case of the non-cooperative approach, they start dropping out of the auction and hence in a long run, the cumulative revenue starts declining compared to the cooperative bid optimizer. This is the adverse effect of the dropping out of the advertisers leading to the retention problem.
+
+Figure 3: Number of bidders retained in the system
+
+### _Number of Bidders Retained in the System_
+
+After each auction, we compute the average number of advertisers retained, to analyze the retention dynamics in the system. Figure 3 gives a comparison between using the cooperative bid optimizer and the GSP approach.
+
+In Figure 3, it can be observed that there are considerably more number of advertisers retained in the system when using the solution vector suggested by the cooperative bid optimizer in comparison to the GSP approach. This explains the reason for the dip in the non-cooperative cumulative revenue of the search engine.
+
+Through experimentation, we are able to demonstrate revenue increase for the search engine in the long run and also reduction of the retention problem considerably. It should be noted that even though the raise is not substantial, its impact on retaining and attracting the advertisers and thereby other indirect advantages are immense.
+
+## IV Summary and Future Work
+
+We have proposed a bid optimizer for sponsored keyword search auctions which leads to better retention of advertisers by yielding attractive utilities to the advertisers without decreasing the long-run revenue to the search engine. The bid optimizer is a value added tool the search engine provides to the advertisers which transforms the reported values of the advertisers for a keyword into a correlated bid profile. The correlated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of the underlying game. Through detailed simulation experiments, we have shown that the proposed bid optimizer retains more customers than a plain GSP mechanism and also yields better long-run utilities to the search engine and the advertisers.
+
+The experiments were carried out with a model that captures the phenomenon of customer drop outs and showed that our approach produces a better long run utility to the search engine and all the advertisers. The proposed bid optimizer is beneficial for both the bidders and the search engine in the long run.
+
+We considered GSP auction, which is popularly run in most of the search engines, in our analysis. However, it would be interesting to look at the effects of other auction mechanisms like VCG auctions on the overall process.
+
+The other important components like budget optimization and ad scheduling are involved in sponsored search auctions. We would further like to combine these components with our cooperative bid optimizer.
+
+## References
+
+* [1] M Cary, A Das, B Edelman and I Giotis, K Heimerl, A R Karlin, C Mathieu, and M Schwarz.
+Greedy bidding strategies for keyword auctions.
+In _Proceedings of the Eighth ACM Conference on Electronic Commerce_, pages 57–58, 2007.
+* [2] B Edelman and M Ostrovsky.
+Strategic bidder behavior in sponsored search auctions.
+_Decision Support Systems_, 43(1):192–198, 2007.
+* [3] J Feng and X Zhang.
+Dynamic price competition on the Internet: advertising auctions.
+In _Proceedings of the Eighth ACM Conference on Electronic Commerce_, pages 57–58, 2007.
+* [4] A Mas-Colell, M D Whinston, and J R Green.
+_Microeconomic Theory_.
+Oxford University Press, Oxford, 1995.
+* [5] R B Myerson.
+_Game Theory: Analysis of Conflict_.
+Harvard University Press, Cambridge, Massachusetts, 1997.
+* [6] N Nisan, T Roughgarden, E Tardos and V V Vazirani.
+_Algorithmic Game Theory_.
+Cambridge University Press, 2007.
+* [7] C H Papadimitriou.
+Computing correlated equilibria in multi-player games.
+In _Proceedings of the thirty-seventh annual ACM symposium on Theory of computing_, Baltimore, MD, USA, May 22-24 2005.
+* [8] K Ramakrishnan, D Garg, K Subbian, and Y Narahari.
+A Nash bargaining approach to retention enhancing bid optimization in sponsored search auctions with discrete bids.
+In _Fourth Annual IEEE Conference on Automation Science and Engineering (IEEE CASE)_, Arizona, USA, 2008.
+* [9] P D Straffin.
+_Game Theory and Strategy_.
+Mathematical Association of America, New York, 1993.
+* [10] Y Vorobeychik and D M Reeves.
+Equilibrium analysis of dynamic bidding in sponsored search auctions.
+In _Proceedings of Workshop on Internet and Network Economics (WINE)_, 2007.
+* [11] Y Narahari, D Garg, R Narayanam, H Prakash.
+_Game Theoretic Problems in Network Economics and Mechanism Design Solutions_.
+Springer Series in Advanced Information and Knowledge Processing, 2009.
+* [12] J F Nash Jr.
+_The bargaining problem_.
+Econometrica, 18:155–162, 1950.
+* [13] B Edelman, M Ostrovsky, and M Schwarz.
+_Internet advertising and the generalized second price auction: Selling billons of dollars worth of keywords_.
+American Economic Review, 97(1):242–259, 2007.
+
+## Appendix
+
+Nash [12] proposed that there exists a unique solution function \(f(F,v)\) for every two person bargaining problem, that satisfies the following 5 axioms - _Pareto strong efficient, Individual Rationality, Symmetry, Scale Covariance, and Independence of Irrelevant Alternatives. The solution function is_
+
+\[f(F,v)\in{\rm argmax}_{(x_{1},x_{2})\in F}((x_{1}-v_{1})(x_{2}-v_{2}))\]
+
+_where, \(x_{1}\geq v_{1}\) and \(x_{2}\geq v_{2}\) and the point \(v=(v_{1},v_{2})\) is known as the point of disagreement. There are several possibilities for choosing the disagreement point \(v\). The three popular choices are those based on (1) a minimax criterion, (2) focal equilibrium, and (3) rational threats. As part of this paper, we use the rational threats to identify disagreement point \(v\)[5]. For more details please refer to the books [5][9]._
+

arxiv_ground_truth/0906.5489.md

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+# Improved and Developed Upper Bound of Price of Anarchy in Two Echelon Case
+
+Takashi Shinzato and Ikou Kaku
+Department of Management Science and Engineering
+Graduate School of Systems Science and Technology, Akita Prefectual University
+shinzato@akita-pu.ac.jp  ikou\(\_\)kaku@akita-pu.ac.jp
+
+###### Abstract
+
+Price of anarchy, the performance ratio, which could characterize the loss of efficiency of the distributed supply chain management compared with the integrated supply chain management is discussed by utilizing newsvendor problem in single period which is well-known. In particular, some of remarkable distributed policies are handled, the performance ratios in each case which have been investigated in the previous works are analyzed theoretically and the tighter upper bound of price of anarchy and the lower bound are presented. Furthermore our approach is developed based on a generalized framework and a geometric interpretation of price of anarchy is appeared via the literature of convex optimization.
+
+Index Terms: newsvendor problem, price of anarchy, convex optimization, inequalities, geometric interpretation, autoregression model with \(\chi\) square noise
+
+## I Introduction
+
+Measuring the efficiency of supply chains plays an important role in operations management because there exit many complicated factors (various contracts, order policies and physical structures) which could influence the decision making process. In practice, several formulae have been reported in most of the earlier studies [3, 5, 8, 14, 19, 20]. In history, bullwhip effect, which could compare the variance of orders with that of demands, has been widely used to evaluate the efficiency of supply chain management in the stationary market [5]. Although it is straightforward to forecast the optimal order in the following term via finite instance of demand and order observed according to the manner of bullwhip effect, if the market behaves in equilibrium, it is too hard to determine the optimal order of goods in the typical case with respect to the given supply chain. While, as another approach, recently numerical experiments in Cachon (2004) indicate that the relative efficiency of a two-stage decentralized supply chain could be as low as 70-90\(\%\) for either push or pull configurations with price-only contract policy [4]. Perakis and Roels (2007) mathematically extends Cachon’s works into several different supply chain configurations such as push or pull inventory positioning, two or more stages, serial or assembly systems, single or multiple competing suppliers, and single or multiple retailers [14]. By introducing the concept of price of anarchy into supply chain, which can measure the ratio of the performance of the centralized system to the worst performance of the decentralized system (Koutsoupias and Papadimitriou 1999 and Papadimitriou 2001), they found that even in a two-stage supply chain the loss of efficiency might be more than 42\(\%\) under the same conditions of price-only contract, and pointed that a pull inventory configuration generally outperforms a push configuration [11, 13, 20].
+
+The contribution of Perakis and Roels (2007) is one of the most pioneer investigations in quantifying the performance of supply chain. However, since their argument is hard to be guaranteed mathematically, rigorously and sufficiently anywhere, we need to improve several points in their logic for more practical use. In this paper, tighter upper bound of price of anarchy is theoretically proved and a lower bound of price of anarchy is presented firstly due to more accurate treatment. Furthermore a geometric interpretation of price of anarchy is also demonstrated which can give a clear illustration of the loss of efficiency of the given distributed supply chain. We only consider the both bounds of price of anarchy in two echelon case because of the most fundamental case, however it turns out that our approach would be simply applied in more complicated case.
+
+This remainder of the present paper is organized as follows; in the next section, the newsvendor problem is presented for simplicity of our discussion, and the derived results in the previous work [14] is introduced briefly. Furthermore an improved upper bound and a lower bound of the loss of efficiency are explained. Section III addresses a generalization of newsvendor problem and assesses the optimal inventory levels and the performance ratios in each mode. In Section IV, a geometric interpretation of price of anarchy is intuitively provided and we confirm that the performance ratio and the analytical procedure handled here are possible to be one of the most unbeatable frameworks. The final section is devoted to a summary and future work.
+
+## II Model setting
+
+Concerned with distribution of goods and information sharing, supply chain management is one of the most vital interests in the cross-disciplinary fields. As one of the most pivotal topics, we discuss here the manner of the inventory management theoretically, in particular, (1) how to determine the optimal inventory level with respect to the given supply chain management and (2) how to assess the loss of efficiency of suboptimal policy. For simplicity of our argument, we restrict the model which is well-defined and is introduced below. Therefore applying our demonstration, one could indeed improve and develop this approach so as to resolve more practical case.
+
+### _Newsvendor problem_
+
+Newsvendor problem is modeled as follows; if the firm prepares the inventory of goods \(Q\) and the order in the market is \(\xi\) , his profit in the single period is expected as follows;
+
+\[\pi(\xi): = -cQ+p\min(Q,\xi),\] (1)
+
+where the purchasing cost and the selling price describe \(c\) and \(p\), respectively, furthermore \(\min(Q,\xi)\) denotes the lesser value of \(Q\) and \(\xi\). Note that \(c\leq p\) is needed in nature since the firm won’t prefer to stock and buy the goods in the case \(c>p\) (briefly the profit \(\pi(\xi)\leq 0\) at \(c>p\) in other words), and notice that \(Q\) and \(\xi\) are assumed as nonnegative and real numbers without the loss of generality. Here the opportunity loss is not handled and the inventory space is large enough as a matter of convenience, however it turns out that our approach would be simply applied in the case with opportunity loss and the upper bound of inventory level. Generally speaking, it is too hard to estimate that the demand in each single period is fixed. Hence, let us propose that the demand \(\xi\) is stochastically drawn from the given density function \(f(\xi)\) with the cumulative probability. Now, the expected aggregate profit is represented as follows;
+
+\[\Pi:=\int_{0}^{\infty}d\xi f(\xi)\pi(\xi)=-cQ+p\int_{0}^{Q}d\xi\overline{F}\left(\xi\right),\] (2)
+
+where \(\overline{F}(\xi):=\int_{\xi}^{\infty}dxf(x)\) describes the cumulative probability whose stochastic variable is greater than or equal to \(\xi\) and \(\Pi\) is a concave function of the inventory quantity strictly (show appendix A-B). Note that \(\Pi=0\) at \(Q=0\) is required for any distribution of demand by definition and it implies that no stock is no benefit. Moreover as trivial, the supremum of the expected entire benefit \(\Pi\) is indeed greater than or equal to zero because of the previous notice.
+
+The production planner’s purpose in general is to maximize his expected gross benefit \(\Pi\) by adjusting the inventory level \(Q\) in typical situation. However, nowadays the logistics, the distribution of goods in supply chain behaves like bloodstream in the human society, the necessity not only of the integrated inventory management but also of the distributed inventory management has been recognized. In the given distributed configurations, the problem who makes to store the inventory and/or who needs to decide the wholesale price in supply chain is one of the most vital issues, furthermore it is also important how one examines the loss of efficiency of the distributed management in some of remarkable configurations compared to the integrated management.
+
+### _Centralized supply chain_
+
+Let us review here the problem how the inventory level \(Q\) is derived in order to maximize the expected entire profit of the given integrated (or centralized) supply chain. From eq. (2), the unique optimal solution which can be desired is intuitively derived as follows; \({Q_{c}}:=\overline{F}^{-1}\left(r\right)\) where \(r:=c/p\) and \(\overline{F}^{-1}(y)(=x)\) represents the inverse function of \(\overline{F}(x)(=y)\). Since the second term in eq. (2) is a concave function of \(Q\) (show appendix A-B), it turns out that the unique optimal solution is strictly determined.
+
+### _The profits in two echelon case_
+
+The counterpart of the centralized case in supply chain management, the distributed (or decentralized) supply chain management is explained here. In the distributed case the expected profit could be divided into two distinguished profit functions as follows; \(\Pi=\Pi^{\rm M}+\Pi^{\rm R}\) in push serial supply chain and \(\Pi=\Xi^{\rm M}+\Xi^{\rm R}\) in pull serial supply chain. Firstly, \(\Pi^{\rm M}:=(w-c)Q\) and \(\Pi^{\rm R}:=-wQ+p\int_{0}^{Q}d\xi\overline{F}\left(\xi\right)\) indicate the manufacturer’s entire profit and the retailer’s whole benefit, respectively, in the case that the retailer makes to stock the inventory in push serial supply chain. While \(\Xi^{\rm M}:=-cQ+w\int_{0}^{Q}d\xi\overline{F}\left(\xi\right)\) and \(\Xi^{\rm R}:=(p-w)\int_{0}^{Q}d\xi\overline{F}\left(\xi\right)\) describe the manufacturer’s aggregate profit and the retailer’s total benefit, respectively, in the situation that the manufacturer makes to store the inventory in pull serial supply chain. In each case, the leader decides the wholesale price \(w\) (note that \(c\leq w\leq p\) is required in practice, because these profits are satisfied with positivity) in order to maximize the leader’s expected whole profit, while the follower should choose selfishly the optimal inventory level \(Q\) by employing the optimization problem of the follower’s expected benefit with respect to the given wholesale price, that is, it is the scenario of Stackelberg leadership game [16] . In the previous investigations, the analysis of the class of increasing generalized failure rate distribution, where one ensures that the optimization problem in the decentralized configurations could possess the well-defined optimal solution, has been reported comparatively well with regard to several concrete distributed configurations in two echelon case as follows;
+
+ **(a)**: The manufacturer is the decision maker in push serial supply chain. \({Q_{d}}\) is determined by the following equation; \(\overline{F}\left({Q_{d}}\right)\left(1-g\left({Q_{d}}\right)\right)=r\) where \(g(Q):=\frac{Qf(Q)}{\overline{F}(Q)}\) is utilized. In this paper \(g(Q)\) is assumed as a nondecreasing function of \(Q\) and \(0\leq g(Q)\leq 1\) because it is guaranteed that the optimal solution of the follower’s optimization problem is unique. Thus \(g(Q)\) is termed as increasing generalized failure rate. In addition, the desirable wholesale price is yielded as \(w=p\overline{F}({Q_{d}})\).
+ **(b)**: The retailer is the decision maker in push serial supply chain. \({Q_{d}}\) is consistent with the inventory level in the integrated supply chain management, because the inventory is stored at the downstream site and it is to be expected that the wholesale price is equal to the purchasing cost.
+ **(c)**: The manufacturer is the decision maker in pull serial supply chain. \({Q_{d}}\) is also consistent with the inventory level in integrated supply chain, because the inventory is stored at the leader’s site and it is to be desired that the wholesale price corresponds to the selling price.
+ **(d)**: The retailer is the decision maker in pull serial supply chain. \({Q_{d}}\) is decided by the following equation; \(\overline{F}\left({Q_{d}}\right)\left(1+l\left({Q_{d}}\right)\right)^{-1}=r\) where \(l(Q):=\frac{{f(Q)}}{{\overline{F}^{2}(Q)}}\int_{0}^{Q}d\xi\overline{F}(\xi)\) is employed and the optimal wholesale price is derived as \(w=c/\overline{F}({Q_{d}})\). If \(g(Q)\) is increasing generalized failure rate, then \(l(Q)\) is nondecreasing function of \(Q\) strictly [4, 14].
+
+In practice, the production planner should choose the most appropriate distributed inventory management in some remarkable configurations by some means. For that reason, our purpose is here to examine the following measurement so as to assess the loss of efficiency of each configuration. Price of anarchy, the performance ratio which can characterize the loss of efficiency of the expected whole benefit of the given decentralized inventory management compared with that of the given centralized inventory management and which is well-known, is denoted as follows;
+
+\[{\rm PoA}: = \frac{{-c{Q_{c}}+p\int_{0}^{{Q_{c}}}d\xi\overline{F}\left(\xi\right)}}{{-c{Q_{d}}+p\int_{0}^{{Q_{d}}}d\xi\overline{F}\left(\xi\right)}},\] (3)
+
+where eq. (2) is utilized and it turns out that \({\rm PoA}\) is greater than or equal to unit in general (show appendix A-A).
+
+### _Rigorous results derived in the previous works_
+
+With respect to the ensemble of increasing generalized failure rate, the rigorous results of price of anarchy were presented in the previous work as follows [14]; (a) \({\rm PoA}\leq(1-k)^{-\frac{1}{k}}-(1-k)^{-1}\), (b) \({\rm PoA}=1\), (c) \({\rm PoA}=1\) and (d) \({\rm PoA}\leq(1+l)^{1+\frac{1}{l}}-(1+l)\), where \(k:=g\left({Q_{d}}\right)\) and \(l:=l\left({Q_{d}}\right)\) are utilized (the indices are mentioned in the previous subsection). Indeed, although the previous work explained that the upper bounds in (a) and (d) are comparatively tight, it is hard to assess the loss of efficiency in precision and in good faith, therefore we discuss more conscientiously and obtain a tighter upper bound compared with the derived one in Perakis and Roels (2007), and a tighter lower bound with respect to the class of increasing generalized failure rate exactly.
+
+### _An upper and lower bound of cumulative probability_
+
+From the definition of increasing generalized failure rate, \(\log\overline{F}(\xi)=-\int_{0}^{\xi}dy\frac{g(y)}{y}\) is analytically yielded where \(\overline{F}(0)=1\). Moreover with regard to the ensemble of increasing generalized failure rate, \(\max\left(\overline{F}\left({Q_{c}}\right),\overline{F}\left({Q_{d}}\right)\left({Q_{d}}\right)^{s}\xi^{-s}\right)\leq\overline{F}\left(\xi\right)\leq\max\left(\overline{F}\left({Q_{c}}\right),\overline{F}\left({Q_{d}}\right)\left({Q_{d}}\right)^{k}\xi^{-k}\right)\) is obtained where \({Q_{d}}\leq\xi\leq{Q_{c}}\) and \(k=g\left({Q_{d}}\right)\) and \(s:=g\left({Q_{c}}\right)\) are employed. According to the ratio of the inventory level in the integrated case to the one in the distributed case, \(\alpha:={Q_{c}}/{Q_{d}}\), we obtain an upper bound and a lower bound of the integration as follows;
+
+\[{\cal L}(\alpha,s)\leq{\int_{{Q_{d}}}^{{Q_{c}}}d\xi\overline{F}\left(\xi\right)}\qquad\qquad\qquad\qquad\qquad\qquad\\]
+\[\leq\left\{\begin{array}[]{ll}{Q_{d}}\overline{F}\left({Q_{d}}\right)\frac{\alpha^{1-k}-1}{1-k}&(1-k)^{-\frac{1}{k}}\leq\alpha\\ {\cal L}(\alpha,k)&(1-k)^{-\frac{1}{s}}\leq\alpha\leq(1-k)^{-\frac{1}{k}}\end{array}\right.\] (6)
+
+where \({\cal L}(\alpha,t):={Q_{d}}\overline{F}\left({Q_{d}}\right)\left[(1-k)\alpha+\frac{t\left(1-k\right)^{1-\frac{1}{t}}-1}{1-t}\right]\) is used. Notice that \(\alpha<(1-k)^{-\frac{1}{s}}\) is not satisfied with the boundary condition, \(\overline{F}({Q_{c}})=r\) and \({\cal L}(\alpha,t)\) is a nonincreasing and convex function of \(t\) for any \(\alpha\). Since the integration in eq. (6) is evaluated more accurately compared with their insufficient discussion in [14], the comparative tight upper bound of price of anarchy and the lower bound are expected fortunately. Furthermore, since the cumulative probability is a nonincreasing function, the upper bound in this interval \({Q_{d}}\leq\xi\) is not possible to exceed the probability at \(Q=0\) in nature, the inequality \(\overline{F}\left({Q_{d}}\right)\leq\overline{F}(\xi)\leq\min\left(1,\overline{F}\left({Q_{d}}\right)\left({Q_{d}}\right)^{k}\xi^{-k}\right)\) is derived, and an upper bound and a lower bound of the integration,
+
+\[{Q_{d}}\overline{F}\left({Q_{d}}\right)\leq\int_{0}^{{Q_{d}}}d\xi\overline{F}(\xi)\] (7)
+\[\leq {Q_{d}}\overline{F}\left({Q_{d}}\right)\left[1+\frac{k\left(1-\overline{F}^{\frac{1}{k}-1}\left({Q_{d}}\right)\right)}{1-k}\right],\]
+
+is obtained.
+
+### _Both bounds of price of anarchy; the manufacturer is the leader in push serial supply chain_
+
+According to the argument in the previous work [14], by definition, one can replace price of anarchy as follows; \({\rm PoA}=1+\frac{{\int_{{Q_{d}}}^{{Q_{c}}}d\xi\left(\overline{F}(\xi)-r\right)}}{{-r{Q_{d}}+\int_{0}^{{Q_{d}}}d\xi\overline{F}(\xi)}}\). Thus employing eq. (6) and eq. (7), an upper bound
+
+\[{\rm PoA}\leq\left\{\begin{array}[]{ll}(1-k)^{-\frac{1}{k}}-(1-k)^{-1}&(1-k)^{-\frac{1}{k}}\leq\alpha\\ \frac{\alpha^{1-k}-(1-k)^{2}\alpha}{k(1-k)}-(1-k)^{-1}&{\rm otherwise}\end{array}\right.\] (10)
+
+and a lower bound
+
+\[{\rm PoA}\geq\frac{{\frac{s(1-k)^{1-\frac{1}{s}}-s}{1-s}-\frac{{kr^{\frac{1}{k}-1}(1-k)^{1-\frac{1}{k}}-k}}{{1-k}}}}{k+\frac{{k-kr^{\frac{1}{k}-1}(1-k)^{1-\frac{1}{k}}}}{{1-k}}},\] (11)
+
+are simply yielded. It turns out that the numerator of the first term of the upper bound in \((1-k)^{-\frac{1}{s}}\leq\alpha\leq(1-k)^{-\frac{1}{k}}\) in eq. (10) is a nondecreasing and concave function of \(\alpha\). The supremum of the right hand side in eq. (10) was already presented in [14], while the comparative tight upper bound is to be desired in \((1-k)^{-\frac{1}{s}}\leq\alpha\leq(1-k)^{-\frac{1}{k}}\). Fig. 2 shows that in the limit of \(k\to 0\) for any \(s>k\), both bounds at \(\alpha\to(1-k)^{-\frac{1}{s}}\) are close to unit and if \(s=k\), the lower bound \(\left(1+\frac{2-k}{\left(1-r^{\frac{1}{k}-1}\right)(1-k)^{1-\frac{1}{k}}}\right)^{-1}\) is greater than or equal to unit at \(\alpha\geq(1-k)^{-\frac{1}{k}}\). While Fig. 2 indicates the behavior at \(k=0.20\) and \(r=0.40\) and it is found that both bounds are greater than unit at any ratio \(\alpha\).
+
+Figure 1: The ratio \(\alpha\) v.s. both bounds of price of anarchy at \(k=0.01\), \(\overline{F}({Q_{d}})=0.5\) and \(s\simeq 1.0\)
+Figure 2: The ratio \(\alpha\) v.s. both bounds of price of anarchy at \(k=0.20\), \(\overline{F}({Q_{d}})=0.5\) and \(s\simeq 1.0\)
+
+### _Both bounds of price of anarchy; the retailer is the leader in pull serial supply chain_
+
+In this case, compared with the previous subsection, \(k\) and \(s\) are rewritten as \(1-k=(1+l)^{-1}\) and \(1-s=(1+t)^{-1}\), respectively, where \(l=l\left({Q_{d}}\right)\) and \(t\geq l\), then an upper bound and a lower bound;
+
+\[{\rm PoA} \leq \left\{\begin{array}[]{ll}(1+l)^{1+\frac{1}{l}}-(1+l)&(1+l)^{1+\frac{1}{l}}\leq\alpha\\ {\frac{(1+l)^{2}\alpha^{\frac{1}{1+l}}-\alpha}{l}-(1+l)}&{\rm otherwise}\end{array}\right.\] (14)
+\[{\rm PoA} \geq \frac{t(1+l)^{\frac{1}{t}}-t+l-lr^{\frac{1}{l}}(1+l)^{\frac{1}{l}}}{(1+l)^{-1}+l-lr^{\frac{1}{l}}(1+l)^{\frac{1}{l}}}\]
+
+are also yielded.
+
+### _Example: Nonnegative order drawn from normal Gaussian distribution_
+
+Let us confirm the effectiveness of our approach with a novel toy model. For simplicity, it is assumed that the demands are independently and identically distributed according to most of the previous works. Here the density function of demand, \(f(\xi)\) is satisfied with \(\frac{2}{\sqrt{2\pi}}e^{-\frac{\xi^{2}}{2}}\) for \(\xi\geq 0\) and \(0\) otherwise. One can easily validate that \(g(Q)\) of this model is increasing generalized failure rate. Thus \({Q_{c}}\) and \({Q_{d}}\) in the two cases of \({\rm PoA}\neq 1\), are illustrated in Fig. 5. Furthermore, as shown in Fig. 5 and in Fig. 5 that the numerical results of \({\rm PoA}\) and the derived bounds are compared with each other in the case that the manufacturer is the decision maker in push serial supply chain and in the case that the retailer is the decision maker in pull serial supply chain, respectively. In conclusion, it turns out that our approach is valid in this model.
+
+Figure 3: The ratio \(r=c/p\) v.s. \({Q_{c}}\) and \({Q_{d}}\).
+Figure 4: The ratio \(r\) v.s. price of anarchy and the bounds. In push serial supply chain, the manufacturer is the decision maker.
+Figure 5: The ratio \(r\) v.s. price of anarchy and the bounds. In pull serial supply chain, the retailer is the decision maker.
+
+### _Example: Autoregression model with \(\chi\) square noise_
+
+As the real-world data analysis, it is natural that the demands are correlated with one another, for instance, a market trend, rather than that they are independently and identically distributed [10, 17, 18]. However, in most of the earlier studies, for simplicity, the demands were independently and identically distributed and this assumption is obviously not practical since real-world date are usually somewhat biased and correlated across the instance. In general, the demand might consist of the trend effect which implies correlation and an uncertain factor which connotes noise (c.f. bullwhip effect). Therefore as the first step, let us handle a novel toy model in order to confirm whether or not these bounds of price of anarchy which have been presented in this paper are influenced by the correlation in the market. Well, the demand at discrete time \(T\) (denoted by \(\xi_{T}\)) is generated from the following autoregression model; \(\xi_{T+1}=\beta\xi_{T}+\sigma^{2}\chi_{T}^{2}\), where \(\beta\) indicates a dumping coefficient in \(0\leq\beta\leq 1\) and \(\chi_{T}^{2}\), the random variable at time \(T\), is drawn from \(\chi\) square distribution with one degree of freedom (in addition, for simplicity each noise is assumed to be independently and identically distributed and \(\sigma^{2}\) implies a noise intensity). Although there is not an aggressive premise, because the demand is guaranteed to be always nonnegative, this toy model is accepted here.
+
+Figure 6: It is well-known that the density function \(f(\xi)\) is analytically yielded via the cumulant generating function, the logarithm function of characteristic function, \(\log\int_{0}^{\infty}d\xi f(\xi)e^{i\theta\xi}=-\frac{1}{2}\sum_{k=1}^{\infty}\log\left(1-2i\theta\sigma^{2}\beta^{k}\right)\)[7]. However we estimate here the density function utilizing the histogram numerically evaluated by \(0.5\times 10^{9}\) demands which are consisted of the sequences which are randomly chosen from the times series, which is stable, and we fit the logarithm of the density function derived numerically into the fifth-degree polynomial function of \(\log\xi\), which is supported by minimization of the the leave-one-out cross validation error [18]. \(f(\xi)\) at \(\beta=0.9\) and \(\sigma^{2}=100.0\) is illustrated in this figure.
+
+Now, given \(\beta\) and \(\sigma^{2}\), and the density function of demand \(f(\xi)\) is stable, one can evaluate the loss of efficiency, \({\rm PoA}\) with respect to the ratio \(r=c/p\), furthermore, we can confirm whether or not both bounds are valid. The density function of demand, \(f(\xi)\) at \(\beta=0.9\) and \(\sigma^{2}=100.0\) is illustrated in Fig. 6, \({Q_{c}}\) and \({Q_{d}}\) in the two cases are depicted in Fig. 9. As shown in Fig. 9 and in Fig. 9 that the numerical result of \({\rm PoA}\) and the upper and lower bounds are compared with one another at \(\beta=0.9\) and \(\sigma^{2}=100.0\) in the case that the manufacturer is the leader in push serial supply chain and in the case that the retailer is the leader in pull serial supply chain, respectively. Likewise, it turns out that our procedure is validly supported in this correlated model.
+
+Figure 7: The ratio v.s. the inventory levels at \(\beta=0.9\) and \(\sigma^{2}=100.0\).
+Figure 8: The ratio \(r\) v.s. price of anarchy and the derived bounds. In push serial supply chain, the manufacturer is the decision maker.
+Figure 9: The ratio \(r\) v.s. price of anarchy and the derived bounds. In pull serial supply chain, the retailer is the decision maker.
+
+## III Generalized Framework
+
+We have discussed price of anarchy, the performance ratio which can characterize the loss of efficiency of the given distributed supply chain management compared with the integrated management policy, based on newsvendor problem which is well-known. In particular, some of remarkable distributed policies are handled, the performance ratios in each case, for instance, who makes to store the inventory and/or who needs to decide the wholesale price, are analyzed theoretically, and the tighter upper bound of price of anarchy and the lower bound are presented. However, two points should be noted here that these results are not restricted to newsvendor problem (hereafter it is termed as original newsvendor problem) and price of anarchy is possible to be regarded as one of the most unbeatable feature quantity with respect to a broad class of optimization problems.
+
+In this section, let us propose a generalization of the approach treated in the previous section (hereafter generalized newsvendor problem is handled intuitively, however it is also demonstrated in the literature of supply chain management as a matter of convenience). According to our argument, one could replace the profit function in eq. (1) as \(\pi(\xi)=-cQ+pm(\xi,Q)\), i.e. \(m(\xi,Q)=\min(\xi,Q)\); it turns out that \(m(\xi,Q)\) can describe a more complicated policy of price contracts, which could sufficiently characterize a market trend, for instance, the order of hard-to-find items in practice. Now, for simplicity of the discussion, we prepare the expected aggregate profit,
+
+\[\Pi: = -cQ+p{\cal M}(Q),\] (15)
+
+where \({\cal M}(Q):=\int_{0}^{\infty}d\xi f(\xi)m(\xi,Q)\) and \({\cal X}(Q):=\frac{\partial{\cal M}(Q)}{\partial Q}\) are defined. In order to extend the method, we should explain \(m(\xi,Q)\), \({\cal M}(Q)\) and \({\cal X}(Q)\) simply. Firstly \(m(\xi,Q)\) is here an arbitrary function which is satisfied with the following conditions, \(\frac{\partial^{2}{\cal M}(Q)}{\partial Q^{2}}=\frac{\partial{\cal X}(Q)}{\partial Q}\leq 0\) and \({\cal M}(0)=0\). Because it is required for the concavity of the expected total profit function (one would not presume that \(m(\xi,Q)\) is a concave function of \(Q\) in general, i.e. \(m(\xi,Q)=\min(\xi,Q)\)) and \({\cal M}(0)=0\) is necessary so as to connote that no stock is no benefit. Furthermore from \({\cal X}(Q)\) and \({\cal M}(0)=0\), \({\cal M}(Q)\) is possible to be replaced as \({\cal M}(Q)=\int_{0}^{Q}d\xi{\cal X}(\xi)\) (it is not required to disclose the density function of demand in generalized newsvendor problem), where \({\cal X}(\xi)\) can be regarded as \(\overline{F}(\xi)\) in the previous section. Further it implies that \({\cal X}(Q)\) is assumed as positive because the optimal inventory level is nonnegative. Lastly, \(c\leq p{\cal X}(0)\) is needed in nature (briefly \(\Pi\leq 0\) for any inventory level is satisfied at \(c\geq p{\cal X}(0)\)).
+
+### _Centralized case and profit functions_
+
+The unique optimal solution in the integrated case of generalized newsvendor problem is denoted as follows; \({Q_{c}}={\cal X}^{-1}(r)\), where \(r=c/p\) (notice that \(0\leq\frac{c}{p{\cal X}(0)}\leq 1\)) and \({\cal X}^{-1}(y)(=x)\) represents the inverse function of \({\cal X}(x)(=y)\). Next, the expected profit function \(\Pi\) in eq. (15) is possible to be divided into two distinguished parts in push and pull configurations as follows; (i) \(\Pi^{\rm M}:=(w-c)Q\), (ii) \(\Pi^{\rm R}:=-wQ+p{\cal M}(Q)\), (iii) \(\Xi^{\rm M}:=-cQ+w{\cal M}(Q)\) and (iv) \(\Xi^{\rm R}:=(p-w){\cal M}(Q)\), where the wholesale price \(w\) is satisfied with \(c\leq w\leq p{\cal X}(0)\) in push serial supply chain and \(c/{\cal X}(0)\leq w\leq p\) in pull serial supply chain, respectively. According to the previous argument, price of anarchy is also denoted as follows;
+
+\[{\rm PoA}:=\frac{{-c{Q_{c}}+p{\cal M}({Q_{c}})}}{{-c{Q_{d}}+p{\cal M}({Q_{d}})}}.\] (16)
+
+In the follows, we explain how the optimal inventory level and the desirable wholesale price of each decentralized case are determined so as to measure the loss of efficiency.
+
+### _Manufacturer is the leader in push serial supply chain_
+
+It is well-known that \({\rm PoA}\) in this situation of original newsvendor problem is not always equal to unit. In practice one can derive the optimal solution iteratively via the saddle point equation as follows;
+
+\[{Q_{d}}={\cal X}^{-1}(\varepsilon)\ \ \left.\begin{array}[]{cc}\Longrightarrow\\ \Longleftarrow\end{array}\right.\ \ \varepsilon=r+g({Q_{d}}){\cal X}({Q_{d}})\] (19)
+
+where \(\varepsilon:=w/p\) and a novel function, \(g(Q):=-Q\frac{\partial}{\partial Q}\log{\cal X}(Q)\) are already used. By definition, \(g(Q)\) is a nonnegative function of \(Q\) in general. Furthermore in order to determine the unique solution of the decentralized management, we assume that \(g(Q)\) is increasing generalized failure rate (note that if \(g(Q)\) is termed as increasing generalized failure rate, then \(g(Q)\) is satisfied with \(0\leq g(Q)\leq 1\) and \(\frac{\partial g(Q)}{\partial Q}\geq 0\)).
+
+In addition, although this iteration connotes the recursive procedure in order to resolve the optimal inventory level systematically in the distributed system, using eq. (19), one can represent also the following relation as \({\cal X}({Q_{d}})\left(1-g({Q_{d}})\right)=r\), where the desirable wholesale price is \(w=p{\cal X}({Q_{d}})\) by the definition of \(\varepsilon\).
+
+### _Retailer is the leader in push serial supply chain_
+
+In this case, \({\rm PoA}\) is equal to unit because the derivative of the leader’s expected profit \(\Pi^{\rm R}\) with respect to the wholesale price \(w\) is nonpositive, the optimal wholesale price is desirable to be consistent with the purchasing cost in \(c\leq w\leq p{\cal X}(0)\), therefore it is possible to be regarded as the integrated system as the follower’s benefit is zero, that is, \({Q_{d}}\) is equivalent to \({Q_{c}}\).
+
+### _Manufacturer is the leader in pull serial supply chain_
+
+Fortunately, \({\rm PoA}\) coincides with the value derived in original newsvendor problem because the derivative of the leader’s expected profit \(\Xi^{\rm M}\) with respect to the wholesale price \(w\) is nonnegative, the optimal wholesale price is desirable to be equal to the selling price in \(c/{\cal X}(0)\leq w\leq p\), therefore it is possible to be regarded as the integrated system as the follower’s benefit is zero, that is, \({Q_{d}}\) is consistent with \({Q_{c}}\).
+
+### _Retailer is the leader in pull serial supply chain_
+
+It is sure that \({\rm PoA}\) in the last case of original newsvendor problem is not less than unit. With respect to generalized newsvendor problem, let us evaluate the optimal solution sequentially utilizing the steepest descent method as follows;
+
+\[{Q_{d}}={\cal X}^{-1}(\delta)\ \ \left.\begin{array}[]{cc}\Longrightarrow\\ \Longleftarrow\end{array}\right.\ \ \frac{1}{\delta}=\frac{1}{r}-\frac{l({Q_{d}})}{{\cal X}({Q_{d}})},\] (22)
+
+where \(\delta:=c/w\) and a novel function \(l(Q):=-\frac{{\frac{\partial}{\partial Q}\log{\cal X}(Q)}}{{\frac{\partial}{\partial Q}\log{\cal M}(Q)}}\) are employed. Under the definition, \(l(Q)\) is a nonnegative function of the inventory level in nature. Moreover, so as to derive the unique solution of the distributed management, we require that \(l(Q)\) is a nondecreasing function of \(Q\) as the sufficient condition (if \(g(Q)\) is assumed as increasing generalized failure rate, then \(l(Q)\) is a nondecreasing function of \(Q\), show appendix A-D).
+
+Additionally, although this iteration derived here implies the algorithmic procedure so as to assess the optimal inventory level in the decentralized system, applying eq. (22), the following relation, \({\cal X}({Q_{d}})\left(1+l({Q_{d}})\right)^{-1}=r\), is obtained exactly where the desirable wholesale price is \(w=c/{\cal X}({Q_{d}})\).
+
+### _Both bounds of price of anarchy_
+
+From the above discussion, it turns out that \({\rm PoA}\) in the two distributed cases that the retailer is the leader in push serial supply chain and the manufacturer is the leader in pull serial supply chain, respectively, namely the inventory is stocked at the leader’s site, is similar to the derived results of original newsvendor problem. Herein the others are discussed.
+
+Firstly we explain the case that the manufacturer is the decision maker in push serial supply chain. Utilizing the definition of increasing generalized failure rate, \(\log{\cal X}(\xi)=\log{\cal X}(0)-\int_{0}^{\xi}dy\frac{g(y)}{y}\) is assessed. Thus with respect to the class of increasing generalized failure rate distribution, \(\max\left({\cal X}({Q_{c}}),{\cal X}({Q_{d}})({Q_{d}})^{s}\xi^{-s}\right)\leq{\cal X}(\xi)\leq\max\left({\cal X}({Q_{c}}),{\cal X}({Q_{d}})({Q_{d}})^{k}\xi^{-k}\right)\) is derived where \({Q_{d}}\leq\xi\leq{Q_{c}}\) and \(k=g({Q_{d}})\) and \(s=g({Q_{c}})\) are represented, respectively. According to the ratio \(\alpha={Q_{c}}/{Q_{d}}\), one can analyze an upper bound and a lower bound of the integration as follows;
+
+\[{\cal L}(\alpha,s)\leq\int_{{Q_{d}}}^{{Q_{c}}}d\xi{\cal X}(\xi)\] (25)
+\[\leq \left\{\begin{array}[]{ll}{Q_{d}}{\cal X}({Q_{d}})\frac{\alpha^{1-k}-1}{1-k}&(1-k)^{-\frac{1}{k}}\leq\alpha\\ {\cal L}(\alpha,k)&(1-k)^{-\frac{1}{s}}\leq\alpha\leq(1-k)^{-\frac{1}{k}}\end{array}\right.\quad\]
+
+where \({\cal L}(\alpha,t):={Q_{d}}{\cal X}({Q_{d}})\left[(1-k)\alpha+\frac{t(1-k)^{1-\frac{1}{t}}-1}{1-t}\right]\) is employed. Moreover \({Q_{d}}{\cal X}({Q_{d}})\leq\int_{0}^{{Q_{d}}}d\xi{\cal X}(\xi)\leq{Q_{d}}{\cal X}({Q_{d}})\left[1+\frac{k}{1-k}\left(1-\left(\frac{{\cal X}({Q_{d}})}{{\cal X}(0)}\right)^{\frac{1}{k}-1}\right)\right]\) is calculated since \({\cal X}({Q_{d}})\leq{\cal X}(\xi)\leq\min\left({\cal X}(0),{\cal X}({Q_{d}})({Q_{d}})^{k}\xi^{-k}\right)\) is obtained in \(\xi\leq{Q_{d}}\). Therefore an upper bound and a lower bound of price of anarchy in generalized newsvendor problem are evaluated as follows;
+
+\[{\rm PoA} \leq \left\{\begin{array}[]{ll}(1-k)^{-\frac{1}{k}}-(1-k)^{-1}&(1-k)^{-\frac{1}{k}}\leq\alpha\\ \frac{\alpha^{1-k}-(1-k)^{2}\alpha}{k(1-k)}-(1-k)^{-1}&{\rm otherwise}\end{array}\right.\ \\] (28)
+\[{\rm PoA} \geq \frac{{\frac{s(1-k)^{1-\frac{1}{s}}-s}{1-s}-\frac{{k\tilde{r}^{\frac{1}{k}-1}(1-k)^{1-\frac{1}{k}}-k}}{{1-k}}}}{k+\frac{{k-k\tilde{r}^{\frac{1}{k}-1}(1-k)^{1-\frac{1}{k}}}}{{1-k}}}\]
+
+where we replace the ratio \(r\) as the rescaled ratio of the purchasing cost to the selling price, \(\tilde{r}:=\frac{c}{p{\cal X}(0)}\in[0,1]\). It turns out that the derived bounds in generalized newsvendor problem are as well as the ones in original newsvendor problem. Lastly, both bounds are also simply derived in the case that the retailer is the leader in pull serial supply chain.
+
+### _Example: A toy model_
+
+The previous argument has indicated only if one validates both bounds of the performance ratio in resolving generalized newsvendor problem, we need not to restrict to the literature of supply chain management. As a matter of course, with respect to the ensemble of increasing generalized failure rate without the context of operations management, one also needs to vindicate the improved and developed bounds. Hence we would apply \({\cal M}(Q):=\tanh(Q)\) for simplicity of the discussion because \({\cal X}(Q)=1-\tanh^{2}(Q)\), \(g(Q)=2Q\tanh(Q)\) and \(l(Q)={2}{\sinh^{2}(Q)}\) are derived briefly and analytically.
+
+Firstly, in the case that the manufacturer is the leader in push serial supply chain (as a matter of convenience we address so), it is comparatively easy to execute the algorithm based on eq. (19), then \({Q_{c}}\) and \({Q_{d}}\) are illustrated in Fig. 12 and \({\rm PoA}\) and the derived bounds are indicated in Fig. 12. Conclusionally, it turns out that our approach is valid in this case. While, in the case that the retailer is the leader in pull serial supply chain, it is also to perform the iteration based on eq. (22), then \({Q_{c}}\) and \({Q_{d}}\) are shown in Fig. 12 and \({\rm PoA}\) and the derived bounds are presented in Fig. 12. Similarly it turns out that our procedure is supported validly in this case.
+
+Figure 10: The ratio \(r=c/p\) v.s. \({Q_{c}}\) and \({Q_{d}}\).
+Figure 11: The ratio \(r\) v.s. price of anarchy and the bounds. In push serial supply chain, the manufacturer is the decision maker.
+Figure 12: The ratio \(r\) v.s. price of anarchy and the bounds. In pull serial supply chain, the retailer is the decision maker.
+
+## IV Geometric interpretation
+
+In the previous section, we have examined price of anarchy with respect to generalized newsvendor problem. Furthermore here, in order to comprehend the measure in depth [6], let us provide a novel geometric interpretation with respect to price of anarchy via the property of the convexity of \(-{\cal M}\left(Q\right)\)[2, 15].
+
+### _Geometric interpretation; the integrated supply chain_
+
+First, one can divide two distinguished functions with respect to the expected aggregate profit function, \(\Pi=-cQ+p{\cal M}\left(Q\right)\) as follows; \(y_{1}\left(Q\right)=cQ+\Pi\) and \(y_{2}\left(Q\right)=p{\cal M}\left(Q\right)\) where from \(y_{1}(Q)=y_{2}(Q)\), that is, if there exists intersection point, then \(\Pi=-cQ+p{\cal M}\left(Q\right)\) is derived. As shown in Fig. 13 that \(\Pi\) implies the intercept of \(y_{1}(Q)\). Or as another representation, the linear function of \(Q\) which has slope \(c\) and passes through an intersection point \(\left(Q^{*},p{\cal M}\left(Q^{*}\right)\right)\) is represented as follows; \(y_{1}^{*}\left(Q\right)=c\left(Q-Q^{*}\right)+p{\cal M}\left(Q^{*}\right)\). Indeed the intercept of \(y_{1}^{*}(Q)\) describes also \(\Pi\). Show Fig. 13, in order to maximize the intercept of \(y_{1}(Q)\), both functions should intersect at one point (denoted by \(\left({Q_{c}},p{\cal M}\left({Q_{c}}\right)\right)\)) at least. Since the derivatives of both functions of \(Q\), i.e. \(\frac{\partial y_{1}\left(Q\right)}{\partial Q}=c\) and \(\frac{\partial y_{2}\left(Q\right)}{\partial Q}=p{\cal X}\left(Q\right)\), are yielded easily, \({\frac{\partial y_{1}\left(Q\right)}{\partial Q}=\frac{\partial y_{2}\left(Q\right)}{\partial Q}}\) at \(Q={Q_{c}}\), namely
+
+\[c = p{\cal M}\left({Q_{c}}\right),\] (29)
+
+is possible to be sufficiently satisfied. Thus we can resolve the unique optimal solution of the integrated inventory management.
+
+Figure 13: The maximum of intercept of linear function implies the extremum of the expected total profit function.
+
+One point should be worthy to be noticed here. \({\cal X}\left(Q\right)\), the derivative of \({\cal M}\left(Q\right)\) of \(Q\), is not always necessary for the continuous function of \(Q\) (however by definition, \({\cal M}(Q)\) is satisfied with the continuous function because of the concavity). For example, the given function,
+
+\[p{\cal M}\left(Q\right):=\left\{\begin{array}[]{ll}\log\left(1+Q\right)&0\leq Q\leq{Q_{v}}\\ v\left(Q-{Q_{v}}\right)+\log\left(1+{Q_{v}}\right)&{Q_{v}}<Q\end{array}\right.\] (32)
+
+is defined with constant \({Q_{v}}\) and \(v\) where \(v<\frac{1}{1+{Q_{v}}}\) is required because \({\cal X}(Q)\) is a nonincreasing function of \(Q\). Then with respect to the slope of \(y_{1}(Q)\) in \(v\leq c\leq\frac{1}{1+{Q_{v}}}\), it turns out that the intersection point is \(\left({Q_{v}},p{\cal M}\left({Q_{v}}\right)\right)\), however, this optimal solution is not satisfied with eq. (29) indeed. That is, we need to comprehend that eq. (29) describes the sufficient condition but not the necessary. Without the loss of generality, so as to prevent also us from misleading in practice, we should confirm the behaviors of both functions \(y_{1}(Q)=cQ+\Pi\) and \(y_{2}(Q)=p{\cal M}(Q)\) being supported by the picture such as Fig. 13.
+
+Figure 14: A geometric interpretation of price of anarchy; the manufacturer is the leader in push serial supply chain. Fig. 15 illustrates how \({Q_{d}}\) is determined.
+Figure 15: \({Q_{d}}\) is satisfied with \(p{\cal X}({Q_{d}})\left(1-g({Q_{d}})\right)=p{\cal X}({Q_{c}})\).
+
+Figure 16: A geometric interpretation of price of anarchy; the retailer is the leader in pull serial supply chain. Fig. 17 illustrates how \({Q_{d}}\) is determined.
+Figure 17: \({Q_{d}}\) is satisfied with \(p{\cal X}({Q_{d}})\left(1+l({Q_{d}})\right)^{-1}=p{\cal X}({Q_{c}})\).
+
+### _Geometric interpretation; the manufacturer is the leader in push serial supply chain_
+
+According to the explanation in the previous subsection, with respect to \(\Pi^{\rm R}=-wQ+p{\cal M}(Q)\), in order to solve the optimization problem of the follower, the two functions, \(y_{1}(Q):=wQ+\Pi^{\rm R}\) and \(y_{2}(Q):=p{\cal M}(Q)\) are denoted. As shown in Fig. 15 that the supremum of the intercept of \(y_{1}(Q)\) (the optimal solution is \(\left({Q_{d}},p{\cal M}\left({Q_{d}}\right)\right)\)) describes the desirable whole profit and the expected total profit in the centralized case, \(\Pi\left({Q_{c}}\right)\) is greater than \(\max\Pi^{\rm R}\) because of \(c\leq w\). Furthermore since the leader’s aggregate profit is represented as \(\Pi^{\rm M}=(w-c){Q_{d}}\), Fig. 15 illustrates that Π(Qc)≥(w−c)Qd+(−wQd+pℳ(Qd))=:Π(Qd), where \(\Pi({Q_{d}})\) connotes the expected profit of the decentralized case. Conclusionally, it turns out that PoA is greater than unit in nature.
+
+### _Geometric interpretation; the retailer is the leader in pull serial supply chain_
+
+In the other case, so as to analyze the optimization problem of the follower, with respect to \(\Xi^{\rm M}=-cQ+w{\cal M}(Q)\), the two functions, \(y_{1}(Q):=cQ+\Xi^{\rm M}\) and \(y_{2}(Q):=w{\cal M}(Q)\), are defined. As shown in Fig. 17 that the maximum of the intercept of \(y_{1}(Q)\) (the optimal solution is \(\left({Q_{d}},p{\cal M}\left({Q_{d}}\right)\right)\)) represents the desirable entire profit and the expected aggregate profit in the integrated case \(\Pi\left({Q_{c}}\right)\) is greater than \(\max\Xi^{\rm M}\) because of \(w\leq p\). Moreover if the leader’s whole profit is \(\Xi^{\rm R}=(p-w){\cal M}({Q_{d}})\), Fig. 17 depicts, \(\Pi\left({Q_{c}}\right)\geq-c{Q_{d}}+w{\cal M}\left({Q_{d}}\right)+(p-w){\cal M}\left({Q_{d}}\right)=\Pi\left({Q_{d}}\right)\). In conclusion, it turns out that PoA is greater than unit in general.
+
+## V Conclusions
+
+We discussed price of anarchy, the performance ratio, which could characterize the loss of efficiency of the distributed supply chain management compared with the integrated supply chain management via newsvendor problem and the generalization instead of bullwhip effect. In particular, the performance ratios in some of remarkable decentralized supply chain managements are analyzed theoretically and numerically. Furthermore, with respect to the ensemble of increasing generalized failure rate, which one can ensure that the optimization problem of the follower could possess the well-defined solution; (a) the upper bound which has been investigated in the previous work [14] is improved in this paper utilizing the more accurate evaluation of the integration and (b) the lower bound is derived in the same manner, in the case that the manufacturer could control the wholesale price in push serial supply chain and in the case that the retailer could adjust the wholesale price in pull serial supply chain, namely the two cases that the follower makes to stock the inventory. Moreover the framework handled in section II has been developed and deepened for generalized newsvendor problem, and we indicate that the loss of efficiency is measured as well as original newsvendor problem. Hence price of anarchy is possible to be one of the most unbeatable feature quantity with respect to the convex optimization involving Stackelberg leadership game [16]. While our approach is supported validly in some examples which are satisfied with increasing generalized failure rate. Lastly without the loss of generality, a geometric interpretation of price of anarchy has been provided concretely.
+
+The investigations of geometric interpretation of price of anarchy in multiechelon case, and of the other ensembles which are guaranteed that the optimization problem of the follower can possess the well-defined solution are promising topics for future works.
+
+## Acknowledgment
+
+One of the authors (TS) appreciates T. Kamishima who works in Advanced Industrial Science and Technology (AIST) for his fruitful advice.
+
+## Appendix A Preliminaries
+
+### _Global optimal and local optimal_
+
+In this appendix, let us introduce the relationship between global optimal and local optimal. Well, we assume that \({\cal X}\) and \({\cal Y}\) are convex sets and employ as \(x\in{\cal X}\) and \(y\in{\cal Y}\), respectively. Furthermore two real-valued functions \(f(x,y)\) and \(g(x,y)\) are bounded from above in the region \((x,y)\in{\cal X}\otimes{\cal Y}\). Then a novel function is denoted as follows; \(F(x,y):=f(x,y)+g(x,y)\) where this function is also satisfied with bounded above and let \((x^{*},y^{*})\) be an extremal solution of \(F(x,y)\) in the given finite region. While \((x^{**},y^{**})\) indicates an extremal solution of \(f(x,y)\), that is, one part of \(F(x,y)\), then, \(F(x^{*},y^{*})\geq F(x^{**},y^{**})\) is obtained in general. Thus price of anarchy is greater than or equal to unit by definition.
+
+### _The concavity of \(\int_{0}^{Q}d\xi\overline{F}(\xi)\)_
+
+Because \(\overline{F}(\xi)\) is a nonincreasing function of \(\xi\) firstly, \(\int_{0}^{Q}d\xi\overline{F}(\xi)\leq\int_{0}^{Q_{0}}d\xi\overline{F}(\xi)+\overline{F}(Q_{0})(Q-Q_{0})\) is held for any \(Q\) and \(Q_{0}\) in general, therefore, for any \(Q,Q^{\prime}\) and \(\lambda\in[0,1]\), the concavity, \(\lambda\int_{0}^{Q}d\xi\overline{F}(\xi)+(1-\lambda)\int_{0}^{Q^{\prime}}d\xi\overline{F}(\xi)\leq\int_{0}^{\lambda Q+(1-\lambda)Q^{\prime}}\raisebox{-3.99994pt}{$d\xi\overline{F}(\xi)$}\) is satisfied when \(Q_{0}=\lambda Q+(1-\lambda)Q^{\prime}\) is rewritten. Additionally, if \(Q={Q_{d}}\) and \(Q_{0}={Q_{c}}\), it is also proved that \({\rm PoA}\) is greater than or equal to unit from \(\int_{0}^{Q}d\xi\overline{F}(\xi)\leq\int_{0}^{Q_{0}}d\xi\overline{F}(\xi)+\overline{F}(Q_{0})(Q-Q_{0})\).
+
+### _Young’s inequality_
+
+Because \(\overline{F}(\xi)\) is a nonincreasing function of \(\xi\), \(Q\varphi+\int_{\varphi}^{1}dy\overline{F}^{-1}(y)\geq\int_{0}^{Q}d\xi\overline{F}(\xi)\geq Q\overline{F}(Q)\) is held without the loss of generality for any \(\varphi\geq 0\). The more left inequality is termed as Young’s inequality, iff \(\overline{F}(Q)=\varphi\) gives the equality. Therefore the more right inequality is obtained from \(\int_{0}^{Q}d\xi\overline{F}(\xi)-Q\overline{F}(Q)=\int_{\overline{F}(Q)}^{1}dy\overline{F}^{-1}(y)\geq 0\) at \(\overline{F}(Q)=\varphi\) without the property of increasing generalized failure rate [7, 9].
+
+### _A proof of nondecreasing function \(l(Q)\)_
+
+If \(g(Q)\) is increasing generalized failure rate, then \(l(Q)\) is strictly satisfied with \(\frac{\partial l(Q)}{\partial Q}\geq 0\). Because one can prepare a novel function firstly, \(j(Q):=\frac{{1}}{{\cal X}(Q)}\int_{0}^{Q}d\xi{\cal X}(\xi)\), where \(\frac{\partial j(Q)}{\partial Q}=1+l(Q)\). Here we allow to rewrite \(l(Q)\) as \(l(Q)=\frac{j(Q)g(Q)}{Q}\). From the derivative of \(\log l(Q)\) with respect to \(Q\),
+
+\[\frac{1}{l(Q)}\frac{\partial l(Q)}{\partial Q}=\frac{1}{g(Q)}\frac{\partial g(Q)}{\partial Q}+\frac{{g(Q)-\left(1-\frac{Q}{j(Q)}\right)}}{Q},\]
+
+is calculated where \(Q\), \(j(Q)\), \(g(Q)\) and \(l(Q)\) are nonnegative by definition. Hence as proof by contradiction, \(g(Q)<1-\frac{Q}{j(Q)}\) is assumed. Then the derivative of \(1-\frac{Q}{j(Q)}\) is derived to be negative exactly. Thus \(g(Q)>0>1-\frac{Q}{j(Q)}\) is yielded where \(\lim_{Q\to 0}g(Q)=\lim_{Q\to 0}\left(1-\frac{Q}{j(Q)}\right)=0\), however this result is inconsistent with the assumption \(g(Q)<1-\frac{Q}{j(Q)}\), namely \(g(Q)\geq 1-\frac{Q}{j(Q)}\) is held in nature. Therefore it is proved that \(l(Q)\) is a nondecreasing function of \(Q\).
+
+### _Magnitude relation of price of anarchy_
+
+From Young’s inequality and the discussion of the previous appendix, \(l(Q)\geq g(Q)\) and \(l(Q)g(Q)-l(Q)+g(Q)\geq 0\) are derived, respectively. Therefore the relationship between the derivative of the inventory level \(Q_{d,\rm pull}\) in pull serial supply chain with respect to the rate \(r\) and the derivative of the inventory level \(Q_{d,\rm push}\) in push serial supply chain with respect to the rate is obtained as \(\frac{\partial Q_{d,\rm pull}}{\partial r}\leq\frac{\partial Q_{d,\rm push}}{\partial r}\) strictly. Thus \(Q_{d,\rm pull}=-\int_{r}^{1}dr\frac{\partial Q_{d,\rm pull}}{\partial r}\geq-\int_{r}^{1}dr\frac{\partial Q_{d,\rm push}}{\partial r}=Q_{d,\rm push}\) is held. Moreover because \(\Pi\) is a nondecreasing function of \(Q\) in \(Q<{Q_{c}}\), \({\rm PoA}\) in push configuration is greater than or equal to \({\rm PoA}\) in pull configuration in general.
+
+### _Price of anarchy in the fixed order case_
+
+For any \(Q\in(0,{Q_{c}}]\), the sufficient and necessary condition of the equality \(1-g(Q)=(1+l(Q))^{-1}\) is \({\cal M}(Q)\propto Q\). In original newsvendor problem, it implies that \(\overline{F}(\xi)=1\) for \(\xi\leq Q_{0}\) and \(0\), otherwise, namely the order is fixed at \(\xi=Q_{0}\), then \({Q_{c}}={Q_{d}}=Q_{0}\) is held in each decentralized supply chain. In conclusion, if the order is constant, the performances of the four cases discussed in this paper are consistent with one another.
+
+## Appendix B \(N\) serial supply chain management
+
+Our approach based on generalized newsvendor problem is simply to be extended \(N\) serial supply chain management (the case of \(N=2\) is already mentioned). The optimal inventory level of each distributed management is devoted as follows:
+
+**(a)**: The manufacturer is the decision maker in push supply chain. \({Q_{d}}\) is derived from the following equation; \(\left(1+Q\frac{\partial}{\partial Q}\right)^{N-1}{\cal X}(Q)=r\) where \(r=c/p\) and roughly speaking, \({Q_{d}}\) is probably to be satisfied with the condition \({\cal X}(Q)\left(1-g(Q)\right)^{N-1}\geq r\). Here one point should be noteworthy. The optimal inventory level \({Q_{d}}\) is not always satisfied with the equality, \({\cal X}(Q)\left(1-g(Q)\right)^{N-1}=r\), because \(\frac{\partial^{n}g(Q)}{\partial Q^{n}}=0\) is not always held for each integer in \(1<n<N\) Moreover it is hardly desirable that the comparative tight both bounds of price of anarchy in this configuration are derived by the optimal inventory level which is satisfied with the above inequality.
+**(b)**: The retailer is the decision maker in push supply chain. Since the inventory is stored at the leader’s site, \({Q_{d}}={Q_{c}}\) is desirable.
+**(c)**: The manufacturer is the decision maker in pull supply chain. Nevertheless to say, as the goods is stocked at the upstream site, \({Q_{d}}={Q_{c}}\) is expected.
+**(d)**: The retailer is the decision maker in pull supply chain. \({Q_{d}}\) is derived as the solution of the following relation, \(\left(1+\frac{{\int_{0}^{Q}d\xi{\cal X}(\xi)}}{{\cal X}(Q)}\frac{\partial}{\partial Q}\right)^{N-1}\frac{1}{{\cal X}(Q)}=\frac{1}{r}\) where roughly speaking, \({Q_{d}}\) is possible to be satisfied with the condition \({\cal X}(Q)\left(1+l(Q)\right)^{-(N-1)}\geq r\).
+
+## Appendix C Multiple materials and multiple items
+
+We could develop our approach in the case of the inventory management of \(S\) multiple materials and \(I\) multiple items briefly. Let \(\vec{c}:=\left\{c_{1},c_{2},\cdots,c_{S}\right\}^{\rm T}\in{\bf R}^{S}\) and \(\vec{Q}:=\left\{Q_{1},Q_{2},\cdots,Q_{S}\right\}^{\rm T}\in{\bf R}^{S}\) be the purchasing costs and the inventory levels of the materials, respectively. Furthermore \(\vec{p}:=\left\{p_{1},p_{2},\cdots,p_{I}\right\}^{\rm T}\in{\bf R}^{I}\) and ℳ→:={ℳ1,ℳ2,⋯,ℳI}T∈𝐑I represent the selling prices and the order levels of the items, respectively. The entry of the order levels is assumed to be strictly a convex function of \(\vec{Q}\). The expected whole profit is defined as follows; \(\Pi=-\vec{c}^{\rm T}\vec{Q}+\vec{p}^{\rm T}\vec{{\cal M}}\), where the notation \({\rm T}\) denotes the matrix transpose. First, the optimal inventory levels of the integrated supply chain \(\vec{Q}_{c}\) is held with the following equation,
+
+\[c_{s} = \sum_{\mu=1}^{I}p_{\mu}\left(\frac{\partial{\cal M}_{\mu}}{\partial{Q}_{s}}\right)_{\vec{Q}\to\vec{Q}_{c}}.\]
+
+Next, in the case that the manufacturer is the leader in push serial supply chain, the inventory levels of the decentralized configuration \(\vec{Q}_{d}\) is satisfied as follows;
+
+\[c_{s}=\sum_{\mu=1}^{I}p_{\mu}\left[\frac{\partial{\cal M}_{\mu}}{\partial Q_{s}}+\sum_{t=1}^{S}Q_{t}\frac{\partial^{2}{\cal M}_{\mu}}{\partial Q_{s}\partial Q_{t}}\right]_{\vec{Q}\to\vec{Q}_{d}}.\]
+
+Therefore the wholesale prices \(\vec{w}:=\left\{w_{1},w_{2},\cdots,w_{S}\right\}\in{\bf R}^{S}\) is denoted as follows; \(w_{s}=\sum_{\mu=1}^{I}p_{\mu}\left(\frac{\partial{\cal M}_{\mu}}{\partial Q_{s}}\right)_{\vec{Q}\to\vec{Q}_{d}}\). Lastly, in the case that the retailer is the decision maker in pull serial supply chain, the inventory levels of the distributed configurations \(\vec{Q}_{d}\) is satisfied as follows;
+
+\[p_{\mu}=\sum_{s=1}^{S}c_{s}\left[\frac{\partial Q_{s}}{\partial{\cal M}_{\mu}}+\sum_{\nu=1}^{I}{\cal M}_{\nu}\frac{\partial^{2}Q_{s}}{\partial{\cal M}_{\mu}\partial{\cal M}_{\nu}}\right]_{\vec{Q}\to\vec{Q}_{d}},\]
+
+where the desirable wholesale prices \(\vec{w}\) is held as follows; \(w_{\mu}=\sum_{s=1}^{S}c_{s}\left(\frac{\partial Q_{s}}{\partial{\cal M}_{\mu}}\right)_{\vec{Q}\to\vec{Q}_{d}}\).
+
+Three points should be noted here. Firstly, several specific problems which are possible to be handled via our approach have been already investigated [1, 3, 12]. Especially, only if the solution with respect to the given optimization problem is definite in several types, utilizing such as Lagrange’s multiplier method, we could resolve in the same way. Next, in the case of the supply chain management in multiperiod (i.e. \(S=I\)), let \(c_{s}\), \(p_{s}\), \(Q_{s}\) and \(\xi_{s}\) be the purchasing cost, the selling price, the inventory level and the demand at term \(s\), respectively. Further the demands \(\vec{\xi}:=\left\{\xi_{1},\xi_{2},\cdots,\xi_{S}\right\}^{\rm T}\) are distributed with the given multivariate density function \(f(\vec{\xi})\) and \({\cal M}_{s}:=\int_{0}^{\infty}d\vec{\xi}f(\vec{\xi})m_{s}(\vec{\xi},\vec{Q})\) indicates the order at term \(s\) where \(m_{s}(\vec{\xi},\vec{Q})\) describes that the order at term \(s\) is depended on the market trend strongly. Likewise, it turns out that one can deal with multiperiod case based on our approach (c.f. bullwhip effect). Lastly, in the case of \(N\) serial supply chain, the optimal inventory levels \(\vec{Q}_{d}\) in the two cases are satisfied with the following equations;
+
+\[c_{s} = \sum_{\mu=1}^{I}p_{\mu}\left[\mathop{\rm Tr}_{\vec{t}}\prod_{i=1}^{N-1}\left(\delta_{t_{i}s}+Q_{t_{i}}\frac{\partial}{\partial Q_{t_{i}}}\right)\frac{\partial{\cal M}_{\mu}}{\partial Q_{s}}\right]_{\vec{Q}\to\vec{Q}_{d}},\]
+\[p_{\mu} = \sum_{s=1}^{S}c_{s}\left[\mathop{\rm Tr}_{\vec{\nu}}\prod_{i=1}^{N-1}\left(\delta_{\nu_{i}\mu}+{\cal M}_{\nu_{i}}\frac{\partial}{\partial{\cal M}_{\nu_{i}}}\right)\frac{\partial Q_{s}}{\partial{\cal M}_{\mu}}\right]_{\vec{Q}\to\vec{Q}_{d}},\]
+
+respectively, where \(\vec{t}:=\left\{t_{1},t_{2},\cdots,t_{N-1}\right\}\) and \(\vec{\nu}:=\left\{\nu_{1},\nu_{2},\cdots,\nu_{N-1}\right\}\) are used, \(\delta_{ab}\) indicates Kronecker’s delta which is the entry of the unit matrix and the notation \(\mathop{\rm Tr}_{\vec{t}}\) and \(\mathop{\rm Tr}_{\vec{\nu}}\) denote the summation over all possible states of \(\vec{t}\) and of \(\vec{\nu}\), respectively.
+
+## References
+
+* [1]F. Bernstein and G. A. DeCroix, Management Science Vol. **50**, No. 9, 1293-1308, (2004)
+* [2]S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, (2004)
+* [3]G. P. Cachon and M. A. Lariviere, Management Science Vol. **47**, No. 5, 629-646, (2001)
+* [4]G. P. Cachon, Management Science, Vol. **50**, No. 2, 222-238, (2004)
+* [5]F. Chen, Z. Drezner, J. K. Ryan and D. Simchi-Levi, Management Science, Vol. **46** No. 3, 436-443, (2000)
+* [6]J. R. Correa, A. S. Schulz and N. E. Stier-Moses, Game and Economic Behavior Vol. **64**, 457-469 (2008)
+* [7]W. Feller, An Introduction to Probability Theory and its Applications, John Wiley and Sons, (1968)
+* [8]S. C. Graves, A. H. G. R. Kan, P. H. Zipkin, Logistics of Production and Inventory, North-Holland, (1993)
+* [9]G. Hardy, J. E. Littlewood and G. P\(\acute{\rm o}\)lya, Inequalities, Cambridge, (1951)
+* [10]Y. Kabashima, J. Phys. Conf. Vol. **95**, No. 1, 012001, (2008)
+* [11]E. Koutsoupias and C. Papadimitriou, Proc. 16th Sympos. Theoretical Aspects of Comput. Sci. Vol. **1563**, 404-413, (1999)
+* [12]M. A. Lariviere and E. L. Porteus, Manufacturing and Service Operations Management, Vol. **3**, No. 4, 293-305, (2001)
+* [13]C. Papadimitriou, Porc. 33rd Annual ACM Sympos. Theory of Computing, 749-753, (2001)
+* [14]G. Perakis and G. Roels, Management Science, Vol. **53**, No. 8, 1249-1268, (2007)
+* [15]R. T. Rockafeller, Convex Analysis, Princeton University Press, (1996)
+* [16]T. Roughgarden, Selfish Routing and the Price of Anarchy, MIT Press (2005)
+* [17]T. Shinzato and Y. Kabashima, J. Phys. A: Math. Theor. Vol. **41**, No. 32, 324013, (2008)
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+* [19]E. A. Silver, D. F. Pyke, R. Peterson, Inventory Management and Production Planning and Scheduling, John Wiley and Sons, (1998)
+* [20]D. Simchi-Levi, X. Chen, J. Bramel, The Logic of Logistics, Springer, (2004)
+

arxiv_ground_truth/0907.1650.md

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+# Structural break models of climatic regime-shifts: claims and forecasts
+
+David R.B. Stockwell and Anthony Cox
+
+http://landshape.org, Email: davids99us@gmail.com
+
+(July 4, 2024)
+
+## 1 Abstract
+
+A Chow test for structural breaks in the surface temperature series is used to investigate two common claims about global warming. Quirk (2009) proposed that the increase in Australian temperature from 1910 to the present was largely confined to a regime-shift in the Pacific Decadal Oscillation (PDO) between 1976 and 1979. The test finds a step change in both Australian and global temperature trends in 1978 (HadCRU3GL), and in Australian rainfall in 1982 with flat temperatures before and after. Easterling & Wehner (2009) claimed that singling out the apparent flatness in global temperature since 1997 is ’cherry picking’ to reinforce an arbitrary point of view. On the contrary, we find evidence for a significant change in the temperature series around 1997, corroborated with evidence of a coincident oceanographic regime-shift. We use the trends between these significant change points to generate a forecast of future global temperature under specific assumptions.
+
+## 2 Introduction
+
+Climatic effects of fluctuations in oceanic regimes are generally studied by decomposing rainfall and temperature into periodic components: e.g. singular spectrum analysis (SSA) (Ghil & Vautard, 1991), and variations on principle components analysis (PCA) (Parker et al., 2007). Such approaches can capture such effects as the influence of short period phenomena like El Ninõ on Australia (Mantua et al., 1997; Guilderson & Schrag, 1998), and the potential for longer term phenomena such as the Pacific Decadal Oscillation (PDO) to ’offset’ putative increases in global temperature (Keenlyside et al., 2008). Methods designed for finding and testing structural breaks address more infrequent regime-shifts. An F-statistic known as the Chow test (Chow, 1960) based on the reduction in the residual sum of squares through adoption of a structural break, relative to an unbroken simple linear regression, is a straightforward approach to modeling regime-shifts with structural breaks.
+
+Two claims are evaluated here. In the first, Quirk (2009) proposed a model of Australian temperature with a regime-shift (herein Q09): slightly increasing to 1976, rapidly increasing to 1979 (the shift), and slowly increasing therafter. The increase in Australian temperature of around 0.9∘C from the start of the readily available records in 1910 (BoM, 2009) is conventionally modeled as a linear trend and, despite the absence of clear evidence, often attributed to increasing concentrations of greenhouse gases (GHGs) e.g. (CSIRO & BoM, 2007). Quirk (2009) proposes a different model, without testing its significance. Regarding only the trends of 0.034∘C and 0.05∘C per decade in the sections before and after the shift substantially lowers the underlying rate of warming potentially attributable to anthropogenic global warming (AGW).
+
+Hartmann & Wendler (2005) noted for the climate of Alaska, that recognition of an abrupt temperature change in climate 1976 profoundly changed the underlying temperature trends from strongly positive, to slightly negative, contradicting a view of increasing greenhouse gases (GHGs) gradually increasing temperatures. The event dubbed the Great Pacific Climate Shift (Kerr, 1992), is clearly seen in a change in the average polarity of the Pacific Decadal Oscillation (PDO) from a cool to a warm phase (Hartmann & Wendler, 2005).
+
+The second example is a claim of lack of statistical significance. Regarding the declining temperature since the El Ninõ event in 1998, Easterling & Wehner (2009) state that using 1998 as a starting point is ’cherry-picking’ to justify a point of view and therefore, statements such as ’global warming stopped in 1997’ have no basis. Structural change methods are used to determine if the period is a significant change point.
+
+From a broader perspective, econometrics research into regime-shifting models indicates that detecting breaks in time series is important and ignoring them can be very costly (Pesaran & Timmermann, 2004). The cost in climate forecasting would be inaccurate forecasts. That is, while a forecast based on a linear model would indicate steadily changing global temperatures, forecasts based on shifts would reflect the moves to relatively static mean values. The choice of underlying model may also impact estimates of the magnitude of climate change, as the Quirk (2009) attribution of a trend of either 0.04∘C or 0.09∘C shows. Further, oceanographic changes have been postulated as the main causes of changes in global temperature in the last 30 years (Gray, 2007), including the recent flat to slightly declining trend in temperatures since 1997.
+
+A larger relevance of structural change models to global temperature data is suggested by the presence of long term persistence (LTP) in complex natural systems (Koutsoyiannis & Cohn, 2008; Halley, 2009). It is known that step changes in the mean at multiple scales can reproduce LTP behaviour (Koutsoyiannis, 2002; Stockwell, 2006). Change points can be found in the mean, or standard deviation (Bai, 1994).
+
+There are well-known difficulties in forecasting with unstable structural breaks models (Hamilton, 1989, 2007). In particular it is known that a least-squares estimator of change points is not reliable for nonstationary \(I(d)\) data with \(0.5<d<1.5\)(Hsu & Kuan, 2007). But when there is a change, the least squares estimator is known to be reliable on stationary \(I(d)\) data (i.e., -0.5 ¡ d ¡ 0.5). Here \(d\) is the fractional differencing parameter as introduced by Granger & Joyeux (1980). We also have some reservations about the data being of adequate length, as models induced from the temperature record can be contingent on subsequent data (Stockwell, 2009). Given these uncertainties, it is important to develop models rigorously, with a transparent development sequence satisfying both significance and optimization of the model space.
+
+## 3 Method
+
+We attempt to verify the optimality and significance of the breakpoints, by reference to both climate data and regime-shifts. Australia-wide annual average rainfall and temperature anomalies were downloaded from the BoM data site (BoM, 2009). Temperature is available as an annual average in anomaly ∘C from 1910 to 2008 while rainfall in anomaly millimeters covers 1900 to 2008. Quirk (2009) only shows temperature data from 1950 to 2003. We also examine the global monthly temperature anomalies from the Hadley Center, HadCRU3GL, for confirmation of the presence of breakpoints at the larger scale (Rayner et al., 2006). Comprehensive evaluation of the full range of available global datasets was beyond the scope of this study.
+
+The approach to developing a structural break model was to calculate the F statistic (Chow test statistic) for potential breaks at all potential change points using the R package \(strucchange\)(Zeileis et al., 2002). In this test, the error sum of squares (ESS) of a linear regression model is compared with the residual sum of squares (RSS) from a model composed of linear sections before and after each potential change point. If \(n\) is the number of observations and \(k\) the number of regressors in the model, the F statistic is:
+
+\(F=(RSS-ESS)/ESS*(n-2*k)\)
+
+Potential breakpoints are indicated by peaks in the value of the F statistic (dashed vertical lines in figures). The red line indicates the height above which a series with a break at that location would be significantly better (at the 95% confidence limit) than a single linear regression.
+
+\begin{table}
+\begin{tabular}{r r r r r r r}
+\hline
+ & R2 & Breakdate & R2b & Pb & P1 & P2 \\
+\hline
+Aus. Temperature & 0.384 & 1978.00 & 0.482 & 0.000 & 0.151 & 0.139 \\
+Aus. Rainfall & 0.041 & 1982.00 & 0.164 & 0.017 & 0.877 & 0.026 \\
+Global Temp. 1910-2008 & 0.721 & 1978.00 & 0.785 & 0.000 & 0.000 & 0.000 \\
+Global Temp. 1976-2008 & 0.613 & 1997.08 & 0.672 & 0.000 & 0.000 & 0.409 \\ \hline
+\end{tabular}
+\end{table}
+Table 1: Fit of models as described in text and shown in Figures 1a,b and 2a,b. R2 is the regression correlation coefficient, R2b is the correlation coefficient with one break, Pb is significance of improvement of the break, over the linear model, P1 and P2 and the significances of the slopes of the first and second segments, respectively.
+
+The relative fit of the models is indicated with a regression coefficient \(R^{2}\), as listed in Table 1. In addition, \(p\) values indicate the significance of model improvements (Pb), and the non-zero slope parameters (P1 and P2).
+
+While the \(strucchange\) software also implements a search for multiple breakpoints, we adopt a conservative approach without undue complexity, mindful of the problematic nature of simultaneous validation of multiple structural break models (Zeileis & Kleiber, 2004). The optimal position of single breaks is clearly indicated with a plot of the F statistic aligned with the x-axis of a plot of the time series.
+
+## 4 Results
+
+As a preliminary check, the estimates of the fractional differencing parameter \(d\) on the Australian temperature and rainfall data were estimated at 0.31 and 0.13 respectively (using R package \(fracdiff\)). Therefore, the Chow test is expected to be reliable. The estimate of \(d\) on HadCRU3GL global temperature series is 0.50, so results must be interpreted more cautiously.
+
+Figure 1: Breakpoints in Australian temperature (a) and rainfall (b). The upper section of each figure is the F statistic from the Chow test for breaks with optimal breaks (vertical, dashed), level of significance (red line) and confidence interval (red bracket). The Australian temperature (below), with the OLS regression model (green) and the optimal break model (blue).
+
+The F statistics, breakpoints, confidence intervals, the simple regression and segmented models for annual Australian temperature from 1910 to 2008, are shown on Figure 1a. The peak in the F statistic above the red line indicates that 1978 is the optimal location for a break, indicated with a dashed vertical line. The 95% confidence interval spans 1974 to 1981, indicated with a red bracket on the x-axis of the F statistic. The structural break model is a significant (\(p<0.001\)) improvement in the fit of the model as shown by the increase in \(R^{2}\) from 0.38 to 0.48. The slope of the two segments is not significant (\(p=0.15\) and \(p=0.14\)). Because of the significance of the break, and the non-significence of slopes in the segments, a model with a single step, or structural change model is justified (Figure 1a blue).
+
+Regarding the precipitation series, a single break-date at 1982 (Figure 1b) significantly improves the fit (\(p=0.02\)) as indicated by the increase in \(R^{2}\) from 0.041 to 0.164 (Table 1). The trend in the segment before the break is not significant (\(p=0.88\)), but a decline in precipitation after the break is significant (\(p=0.03\)), most probably due to the anomalously high rainfall in the years after 1978.
+
+Figure 2: Breakpoints in sea surface monthly temperature anomalies (HadCRU3GL) (a)1910 to present and (b) 1976 to present.
+
+A broad peak in the F statistic for annual global surface temperature from 1910 at 1978 (Figure 2a) (\(p<0.001\)) improves the fit of the model from an \(R^{2}\) of 0.56 to 0.68. The 95% confidence interval spans 1966 to 1985, indicated with a red bracket below. When the monthly global surface temperature series is truncated to the break-date of 1978, the F statistics indicate a strong break-date in 1997(8) (Figure 2b), \(R^{2}\) increasing from 0.61 to 0.67) with a rising trend prior to 1997, and a flat trend thereafter (\(p=0.41\)). The 95% CI spans 1996(8) to 1997(12).
+
+## 5 Discussion
+
+These results confirm a break-date around 1978, as suggested by the general form of Q09 (Quirk, 2009), but differs in respect to one, not two breaks, with non-significant trends outside of the optimal break-date. This is known as a change point model, characterized by abrupt changes in the mean value of the underlying dynamical system, rather than a smoothly increasing or decreasing trend. The confidence in 1978 as a break-date is further strengthened by the results for global temperatures since 1910, indicating the series could be described as gradually increasing to 1978 (\(0.05\pm 0.015^{\circ}\)C per decade), with a steeper trend thereafter (\(0.15\pm 0.04^{\circ}\)C per decade).
+
+The Chow test since 1978 finds another significant break-date in 1997, delineating an increasing trend up to 1997 (\(0.13\pm 0.02^{\circ}\)C per decade) and non-significant trend thereafter (\(-0.02\pm 0.05^{\circ}\)C per decade). Contrary to claims in Easterling & Wehner (2009) that the 10 year trend since 1998 is arbitrary, structural change methods indicate that 1997 was a statistically defensible beginning of a new, and apparently stable regime.
+
+The significance of the dates around 1978 and 1997 to climatic regime-shifts is not in dispute, as they are associated with a range of oceanic, atmospheric and climatic events, whereby thermocline depth anomalies associated with PDO phase shift and ENSO were transmitted globally via ocean currents, winds, Rossby and Kelvin waves (Guilderson & Schrag, 1998; McPhaden & Zhang, 2004; Wainwright et al., 2008). Even though the pattern of response would vary in different parts of the globe, a step-change in temperature remarkably similar to Australia also occurred in Alaska (Hartmann & Wendler, 2005).
+
+Given the assumption that AGW is not a cause of regime-shifts, an assumption that would require further verification, these results suggest a considerably stronger claim for Australian climate, i.e. because the slope of the two segments is not significant (p=0.15 and p=0.14), any trend due to increasing \(CO_{2}\) is statistically insignificant.
+
+Decomposition of global temperature into regime-shifts and AGW is more ambiguous, but it would be argued, similar to Q09, that a large proportion of the increase in temperature increase between 1978 and 1997 was associated with the propagation of a major regime-shift, with low rates of AGW in the segments before and after. Supporting this view are such oceanographic observations as the large reduction in the rate of Pacific equatorial upwelling around 1976 and its resumption in 1998, a reduction of westward volume transport between Australia and Indonesia of 23% from 1976-7 and a general elevation of sea temperature by 1-2∘C (Wainwright et al., 2008). Following two decades of weaker flow coinciding with rising global temperatures from the break-date of 1978, convergence of cold interior ocean pycnocline water towards the equator increased from 13.4 to 24.1\(m^{3}s^{-1}\), and Pacific sea surface temperatures cooled (McPhaden & Zhang, 2004). Cai et al. (2007) asserts a role for strengthening of the global conveyor in global temperatures, by intensifying the rate of heat transfer out of the off-equatorial region and into the subtropics. The relative contribution and interaction of various ocean phases: IPO, PDO and ENSO, the duration of lags, and precipitation effects are more uncertain (Power et al., 1999).
+
+One can still argue for human influence on regime-shifts. Cai et al. (2007) attributes changes in the global conveyor to anthropogenic aerosols, and Vecchi et al. (2006) attributes a weakening of the Walker circulation to anthropogenic forcing. Neither attribute abrupt regional climate changes to non-natural causes. There is also the view that stable temperatures since 1997 may be a result of an ’offset’ or masking of AGW effect by natural variation (Keenlyside et al., 2008). For consistency, this view must also entertain the possibility of putative AGW being amplified by natural variability in the previous regime from 1978 to 1997.
+
+Assuming a regime-shift from 1978-98 reduces the estimates of the underlying rate of AGW warming from around \(0.14^{\circ}\)C to \(0.05^{\circ}\)C per decade. An increase of \(0.5^{\circ}\)C by 2100 is consistent with low-end empirical estimates of climate sensitivity, such as Spencer & Braswell (2008) at \(0.6^{\circ}\)C for \(2XCO_{2}\), but considerably lower than the IPCC projections for the most common AGW scenarios (IPCC, 2007). Our model is consistent in period and timing to the two component model proposed by Akasofu (2009), with a contribution from a natural multidecadal oscillation of \(0.15^{\circ}\)C per decade between 1975 and 2000, and a steady contribution of \(0.05^{\circ}\)C per decade from the upswing of a multi-centennial oscillation, such as the natural cycle of glacial formation and regeneration called the Great Season Climatic Oscillation (Boucenna, 2008). In our case, the model was developed from statistically justified, empirical analysis of temperature data.
+
+Figure 3: Prediction of global temperature to 2100, by projecting the trends of segments delineated by significant regime-shifts. The flat trend in the temperature of the current climate-regime (cyan) breaks upwards around 2050 on meeting the (presumed) underlying AGW warming (green), and increases slightly to about \(0.2^{\circ}\)C above present levels by 2100. The 95% CI for the trend uncertainty is dashed.
+
+Figure 3 illustrates the prediction for temperatures to 2100 following from our structural break model, the assumptions of continuous underlying warming, regime-shift from 1978 to 1997, and no additional major regime-shift. The projections formed by the trend to 1978 (presumed AGW warming, green) and the trend in the current regime (AGW offset by regime-shift, cyan) predicts constant temperatures for fifty years to around 2050, similar to the period of flat temperatures from 1930-80, then increasing to about \(0.2^{\circ}\)C above present by 2100. It must be kept in mind that this extrapolation is based on greatly simplified assumptions regarding regime-shifts and trends, and does not incorporate many of the complexities and natural forcing factors operating to produce climate change in the real world (Marsh, 2009).
+
+These results justify further development of more complex regime-shift models of temperature and rainfall for forecasting purposes, including attempting to decompose global climate into temporally and spatially differentiated regime-shift models.
+
+## 6 Acknowledgements
+
+We gratefully acknowledge the comments of John McLean and David Evans on earlier versions of the manuscript.
+
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+

arxiv_ground_truth/0907.2487.md

@@ -0,0 +1,609 @@
+# Resolutions of Hilbert Modules and Similarity
+
+Ronald G. Douglas
+
+Ciprian Foias
+
+Jaydeb Sarkar
+
+Texas A & M University, College Station, Texas 77843, USA
+rdouglas@math.tamu.edu
+jsarkar@math.tamu.edu
+Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
+jaydeb.sarkar@utsa.edu
+
+###### Abstract.
+
+Let \(H^{2}_{m}\) be the Drury-Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function \((z,w)\in\mathbb{B}^{m}\times\mathbb{B}^{m}\rightarrow(1-\sum\limits_{i=1}^{m}z_{i}\bar{w}_{i})^{-1}\). We investigate for which multipliers \(\theta:\mathbb{B}^{m}\rightarrow\mathcal{L}(\mathcal{E},\mathcal{E}_{*})\) with \(\mbox{ran}\,M_{\theta}\) closed, the quotient module \(\mathcal{H}_{\theta}\), given by
+
+\[\cdots\longrightarrow H^{2}_{m}\otimes\mathcal{E}\stackrel{{ M_{\theta}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{*}\stackrel{{\pi_{\theta}}}{{\longrightarrow}}\mathcal{H}_{\theta}\longrightarrow 0,\]
+
+is similar to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\). Here \(M_{\theta}\) is the corresponding multiplication operator in \(\mathcal{L}(H^{2}_{m}\otimes\mathcal{E},H^{2}_{m}\otimes\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) and \(\mathcal{H}_{\theta}\) is the quotient module \((H^{2}_{m}\otimes\mathcal{E}_{*})/M_{\theta}(H^{2}_{m}\otimes\mathcal{E})\), and \(\pi_{\theta}\) is the quotient map. We show that a necessary condition is the existence of a multiplier \(\psi\) in \(\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) such that
+
+\[\theta\psi\theta=\theta.\]
+
+Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of \(H^{2}_{m}\otimes\mathcal{E}\) for a Hilbert space \(\mathcal{E}\), which is valid for the case of \(m=1\). The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a _finite_ resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator \(m\)-tuple (or algebra) is not necessarily commuting (or commutative). In this case the converse to the similarity result is always valid.
+
+Key words and phrases: quotient module, shift operator, similarity, Commutant lifting theorem, resolutions of Hilbert module 2000 Mathematics Subject Classification: 46E22, 46M20, 46C07, 47A13, 47A20, 47A45, 47B32 This research was partially supported by a grant from the National Science Foundation.
+
+## 1. Introduction
+
+A well known result in operator theory (see [18] and [19]) states that the contraction operator given by a canonical model is similar to a unilateral shift of some multiplicity if and only if its characteristic function has a left inverse. Various approaches to this one-variable result have been given (cf. [21]) but a new one is given in this paper which uses the commutant lifting theorem (CLT). In particular, the proof does not involve, at least explicitly, the geometry of the dilation space for the contraction.
+
+The Drury-Arveson (DA) space \(H^{2}_{m}\) (see [10], [17], [1]) has been intensively studied by many researchers over the past few decades. In particular, the CLT has been extended to this space with a few necessary changes. Using the CLT, we extend to the DA space one direction of the one variable result on the similarity of quotient modules of the Hardy space on the unit disk. We show that the converse is equivalent to the assertion that each complemented submodule of \(H^{2}_{m}\otimes\mathcal{E}\) for a Hilbert space \(\mathcal{E}\) is isomorphic to \(H^{2}_{m}\otimes\mathcal{E}_{*}\) for some Hilbert space \(\mathcal{E}_{*}\). Of course this result follows trivially from the Beurling-Lax-Halmos theorem (BLHT) in case \(m=1\). (Actually, for \(m=1\) the submodule is isometrically isomorphic to \(H^{2}_{1}\otimes\mathcal{E}_{*}=H^{2}(\mathbb{D})\otimes\mathcal{E}_{*}\).)
+
+In Section 2, we recall some definitions and results in multivariable operator theory. In the next section, we consider a characterization of those pure co-spherically contractive Hilbert modules similar to the DA-module of some multiplicity. Using the representation of submodules of the DA-module by inner multipliers [16], we are able to obtain the characterization in terms of inner multiplier associated with a given quotient Hilbert module and the regular inverse of that multiplier.
+
+The quotient modules described above are the simplest case of a resolution by DA-modules for which the connecting maps are all partial isometries or inner multipliers (see [12]). More precisely, using the results of Arveson [1] and Muller and Vasilescu [17] and McCullough and Trent [16], for a given pure co-spherical contractive Hilbert module one can obtain an inner resolution. In [3], Arveson suggested that the inner resolution might not terminate as resolutions do in the algebraic context. In this paper we show that the only isometric inner multiplier, \(V:H^{2}_{m}\otimes\mathcal{E}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), is the trivial one determined by an isometric operator \(V_{0}:1\otimes\mathcal{E}\to 1\otimes\mathcal{E}_{*}\). As a consequence, we show that all finite inner resolutions are trivial in a sense that will be explained in Section 4.
+
+In Section 5, we are able to apply essentially the same proofs to the non-commutative case to obtain an analogous result, except here we need the noncommutative analogue of the BLHT due to Popescu ([24], [23]). More precisely, we show that a quotient of the Fock Hilbert space, \(F^{2}_{m}\otimes\mathcal{E}\), for some Hilbert space \(\mathcal{E}\), by the range of a multi-analytic map \(\Theta\) is similar to \(F^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\) if and only if \(\Theta\) has a multi-analytic regular inverse.
+
+In a concluding section we indicate that many of these results can be extended to complete Nevanlinna-Pick kernel Hilbert spaces and to other Hilbert modules for which the CLT holds.
+
+Acknowledgement: The authors wish to thank the referee for a careful reading of the manuscript and useful remarks which led to an improved paper.
+
+## 2. Preliminaries
+
+We consider two cases, the first one in which the operators commute, or for which the algebra is \(\mathbb{C}[z_{1},\ldots,z_{m}]\) and hence commutative, and the second in which the operators are not assumed to commute or the algebra is \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\). We begin with the commutative case.
+
+Let \(\{T_{1},\ldots,T_{m}\}\) be a commuting \(m\)-tuple of bounded linear operators on a Hilbert space \(\mathcal{H}\); that is, \([T_{i},T_{j}]=T_{i}T_{j}-T_{j}T_{i}=0\) for \(i,j=1,\ldots,m\). A Hilbert module \(\mathcal{H}\) over the polynomial algebra \(\mathbb{C}[z_{1},\ldots,z_{m}]\) of \(m\) commuting variables is defined so that the module multiplication \(\mathbb{C}[z_{1},\ldots,z_{m}]\times\mathcal{H}\rightarrow\mathcal{H}\) is defined by
+
+\[p(z_{1},\ldots,z_{m})\cdot h=p(T_{1},\ldots,T_{m})h,\]
+
+where \(p(z_{1},\ldots,z_{m})\in\mathbb{C}[z_{1},\ldots,z_{m}]\) and \(h\in\mathcal{H}\). We denote by \(M_{1},\ldots,M_{m}\) the operators defined to be module multiplication by the coordinate functions. More precisely,
+
+\[M_{i}h=z_{i}\cdot h=T_{i}h,\quad\quad(h\in\mathcal{H},i=1,\ldots,m).\]
+
+All submodules in this paper are assumed to be closed in the norm topology.
+
+A Hilbert module over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) is said to be _co-spherically contractive_, or define a _row contraction_, if
+
+\[\|\sum_{i=1}^{m}M_{i}h_{i}\|^{2}\leq\sum_{i=1}^{m}\|h_{i}\|^{2},\quad(h_{1},\ldots,h_{m}\in\mathcal{H}),\]
+
+or, equivalently, if
+
+\[\sum_{i=1}^{m}M_{i}M_{i}^{*}\leq I_{\mathcal{H}}.\]
+
+Natural examples of co-spherically contractive Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) are the DA-module, the Hardy module and the Bergman module, all defined on the unit ball \(\mathbb{B}^{m}\) in \(\mathbb{C}^{m}\). These are all reproducing kernel Hilbert spaces over \(\mathbb{B}^{m}\) and, among them, the DA-module plays the key role for the class of co-spherically contractive Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\). In order to be more precise, we briefly recall that a scalar reproducing kernel \(K\) on a set \(X\) is a function \(K:X\times X\rightarrow\mathbb{C}\) which satisfies
+
+\[\sum_{i,j=1}^{l}\bar{c}_{i}c_{j}K(x_{i},x_{j})>0,\]
+
+for \(x_{1},\ldots,x_{l}\in X\), \(c_{1},\ldots,c_{l}\in\mathbb{C}\) with not all \(c_{i}\) zero and \(l\in\mathbb{N}\). The reproducing kernel Hilbert space \(\mathcal{H}_{K}\), corresponding to the kernel \(K\), is the Hilbert space of functions defined on \(X\) with the following reproducing property
+
+\[f(x)=\langle f,K_{x}\rangle,\;f\in\mathcal{H}_{K},\]
+
+where for each \(x\in X\), \(K_{x}:X\rightarrow\mathbb{C}\) is the vector in \(\mathcal{H}_{K}\) defined by \(K_{x}(w)=K(w,x)\), \(w\in X\). The DA-module \(H^{2}_{m}\) is the reproducing kernel Hilbert space corresponding to the kernel \(K:\mathbb{B}^{m}\times\mathbb{B}^{m}\rightarrow\mathbb{C}\) defined by
+
+\[K(z,w)=(1-\sum_{i=1}^{m}z_{i}\bar{w}_{i})^{-1},\;(z,w)\in\mathbb{B}^{m}\times\mathbb{B}^{m}.\]
+
+We identify the Hilbert tensor product \(H^{2}_{m}\otimes\mathcal{E}\) with the \(\mathcal{E}\)-valued \(H^{2}_{m}\) space \(H^{2}_{m}(\mathcal{E})\) or the \(\mathcal{L}(\mathcal{E})\)-valued reproducing kernel Hilbert space with the kernel \((z,w)\mapsto(1-\sum\limits_{i=1}^{m}z_{i}\bar{w}_{i})^{-1}I_{\mathcal{E}}\). Consequently,
+
+\[H^{2}_{m}\otimes\mathcal{E}=\{f\in\mathcal{O}(\mathbb{B}^{m},\mathcal{E}):f(z)=\sum_{\bm{k}\in\mathbb{N}^{m}}a_{\bm{k}}z^{\bm{k}},a_{\bm{k}}\in\mathcal{E},\|f\|^{2}:=\sum_{\bm{k}\in\mathbb{N}^{m}}\frac{\|a_{\bm{k}}\|^{2}}{\gamma_{\bm{k}}}<\infty\},\]
+
+where \(\mathcal{O}(\mathbb{B}^{m},\mathcal{E})\) is the space of \(\mathcal{E}\)-valued holomorphic functions on \(\mathbb{B}^{m}\), \(\bm{k}=(k_{1},\ldots,k_{m})\) and \(\gamma_{\bm{k}}=\frac{(k_{1}+\cdots+k_{m})!}{k_{1}!\cdots k_{m}!}\) are the multinomial coefficients. A function \(\varphi\in\mathcal{O}(\mathbb{B}^{m},\mathcal{L}(\mathcal{E},\mathcal{E}_{*}))\) is said to be a _multiplier_ if \(\varphi f\in H^{2}_{m}\otimes\mathcal{E}_{*}=H^{2}_{m}(\mathcal{E}_{*})\) for all \(f\in H^{2}_{m}\otimes\mathcal{E}=H^{2}_{m}(\mathcal{E})\). By the closed graph theorem, such a multiplier \(\varphi\) defines a bounded module map
+
+\[M_{\varphi}:H^{2}_{m}\otimes\mathcal{E}\to H^{2}_{m}\otimes\mathcal{E}_{*},\quad M_{\varphi}f=\varphi f,\,f\in H^{2}_{m}\otimes\mathcal{E}.\]
+
+Equivalently, we can consider \(\varphi\in\mathcal{O}(\mathbb{B}^{m},\mathcal{L}(\mathcal{E},\mathcal{E}_{*}))\) for which \(M_{\varphi}\) defines a bounded operator from \(H^{2}_{m}\otimes\mathcal{E}\) to \(H^{2}_{m}\otimes\mathcal{E}_{*}\). The set of all such bounded multipliers \(\varphi\in\mathcal{O}(\mathbb{B}^{m},\mathcal{L}(\mathcal{E},\mathcal{E}_{*}))\) will be denoted by \(\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\). A multiplier \(\varphi\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) is said to be _inner_ if \(M_{\varphi}\) is a partial isometry in \(\mathcal{L}(H^{2}_{m}\otimes\mathcal{E},H^{2}_{m}\otimes\mathcal{E}_{*})\).
+
+We recall an analogue of the CLT due to Ball-Trent-Vinnikov (Theorem 5.1 in [4]) on DA-modules which will be used to prove some of the main results of this paper.
+
+Theorem 2.1**.**: _(Ball-Trent-Vinnikov) Let \(\mathcal{N}\) and \(\mathcal{N}_{*}\) be quotient modules of \(H^{2}_{m}\otimes\mathcal{E}\) and \(H^{2}_{m}\otimes\mathcal{E}_{*}\) for some Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), respectively. If \(X:\mathcal{N}\rightarrow\mathcal{N}_{*}\) is a bounded module map, that is,_
+
+\[XP_{\mathcal{N}}(M_{z_{i}}\otimes I_{\mathcal{E}})|_{\mathcal{N}}=P_{\mathcal{N}_{*}}(M_{z_{i}}\otimes I_{\mathcal{E}_{*}})|_{\mathcal{N}_{*}}X,\]
+
+_for \(i=1,\ldots,m\), then there exists a multiplier \(\varphi\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) such that (i) \(\|X\|=\|M_{\varphi}\|\) and_
+
+_(ii) \(P_{\mathcal{N}_{*}}M_{\varphi}=X.\)_
+
+_In the language of Hilbert modules, one has_ \(\pi_{\mathcal{N}_{*}}M_{\varphi}=X\pi_{\mathcal{N}}\)_, where_ \(\pi_{\mathcal{N}}\) _and_ \(\pi_{\mathcal{N}_{*}}\) _are the quotient maps._
+
+The above statement of the CLT for \(\mathbb{C}[z_{1},\ldots,z_{m}]\) is due to Ball-Trent-Vinnikov as indicated. However, Popescu pointed out that the result follows from its noncommutative analogue established earlier by him in [22, 23]. A more recent paper on this topic is due to Davidson and Le ([5]).
+
+We now recall the notion of pureness for a co-spherically contractive Hilbert module \(\mathcal{H}\) over \(\mathbb{C}[z_{1},\ldots,z_{m}]\). Define the completely positive map
+
+\[P_{\mathcal{H}}:\mathcal{L}(\mathcal{H})\rightarrow\mathcal{L}(\mathcal{H})\]
+
+by
+
+\[P_{\mathcal{H}}(A)=\sum_{i=1}^{m}M_{i}AM^{*}_{i},\quad\quad\quad\quad(A\in\mathcal{L}(\mathcal{H})).\]
+
+Now
+
+\[I_{\mathcal{H}}\geq P_{\mathcal{H}}(I_{\mathcal{H}})\geq P^{2}_{\mathcal{H}}(I_{\mathcal{H}})\geq\cdots\geq P^{l}_{\mathcal{H}}(I_{\mathcal{H}})\geq\cdots\geq 0,\]
+
+so that
+
+\[P_{\infty}=\mbox{SOT}-\mbox{lim}_{l\rightarrow\infty}P_{\mathcal{H}}^{l}(I_{\mathcal{H}})\]
+
+exists and \(0\leq P_{\infty}\leq I_{\mathcal{H}}\). The Hilbert module \(\mathcal{H}\) is said to be _pure_ if
+
+\[P_{\infty}=0.\]
+
+A canonical example of a pure co-spherically contractive Hilbert module over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) is the DA-module \(H^{2}_{m}\otimes\mathcal{F}\), where \(\mathcal{F}\) is a Hilbert space. Moreover, quotients of DA-modules characterize all pure co-spherically contractive Hilbert modules.
+
+Theorem 2.2**.**: _Let \(\mathcal{H}\) be a pure co-spherically contractive Hilbert module. Then_
+
+_(i)(Arveson[1], Muller-Vasilescu[17]) \(\mathcal{H}\) is isometrically isomorphic with a quotient module \((H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mathcal{S}\), where \(\mathcal{E}\) is a Hilbert space and \(\mathcal{S}\) is a submodule of \(H^{2}_{m}\otimes\mathcal{E}_{*}\)._
+
+_(ii) (McCullough-Trent[16]) If \(\mathcal{S}\) is a submodule of \(H^{2}_{m}\otimes\mathcal{E}_{*}\), then there exists a multiplier \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for some Hilbert space \(\mathcal{E}\) such that \(M_{\theta}\) is inner and \(\mathcal{S}=M_{\theta}(H^{2}_{m}\otimes\mathcal{E})\)._
+
+_Therefore,_ \(\mathcal{H}\) _is isometrically isomorphic to_ \((H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mbox{ran}\,M_{\theta}\) _for some inner multiplier_ \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) _and Hilbert space_ \(\mathcal{E}\)_._
+
+Note that in the statement of Theorem 2.1, Ball-Trent-Vinnikov [4] made the additional assumption that the submodules \(\mathcal{N}^{\perp}\) and \(\mathcal{N}_{*}^{\perp}\) are invariant under the scalar multipliers. However, that this condition is redundant follows from part (ii) of Theorem 2.2 above due to McCullough-Trent.
+
+We now consider some preliminaries for the case of noncommuting operators. Let \(\mathbb{F}^{+}_{m}\) denote the free semigroup with the \(m\) generators \(g_{1},\ldots,g_{m}\) and let \(F^{2}_{m}\) be the full Fock space of \(m\) variables, which is a Hilbert space. More precisely, if we let \(\{e_{1},\ldots,e_{m}\}\) be the standard orthonormal basis of \(\mathbb{C}^{m}\), then
+
+\[F^{2}_{m}=\bigoplus_{k\geq 0}(\mathbb{C}^{m})^{\otimes k},\]
+
+where \((\mathbb{C}^{m})^{\otimes 0}=\mathbb{C}.\) The creation, or left shift, operators \(S_{1},\ldots,S_{m}\) on \(F^{2}_{m}\) are defined by
+
+\[S_{i}f=e_{i}\otimes f,\]
+
+for all \(f\) in \(F^{2}_{m}\) and \(i=1,\ldots,m\).
+
+Let \(\{T_{1},\ldots,T_{m}\}\) be \(m\) bounded linear operators on a Hilbert space \(\mathcal{K}\) which are not necessarily commuting. One can make \(\mathcal{K}\) into a Hilbert module over the algebra of polynomials \(\mathbb{F}[Z_{1},\ldots Z_{m}]\), in \(m\) noncommuting variables, as follows:
+
+\[\mathbb{F}[Z_{1},\ldots Z_{m}]\times\mathcal{K}\rightarrow\mathcal{K},\quad p(Z_{1},\ldots,Z_{m})\cdot h\mapsto p(T_{1},\ldots,T_{m})h,\,h\in\mathcal{K}.\]
+
+The module \(\mathcal{K}\) over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\) is said to be co-spherically contractive if the row operator given by module multiplication by the coordinate functions is a contraction.
+
+A bounded linear operator \(\Theta\in\mathcal{L}(F^{2}_{m}\otimes\mathcal{E},F^{2}_{m}\otimes\mathcal{E}_{*})\), for some Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), is said to be a multi-analytic operator if it is a module map; that is, if
+
+\[\Theta(S_{i}\otimes I_{\mathcal{E}})=(S_{i}\otimes I_{\mathcal{E}_{*}})\Theta,\;\;i=1,\ldots,m.\]
+
+Given a multi-analytic operator \(\Theta\) as above, one can define a bounded linear operator \(\theta:\mathcal{E}\to F^{2}_{m}\otimes\mathcal{E}_{*}\) by
+
+\[\theta x=\Theta(1\otimes x)\quad\quad\quad\quad(x\in\mathcal{E}).\]
+
+In this correspondence of \(\Theta\) and \(\theta\), each uniquely determines the other. Moreover, one defines the operator coefficients \(\theta_{\bm{\alpha}}\in\mathcal{L}(\mathcal{E},\mathcal{E}_{*})\) of \(\Theta\) by
+
+\[{\langle}\theta_{\bm{\alpha}^{t}}x,y{\rangle}=\langle\theta x,e_{\bm{\alpha}}\otimes y\rangle={\langle}\Theta(1\otimes x),e_{\bm{\alpha}}\otimes y{\rangle}\quad\quad\quad(x\in\mathcal{E},y\in\mathcal{E}_{*})\]
+
+for each \(\bm{\alpha}\in\mathbb{F}^{+}_{m}\), where \(\bm{\alpha}^{t}=g_{i_{p}}\cdots g_{i_{1}}\) for \(\bm{\alpha}=g_{i_{1}}\cdots g_{i_{p}}\). It was proved by Popescu (cf. [24]) that
+
+\[\Theta=\mbox{SOT}-\mbox{lim}_{r\to 1^{-}}\sum_{l=0}^{\infty}\sum_{|\bm{\alpha}|=l}r^{|\bm{\alpha}|}R^{\bm{\alpha}}\otimes\theta_{\bm{\alpha}},\]
+
+where \(R_{i}=U^{*}S_{i}U\) for \(i=1,\ldots,m,\) are the right creation operators on \(F^{2}_{m}\), \(R^{\bm{\alpha}}=R_{g_{i_{1}}}\cdots R_{g_{i_{p}}}\) for \(\bm{\alpha}=g_{i_{1}}\cdots g_{i_{p}}\), and \(U\) is the unitary operator on \(F^{2}_{m}\) defined by \(Ue_{\bm{\alpha}}=e_{\bm{\alpha}^{t}}\) for \(\bm{\alpha}\in\mathbb{F}^{+}_{m}\). The set of all multi-analytic operators in \(\mathcal{L}(F^{2}_{m}\otimes\mathcal{E},F^{2}_{m}\otimes\mathcal{E}_{*})\) coincides with \(R^{\infty}_{m}\bar{\otimes}\mathcal{L}(\mathcal{E},\mathcal{E}_{*})\), the WOT closed algebra generated by the spatial tensor product of \(R^{\infty}_{m}\) and \(\mathcal{L}(\mathcal{E},\mathcal{E}_{*})\), where \(R^{\infty}_{m}=U^{*}F^{\infty}_{m}U\) and \(F^{\infty}_{m}\) is the WOT closed algebra generated by the left creation operators, \(S_{1},\ldots,S_{m},\) and the identity operator on \(F^{2}_{m}\).
+
+Notice that the definition of a pure co-spherically contractive Hilbert module can be extended to the noncommutative case; that is, with appropriate change of notation, the concept of a pure co-spherically contractive Hilbert module \(\mathcal{K}\) over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\) can be defined in a similar way. Popescu proved that any pure co-spherically contractive Hilbert module over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\) can be realized as a quotient module of \(F^{2}_{m}\otimes\mathcal{E}\) for some Hilbert space \(\mathcal{E}\) (see [23] and Theorem 2.10 and references in [24]).
+
+Theorem 2.3**.**: _(Popescu) Given a pure co-spherically contractive Hilbert module \(\mathcal{K}\) over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\), there is a multi-analytic operator \(\Theta\) in \(\mathcal{L}(F^{2}_{m}\otimes\mathcal{E},F^{2}_{m}\otimes\mathcal{E}_{*})\) for some Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), which is isometric such that \(\mathcal{K}\) is isometrically isomorphic to the quotient of \(F^{2}_{m}\otimes\mathcal{E}_{*}\) by the range of \(\Theta\). Moreover, the characteristic operator function \(\Theta\) is a complete unitary invariant for \(\mathcal{K}\)._
+
+Finally we need to extend the analogue of the BLHT to the noncommuting setting which is due to Popescu ([23], [24]).
+
+Theorem 2.4**.**: _(Popescu) If \(\mathcal{S}\) is a closed subspace of \(F^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\), then the following are equivalent:_
+
+_(i) \(\mathcal{S}\) is a submodule of \(F^{2}_{m}\otimes\mathcal{F}\)._
+
+_(ii) There exists a Hilbert space \(\mathcal{E}\) and an (isometric) inner multi-analytic operator \(\Phi:F^{2}_{m}\otimes\mathcal{E}\to F^{2}_{m}\otimes\mathcal{F}\) such that_
+
+\[\mathcal{S}=\Phi(F^{2}_{m}\otimes\mathcal{E}).\]
+
+## 3. Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\)
+
+Let \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) be a multiplier for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) such that \(M_{\theta}\) has closed range and let \(\mathcal{H}_{\theta}\) be the quotient module defined by the sequence
+
+\[\cdots\longrightarrow H^{2}_{m}\otimes\mathcal{E}\stackrel{{ M_{\theta}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{*}\stackrel{{\pi_{\theta}}}{{\longrightarrow}}\mathcal{H}_{\theta}\longrightarrow 0,\]
+
+where \(\pi_{\theta}\) is the quotient map of \(H^{2}_{m}\otimes\mathcal{E}_{*}\) onto the quotient of \(H^{2}_{m}\otimes\mathcal{E}_{*}\) by the range of \(M_{\theta}\). There are several possible relationships between these objects:
+
+Statement 1**.**: _The sequence splits or \(\pi_{\theta}\) is right invertible; that is, there exists a module map \(\sigma_{\theta}:\mathcal{H}_{\theta}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) such that_
+
+\[\pi_{\theta}\sigma_{\theta}=I_{\mathcal{H}_{\theta}}.\]
+
+Statement 2**.**: _The multiplication operator \(M_{\theta}\) has a left inverse. Equivalently, there exists a multiplier \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) satisfying \(\psi(z)\theta(z)=I_{\mathcal{E}}\) for \(z\in\mathbb{B}^{m}\)._
+
+Statement 3**.**: _The multiplier \(\theta\) has a regular inverse. Equivalently, there exists a multiplier \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) satisfying_
+
+\[\theta(z)\psi(z)\theta(z)=\theta(z),\]
+
+_for \(z\in\mathbb{B}^{m}\)._
+
+Note that in case \(\mbox{ker~{}}M_{\theta}=\{0\}\), Statements 2 and 3 are equivalent.
+
+Statement 4**.**: _The range of \(M_{\theta}\) is complemented in \(H^{2}_{m}\otimes\mathcal{E}_{*}\) or there exists a submodule \(\mathcal{S}\) of \(H^{2}_{m}\otimes\mathcal{E}_{*}\) such that_
+
+\[\mbox{ran~{}}M_{\theta}\stackrel{{\bm{.}}}{{+}}\mathcal{S}=H^{2}_{m}\otimes\mathcal{E}_{*}.\]
+
+Statement 5**.**: _The quotient Hilbert module \(\mathcal{H}_{\theta}\) is similar to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\)._
+
+Statement 6**.**: _Suppose \(H^{2}_{m}\otimes\mathcal{E}_{*}\) is a skew direct sum \(\mathcal{S}_{1}\stackrel{{\bm{.}}}{{+}}\mathcal{S}_{2}\), where \(\mathcal{E}_{*}\) is a Hilbert space and \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) are submodules such that \(\mathcal{S}_{1}\) is isomorphic to \(H^{2}_{m}\otimes\mathcal{E}\) for some Hilbert space \(\mathcal{E}\). Then \(\mathcal{S}_{2}\) is isomorphic to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\)._
+
+Note that Statements 4 and 6 imply Statement 5 and would be the converse to Corollary 3.5.
+
+Statement 7**.**: _If \(\mathcal{S}\) is a complemented submodule of \(H^{2}_{m}\otimes\mathcal{E}_{*}\) for some Hilbert space \(\mathcal{E}_{*}\), then \(\mathcal{S}\) is isomorphic to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\)?_
+
+One can reformulate Statement 7 in the following equivalent form.
+
+Statement 8**.**: _Every complemented submodule \(\mathcal{S}\) of \(H^{2}_{m}\otimes\mathcal{E}\), for some Hilbert space \(\mathcal{E}\), is the range of \(M_{\psi}\) for a multiplier \(\psi\in\mathcal{M}(\mathcal{E},\mathcal{F})\) with \(\mbox{ker}\,M_{\psi}=\{0\}\) for some Hilbert space \(\mathcal{F}\)._
+
+Note that one could view an affirmation of this statement as a weak form of the BLHT for DA-modules.
+
+Statement 7 raises an important issue for Hilbert modules: are the complemented submodules \(\mathcal{S}\neq\{0\}\) of \(\mathcal{R}\otimes\mathbb{C}^{n}\) always isomorphic to \(\mathcal{R}\otimes\mathbb{C}^{k}\) for some \(0<k\leq n\). This is certainly not true for a general Hilbert module \(\mathcal{R}\). However, what if \(\mathcal{R}\) belongs to the class of “locally-free” Hilbert modules of multiplicity one which is the case for the DA-module \(H^{2}_{m}\)? For \(m=1\), an affirmation follows trivially from the BLHT. A less obvious argument shows that the result holds for more general “locally-free” Hilbert modules over the unit disk such as the Bergman module. (Although the language is different, this result was proved by J. S. Fang, C. L. Jiang, X. Z. Guo, K. Ti and H. He. The study of the relationship between the eight statements in the one-variable case is close to the theme of the book by C. L. Jiang and F. Wang [15], where details can be found.) Further, one can establish an affirmation to Statement 6 if one assumes that the multiplier \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) is holomorphic on a neighborhood of the closure of \(\mathbb{B}^{m}\), at least if \(\mathcal{E}\) and \(\mathcal{E}_{*}\) are finite dimensional. However, what happens in general for “locally-free” Hilbert modules over \(\mathbb{B}^{m}\), such as the DA-module, is not clear at this point.
+
+Moreover, a necessary condition for \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) with \(H^{2}_{m}\otimes\mathbb{C}^{n}=\mathcal{S}_{1}\stackrel{{\bm{.}}}{{+}}\mathcal{S}_{2}\) to be isomorphic to \(H^{2}_{m}\otimes\mathbb{C}^{k}\) and \(H^{2}_{m}\otimes\mathbb{C}^{n-k}\), respectively, is the existence of a generating set \(\{f_{1},\ldots,f_{n}\}\) for \(H^{2}_{m}\otimes\mathbb{C}^{n}\) with \(\{f_{1},\ldots,f_{k}\}\) in \(\mathcal{S}_{1}\) and \(\{f_{k+1},\ldots,f_{n}\}\) in \(\mathcal{S}_{2}\). Note one can view each \(f_{i}\in\mathcal{O}(\mathbb{B}^{m},\mathbb{C}^{n})\) for \(i=1,\ldots,n\). If one assumes in addition that the vectors \(\{f_{i}\}\) are in \(\mathcal{M}(\mathbb{C}^{n},\mathbb{C}^{m})\), then that is also sufficient.
+
+Note that given a complemented submodule \(\mathcal{S}\) of \(H^{2}_{m}\otimes\mathcal{E}\), that is, for some submodule \(\tilde{\mathcal{S}}\) one has \(\mathcal{S}\stackrel{{\bm{.}}}{{+}}\tilde{\mathcal{S}}=H^{2}_{m}\otimes\mathcal{E}\), there are many choices \(\tilde{\tilde{\mathcal{S}}}\) so that the skew direct sum \(\mathcal{S}\stackrel{{\bm{.}}}{{+}}\tilde{\tilde{\mathcal{S}}}\) is isomorphic to \(H^{2}_{m}\otimes\mathcal{E}_{*}\) for some Hilbert space \(\mathcal{E}_{*}\). (Here we allow a different space \(\mathcal{E}_{*}\).) It is not clear, but seems unlikely that there exists a canonical choice of \(\mathcal{E}_{*}\) and \(\tilde{\tilde{\mathcal{S}}}\), in some sense, or what the “simplest” choice might be. Such ideas are related to the \(K\)-theory group introduced in [15].
+
+In commutative algebra one shows that a short exact sequence of modules
+
+\[0\longrightarrow A\stackrel{{\varphi_{1}}}{{\longrightarrow}}B\stackrel{{\varphi_{2}}}{{\longrightarrow}}C\longrightarrow 0,\]
+
+splits, or \(\varphi_{2}\) has a right inverse, if and only if \(\varphi_{1}\) has a left inverse. Using the closed graph theorem, we can extend this result to Hilbert modules. Moreover, with the CLT we can extend the result to the case where \(\varphi_{1}\) has a kernel. We begin with the simpler result.
+
+Theorem 3.1**.**: _Let \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) be a multiplier for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) such that \(\mbox{ran}\,M_{\theta}\) is closed. Then \(\mbox{ran}\,M_{\theta}\) is complemented in \(H^{2}_{m}\otimes\mathcal{E}_{*}\) if and only if there exists a module map \(\sigma_{\theta}:\mathcal{H}_{\theta}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) which is a right inverse for \(\pi_{\theta}\)._
+
+Proof. If \(H^{2}_{m}\otimes\mathcal{E}_{*}=\mbox{ran}\,M_{\theta}\stackrel{{\cdot}}{{+}}\mathcal{S}\) for a (closed) submodule \(\mathcal{S}\), then \(Y=\pi_{\theta}|_{\mathcal{S}}\) is one-to-one and onto. Hence \(Y^{-1}:(H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mbox{ran}\,M_{\theta}\rightarrow\mathcal{S}\) is bounded by the closed graph theorem and \(\sigma_{\theta}=i~{}Y^{-1}\) is a right inverse for \(\pi_{\theta}\), where \(i:\mathcal{S}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) is the inclusion map.
+
+Conversely, if there exists a right inverse \(\sigma_{\theta}:\mathcal{H}_{\theta}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) for \(\pi_{\theta}\), then \(\sigma_{\theta}\pi_{\theta}\) is an idempotent on \(H^{2}_{m}\otimes\mathcal{E}_{*}\) such that \(\mathcal{S}=\mbox{ran}\,\sigma_{\theta}\pi_{\theta}\) is a complementary submodule for the closed submodule \(\mbox{ran}\,M_{\theta}\) in \(H^{2}_{m}\otimes\mathcal{E}_{*}\).
+
+For a multiplier \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) one could consider the quotient of \(H^{2}_{m}\otimes\mathcal{E}_{*}\) by the closure of \(\mbox{ran~{}}M_{\theta}\). Examples in the case \(m=1\) show that the existence of a right inverse for \(\pi_{\theta}\) does not imply that \(\mbox{ran}\,M_{\theta}\) is closed. Theorem 3.1 shows that the Statements 1 and 4 are equivalent but the preceding comment implies the necessity of the assumption that \(\mbox{ran~{}}M_{\theta}\) is closed. However, we do have the following folklore result which helps clarify matters.
+
+Remark 3.2**.**: _If \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) and \(\mbox{ran}\,M_{\theta}\) is complemented in \(H^{2}_{m}\otimes\mathcal{E}_{*}\), then \(\mbox{ran}\,M_{\theta}\) is closed._
+
+As one knows, by considering the \(m=1\) case, there is more than one multiplier \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) with the same range and thus yielding the same quotient. While things are even more complicated for \(m>1\), the following result using the CLT introduces some order.
+
+Theorem 3.3**.**: _Let \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) be an inner multiplier for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) and \(\varphi\in\mathcal{M}(\mathcal{F},\mathcal{E}_{*})\) for some Hilbert space \(\mathcal{F}\). Then there exists a multiplier \(\psi\in\mathcal{M}(\mathcal{F},\mathcal{E})\) such that \(\varphi=\theta\psi\) if and only if_
+
+\[\mbox{ran~{}}M_{\varphi}\subseteq\mbox{ran~{}}M_{\theta}.\]
+
+Proof. If \(\psi\in\mathcal{M}(\mathcal{F},\mathcal{E})\) such that \(\varphi=\theta\psi\), then \(M_{\varphi}=M_{\theta}M_{\psi}\) and hence
+
+\[\mbox{ran~{}}M_{\varphi}=\mbox{ran~{}}M_{\theta}M_{\psi}\subseteq\mbox{ran~{}}M_{\theta}.\]
+
+Suppose \(\mbox{ran~{}}M_{\varphi}\subseteq\mbox{ran~{}}M_{\theta}\) and \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) is an inner multiplier. This implies that \(\mbox{ran}\,M_{\theta}\) is closed. Consider the module map
+
+\[\hat{M_{\theta}}:(H^{2}_{m}\otimes\mathcal{E})~{}/~{}\mbox{ker~{}}M_{\theta}\longrightarrow\mbox{ran~{}}M_{\theta}\]
+
+defined by
+
+\[\hat{M_{\theta}}\gamma_{\theta}=M_{\theta},\]
+
+which is invertible since \(\mbox{ran~{}}M_{\theta}\) is closed. Let \(\gamma_{\theta}:H^{2}_{m}\otimes\mathcal{E}\longrightarrow(H^{2}_{m}\otimes\mathcal{E})/~{}\mbox{ker~{}}M_{\theta}\) be the quotient module map. Set \(\hat{X}=\hat{M_{\theta}}^{-1}\). Then \(\hat{X}:\mbox{~{}ran~{}}M_{\theta}\rightarrow(H^{2}_{m}\otimes\mathcal{E})/\mbox{~{}ker~{}}M_{\theta}\) is bounded by the closed graph theorem and so is \(\hat{X}M_{\varphi}:H^{2}_{m}\otimes\mathcal{F}\rightarrow(H^{2}_{m}\otimes\mathcal{E})\,/\,\mbox{ker~{}}M_{\theta}\). Appealing to the CLT yields a multiplier \(\psi\in\mathcal{M}(\mathcal{F},\mathcal{E})\) so that
+
+\[\gamma_{\theta}M_{\psi}=\hat{X}M_{\varphi},\]
+
+and hence
+
+\[M_{\theta}M_{\psi}=(\hat{M_{\theta}}\gamma_{\theta})M_{\psi}=\hat{M_{\theta}}(\hat{X}M_{\varphi})=M_{\varphi},\]
+
+or \(\varphi=\theta\psi\) which completes the proof.
+
+Note that the result holds for a multiplier \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) so long as \(\mbox{ran~{}}M_{\theta}\) is closed since that is all that the proof uses.
+
+We now consider our principal result on multipliers and regular inverses.
+
+Theorem 3.4**.**: _Let \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) be a multiplier for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\). Then there exists \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) such that_
+
+\[M_{\theta}M_{\psi}M_{\theta}=M_{\theta}\]
+
+_if and only if \(\mbox{ran}\,M_{\theta}\) is complemented in \(H^{2}_{m}\otimes\mathcal{E}_{*}\), or_
+
+\[H^{2}_{m}\otimes\mathcal{E}_{*}=\mbox{ran}\,M_{\theta}\stackrel{{\cdot}}{{+}}\mathcal{S},\]
+
+_for some submodule \(\mathcal{S}\) of \(H^{2}_{m}\otimes\mathcal{E}_{*}\)._
+
+Proof. If \(H^{2}_{m}\otimes\mathcal{E}_{*}=\mbox{ran}\,M_{\theta}\stackrel{{\cdot}}{{+}}\mathcal{S}\) for some (closed) submodule \(\mathcal{S}\), then \(\mbox{ran~{}}M_{\theta}\) is closed by Remark 3.2. Consider the module map
+
+\[\hat{M_{\theta}}:(H^{2}_{m}\otimes\mathcal{E})\,/\,\mbox{ker}\,M_{\theta}\longrightarrow(H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mathcal{S},\]
+
+defined by
+
+\[\hat{M_{\theta}}\gamma_{\theta}=\pi_{\mathcal{S}}M_{\theta},\]
+
+where \(\gamma_{\theta}:H^{2}_{m}\otimes\mathcal{E}\rightarrow(H^{2}_{m}\otimes\mathcal{E})\,/\,\mbox{ker}\,M_{\theta}\) and \(\pi_{\mathcal{S}}:H^{2}_{m}\otimes\mathcal{E}_{*}\rightarrow(H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mathcal{S}\) are quotient maps. This map is one-to-one and onto and thus has a bounded inverse \(\hat{X}=\hat{M_{\theta}}^{-1}:(H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mathcal{S}\rightarrow(H^{2}_{m}\otimes\mathcal{E})\,/\,\mbox{ker~{}}M_{\theta}\) by the closed graph theorem. Since \(\hat{X}\) satisfies the hypotheses of the CLT, there exists \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) such that
+
+\[\gamma_{\theta}M_{\psi}=\hat{X}\pi_{\mathcal{S}}.\]
+
+Further, \(\hat{M_{\theta}}\gamma_{\theta}=\pi_{\mathcal{S}}M_{\theta}\) yields
+
+\[\pi_{\mathcal{S}}M_{\theta}M_{\psi}=\hat{M_{\theta}}\gamma_{\theta}M_{\psi}=\hat{M_{\theta}}\hat{X}\pi_{\mathcal{S}}=\pi_{\mathcal{S}},\]
+
+and therefore
+
+\[\pi_{\mathcal{S}}(M_{\theta}M_{\psi}M_{\theta}-M_{\theta})=0.\]
+
+Since \(\pi_{\mathcal{S}}\) is one-to-one on \(\mbox{ran}\,M_{\theta}\), it follows that \(M_{\theta}M_{\psi}M_{\theta}=M_{\theta}\).
+
+Now suppose there exists \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) such that \(M_{\theta}M_{\psi}M_{\theta}=M_{\theta}\). This implies that
+
+\[(M_{\theta}M_{\psi})^{2}=M_{\theta}M_{\psi},\]
+
+and hence \(M_{\theta}M_{\psi}\) is an idempotent. From the equality \(M_{\theta}M_{\psi}M_{\theta}=M_{\theta}\) we obtain both that \(\mbox{ran}\,M_{\theta}M_{\psi}\) contains \(\mbox{ran}\,M_{\theta}\) and that \(\mbox{ran}\,M_{\theta}M_{\psi}\) is contained in \(\mbox{ran}\,M_{\theta}\). Therefore,
+
+\[\mbox{ran}\,M_{\theta}M_{\psi}=\mbox{ran}\,M_{\theta},\]
+
+and
+
+\[\mathcal{S}=\mbox{ran}\,(I-M_{\theta}M_{\psi}),\]
+
+is a complementary submodule of \(\mbox{ran~{}}M_{\theta}\) in \(H^{2}_{m}\otimes\mathcal{E}_{*}\).
+
+Note that Theorem 3.4 implies that Statements 3 and 4 are equivalent.
+
+Corollary 3.5**.**: _Assume \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) such that \(\mbox{ran}\,M_{\theta}\) is closed and \(\mathcal{H}_{\theta}\) is defined by_
+
+\[H^{2}_{m}\otimes\mathcal{E}\stackrel{{ M_{\theta}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{*}\longrightarrow\mathcal{H}_{\theta}\longrightarrow 0.\]
+
+_If \(\mathcal{H}_{\theta}\) is similar to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\), then the sequence splits._
+
+Proof. First, assume that there exists an invertible module map \(X:H^{2}_{m}\otimes\mathcal{F}\rightarrow\mathcal{H}_{\theta}\), and let \(\varphi\in\mathcal{M}(\mathcal{F},\mathcal{E}_{*})\) be defined by the CLT so that \(\pi_{\theta}M_{\varphi}=X\), where \(\pi_{\theta}:H^{2}_{m}\otimes\mathcal{E}_{*}\rightarrow(H^{2}_{m}\otimes\mathcal{E}_{*})\,/\,\mbox{ran~{}}M_{\theta}\) is the quotient map. Since \(X\) is invertible we have
+
+\[H^{2}_{m}\otimes\mathcal{E}_{*}=\mbox{ran~{}}M_{\varphi}\stackrel{{\bm{.}}}{{+}}\mbox{ran}M_{\theta}.\]
+
+Thus \(\mbox{ran~{}}M_{\theta}\) is complemented and hence it follows from the previous corollary that the sequence splits.
+
+Corollary 3.5 shows that Statement 6 implies Statement 1. If Statement 6 is valid, then the converse to Corollary 3.5 holds. Moreover, we see that Statement 8 implies that \(\mathcal{H}_{\theta}\) is similar to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\) or that Statement 4 is valid. Finally, the following weaker converse to Corollary 3.5 always holds.
+
+Corollary 3.6**.**: _Let \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), and set \(\mathcal{H}_{\theta}=(H^{2}_{m}\otimes\mathcal{E}_{*})/\mbox{~{}clos~{}}[\mbox{ran~{}}M_{\theta}]\). Then the following statements are equivalent:_
+
+_(i) there exists_ \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) _such that_ \(\psi(x)\theta(z)=I_{\mathcal{E}}\) _for_ \(z\in\mathbb{B}^{m}\)_, and_
+
+_(ii)_ \(\mbox{ran~{}}M_{\theta}\) _is closed,_ \(\mbox{ker~{}}M_{\theta}=\{0\}\) _and_ \(\mathcal{H}_{\theta}\) _is similar to a complemented submodule_ \(\mathcal{S}\) _of_ \(H^{2}_{m}\otimes\mathcal{E}_{*}\)_._
+
+Proof. If (i) holds, then \(\mbox{ran~{}}M_{\theta}\) is closed and \(\mbox{ker~{}}M_{\theta}=\{0\}\). Further, \(M_{\theta}M_{\psi}\) is an idempotent on \(H^{2}_{m}\otimes\mathcal{E}_{*}\) such that \(\mbox{ran~{}}M_{\theta}M_{\psi}=\mbox{ran~{}}M_{\theta}\) and \(\mathcal{H}_{\theta}\) is isomorphic to \(\mathcal{S}=\mbox{ran~{}}(I-M_{\theta}M_{\psi})\subseteq H^{2}_{m}\otimes\mathcal{E}_{*}\) and \(H^{2}_{m}\otimes\mathcal{E}_{*}=\mbox{ran~{}}M_{\theta}\stackrel{{\bm{.}}}{{+}}\mathcal{S}\) so \(\mathcal{S}\) is complemented.
+
+Now assume that (ii) holds and there exists an isomorphism \(X:\mathcal{H}_{\theta}\rightarrow\mathcal{S}\subseteq H^{2}_{m}\otimes\mathcal{E}_{*}\), where \(\mathcal{S}\) is a complemented submodule of \(H^{2}_{m}\otimes\mathcal{E}_{*}\). Then \(Y=X\pi_{\theta}:H^{2}_{m}\otimes\mathcal{E}_{*}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) is a module map and hence there exists a multiplier \(\omega\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E}_{*})\) so that \(Y=M_{\omega}\). Since \(X\) is invertible, \(\mbox{ran~{}}M_{\omega}=\mathcal{S}\), which is complemented by assumption, and hence by Theorem 3.4 there exists \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E}_{*})\) such that \(M_{\omega}=M_{\omega}M_{\psi}M_{\omega}\) or \(M_{\omega}(I-M_{\psi}M_{\omega})=0\). Therefore,
+
+\[\mbox{ran~{}}(I-M_{\psi}M_{\omega})\subseteq\mbox{ker~{}}M_{\omega}=\mbox{ker~{}}Y=\mbox{ker~{}}\pi_{\theta}=\mbox{ran~{}}M_{\theta}.\]
+
+Applying Theorem 3.3, we obtain \(\varphi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) so that
+
+\[I-M_{\psi}M_{\omega}=M_{\theta}M_{\varphi}.\]
+
+Thus using \(M_{\omega}M_{\theta}=0\) we see that
+
+\[M_{\theta}M_{\varphi}M_{\theta}=(I-M_{\psi}M_{\omega})M_{\theta}=M_{\theta},\]
+
+or
+
+\[M_{\theta}=M_{\theta}M_{\varphi}M_{\theta}.\]
+
+Since \(\mbox{ker~{}}M_{\theta}=\{0\}\), we have
+
+\[M_{\varphi}M_{\theta}=I_{H^{2}_{m}\otimes\mathcal{E}},\]
+
+which completes the proof.
+
+Combining Theorem 3.4 and Corollary 3.5 yields our main result in the commutative setting for similarity.
+
+Corollary 3.7**.**: _Given \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) such that \(\mbox{ran~{}}M_{\theta}\) is closed, consider the quotient Hilbert module \(\mathcal{H}_{\theta}\) defined as above. If \(\mathcal{H}_{\theta}\) is similar to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\), then there exists a multiplier \(\psi\in\mathcal{M}(\mathcal{E}_{*},\mathcal{E})\) satisfying_
+
+\[\theta(z)\psi(z)\theta(z)=\theta(z),\quad\mbox{for}\;z\in\mathbb{B}^{m}.\]
+
+In conclusion, Corollary 3.7 shows that Statement 5 implies Statement 3.
+
+## 4. Resolutions of Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\)
+
+Consideration of resolutions such as those in the preceding section and the ones given in Theorem 2.2 raises the question of what kind of resolutions exist for pure co-spherically contractive Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\). In particular, Theorem 2.2 yields a unique resolution of an arbitrary pure co-spherically contractive Hilbert module \(\mathcal{M}\) over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) in terms of DA-modules and inner multipliers. More specifically, consider DA-modules \(\{H^{2}_{m}\otimes\mathcal{E}_{k}\}\) for Hilbert spaces \(\{\mathcal{E}_{k}\}\) and inner multipliers \(\varphi_{k}\in\mathcal{M}(\mathcal{E}_{k},\mathcal{E}_{k-1})\), or partially isometric module maps \(\{M_{\varphi_{k}}\}\) for \(k\geq 1\); set
+
+\[X_{k}=M_{\varphi_{k}}:H^{2}_{m}\otimes\mathcal{E}_{k}\to H^{2}_{m}\otimes\mathcal{E}_{k-1},\;\;k\geq 1;\]
+
+and a co-isometric module map
+
+\[X_{0}=\pi_{\mathcal{M}}:H^{2}_{m}\otimes\mathcal{E}_{0}\rightarrow\mathcal{M},\]
+
+which is exact. That is, \(\mbox{ran~{}}X_{k+1}=\mbox{ker~{}}X_{k}\) for \(k\geq 1\). Here \(k=0,1,\ldots,N\), with the possibility of \(N=+\infty\). A basic question is whether such a resolution can have finite length or, equivalently, whether we can take \(\mathcal{E}_{N}=\{0\}\) for some finite \(N\). That will be the case if and only if some \(X_{k}\) is an isometry or, equivalently, if \(\mbox{ker}X_{k}=\{0\}\). Unfortunately, the following result shows that this is not possible when \(m>1\), unless \(\mathcal{M}\) is a DA-module and the resolution is a trivial one.
+
+Theorem 4.1**.**: _For \(m>1\), if \(V:H^{2}_{m}\otimes\mathcal{E}\to H^{2}_{m}\otimes\mathcal{E}_{*}\) is an isometric module map for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), then there exists an isometry \(V_{0}:\mathcal{E}\rightarrow\mathcal{E}_{*}\) such that_
+
+\[V(\bm{z}^{\bm{k}}\otimes x)=\bm{z}^{\bm{k}}\otimes V_{0}x,\;\;\mbox{for~{}}\bm{k}\in\mathbb{N}^{m},x\in\mathcal{E}_{*}.\]
+
+_Moreover, \(\mbox{ran~{}}V\) is a reducing submodule of \(H^{2}_{m}\otimes\mathcal{E}_{*}\) of the form \(H^{2}_{m}\otimes(\mbox{ran}\,V_{0})\)._
+
+Proof. For \(x\in\mathcal{E}\), \(\|x\|=1\), we have
+
+\[V(1\otimes x)=f(\bm{z})=\sum_{\bm{k}\in\mathbb{N}^{m}}a_{\bm{k}}\bm{z}^{\bm{k}},\quad\quad\mbox{for}\,\{a_{k}\}\subseteq\mathcal{E}.\]
+
+Then
+
+\[V(z_{1}\otimes x)=VM_{z_{1}}(1\otimes x)=M_{z_{1}}V(1\otimes x)=M_{z_{1}}f=z_{1}f,\]
+
+and
+
+\[\|z_{1}f\|^{2}=\|z_{1}V(1\otimes x)\|^{2}=\|z_{1}\otimes x\|^{2}=1=\|f\|^{2}.\]
+
+Therefore, we have
+
+\[\sum_{\bm{k}\in\mathbb{N}^{m}}\|a_{\bm{k}}\|^{2}_{\mathcal{E}_{*}}\|\bm{z}^{\bm{k}}\|^{2}=\sum_{\bm{k}\in\mathbb{N}^{m}}\|a_{\bm{k}}\|^{2}_{\mathcal{E}_{*}}\|\bm{z}^{\bm{k}+e_{1}}\|^{2},\;\;\mbox{where~{}}\bm{k}+e_{1}=(k_{1}+1,\ldots,k_{m}),\]
+
+or
+
+\[\sum_{\bm{k}\in\mathbb{N}^{m}}\|a_{\bm{k}}\|^{2}_{\mathcal{E}_{*}}\{\|\bm{z}^{\bm{k}+e_{1}}\|^{2}-\|\bm{z}^{\bm{k}}\|^{2}\}=0.\]
+
+If \(\bm{k}=(k_{1},\ldots k_{m})\), then
+
+\[\begin{split}\|\bm{z}^{\bm{k}+e_{1}}\|^{2}&=\frac{(k_{1}+1)!\cdots k_{m}!}{(k_{1}+\cdots+k_{m}+1)!}={\frac{k_{1}!\cdots k_{m}!}{(k_{1}+\cdots+k_{m})!}}\frac{k_{1}+1}{k_{1}+\cdots+k_{m}+1}\\ &<{\frac{k_{1}!\cdots k_{m}!}{(k_{1}+\cdots+k_{m})!}}=\|\bm{z}^{\bm{k}}\|^{2},\end{split}\]
+
+unless \(k_{2}=k_{3}=\ldots=k_{m}=0\). Since, \(a_{\bm{k}}\neq 0\) implies \(\|\bm{z}^{\bm{k}+e_{1}}\|=\|\bm{z}^{\bm{k}}\|\) we have \(k_{2}=\cdots=k_{m}=0\). Repeating this argument using \(i=2,\ldots,m\), we see that \(a_{\bm{k}}=0\) unless \(\bm{k}=(0,\ldots,0)\) and therefore, \(f(\bm{z})=1\otimes y\) for some \(y\in\mathcal{E}_{*}\). Set \(V_{0}x=y\) to complete the first part of the proof.
+
+Finally, since \(\mbox{ran~{}}V=H^{2}_{m}\otimes(\mbox{ran}V_{0})\), we see that \(\mbox{ran~{}}V\) is a reducing submodule, which completes the proof.
+
+Note that this result generalizes Corollary 3.3 of [9] and is related to an earlier result of Guo, Hu and Xu [14].
+
+The theorem implies that all resolutions by DA-modules with partially isometric maps are trivial in a sense we will make precise. We start with a definition.
+
+Definition 4.2**.**: _An inner resolution of length \(N\), for \(N=1,2,3,\ldots,\infty\), for a pure co-spherical contractive Hilbert module \(\mathcal{M}\) is given by a collection of Hilbert spaces \(\{\mathcal{E}_{k}\}_{k=0}^{N}\), inner multipliers \(\varphi_{k}\in\mathcal{M}(\mathcal{E}_{k},\mathcal{E}_{k-1})\) for \(k=1,\ldots,N\) with \(X_{k}=M_{\varphi_{k}}\) and a co-isometric module map \(X_{0}:H^{2}_{m}\otimes\mathcal{E}_{0}\rightarrow\mathcal{M}\) so that_
+
+\[\mbox{ran}\,X_{k}=\mbox{ker}\,X_{k-1},\]
+
+_for \(k=0,1,\ldots,N\). To be more precise, for \(N<\infty\) one has the finite resolution_
+
+\[0\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{N}\stackrel{{ X_{N}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{N-1}\longrightarrow\cdots\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{1}\stackrel{{ X_{1}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{0}\stackrel{{ X_{0}}}{{\longrightarrow}}\mathcal{M}\longrightarrow 0,\]
+
+_and for \(N=\infty\), the infinite resolution_
+
+\[\cdots\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{N}\stackrel{{ X_{N}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{N-1}\longrightarrow\cdots\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{1}\stackrel{{ X_{1}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{0}\stackrel{{ X_{0}}}{{\longrightarrow}}\mathcal{M}\longrightarrow 0.\]
+
+Theorem 4.3**.**: _If the pure, co-spherically contractive Hilbert module \(\mathcal{M}\) possesses a finite inner resolution, then \(\mathcal{M}\) is isometrically isomorphic to \(H^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\)._
+
+Proof. Applying the previous theorem to \(M_{\varphi_{N}}\), we decompose \(\mathcal{E}_{N-1}=\mathcal{E}^{1}_{N-1}\oplus\mathcal{E}^{2}_{N-1}\) so that \(\tilde{M}_{\psi_{N-1}}=M_{N-1}|_{H^{2}_{m}\otimes\mathcal{E}^{2}_{N-1}}\in\mathcal{L}(H^{2}_{m}\otimes\mathcal{E}^{2}_{N-1},H^{2}_{m}\otimes\mathcal{E}_{N-2})\) is an isometry onto \(\mbox{ran}\,M_{N-1}\). Hence, we can apply the theorem to \(\tilde{M}_{N-1}\). Therefore, using induction we obtain the desired conclusion.
+
+The following statement proceeds directly from the theorem.
+
+Corollary 4.4**.**: _If \(\theta\in\mathcal{M}(\mathcal{E},\mathcal{E}_{*})\) is an inner multiplier for the Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) with \(\mbox{ker~{}}M_{\theta}=\{0\}\), then the quotient module \(\mathcal{H}_{\theta}=(H^{2}_{m}\otimes\mathcal{E}_{*})/\,\mbox{ran}\,M_{\theta}\) is isometrically isomorphic to \(H^{2}_{m}\otimes\mathcal{F}\) for a Hilbert space \(\mathcal{F}\). Moreover, \(\mathcal{F}\) can be identified with \((\mbox{ran}\,V_{0})^{\perp}\), where \(V_{0}\) is the isometry from \(\mathcal{E}\) to \(\mathcal{E}_{*}\) given in Theorem 4.1._
+
+Note that in the preceding corollary, one has \(\mbox{dim~{}}\mathcal{E}_{*}=\mbox{dim~{}}\mathcal{E}+\mbox{dim~{}}\mathcal{F}\).
+
+A resolution of \(\mathcal{M}\) can always be made longer in a trivial way. Suppose we have the resolution
+
+\[0\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{N}\stackrel{{ X_{N}}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}_{N-1}\longrightarrow\cdots\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{0}\stackrel{{ X_{0}}}{{\longrightarrow}}\mathcal{M}\longrightarrow 0.\]
+
+If \(\mathcal{E}_{N+1}\) is a nontrivial Hilbert space, then define \(X_{N+1}\) as the inclusion map of \(H^{2}_{m}\otimes\mathcal{E}_{N+1}\subseteq H^{2}_{m}\otimes(\mathcal{E}_{N}\oplus\mathcal{E}_{N+1})\). Further, set \(\tilde{X}_{N}\) equal to \(X_{N}\) on \(H^{2}_{m}\otimes\mathcal{E}_{N}\subseteq H^{2}_{m}\otimes(\mathcal{E}_{N+1}\oplus\mathcal{E}_{N})\) and equal to \(0\) on \(H^{2}_{m}\otimes\mathcal{E}_{N+1}\subseteq H^{2}_{m}\otimes(\mathcal{E}_{N}\oplus\mathcal{E}_{N+1})\). Extending \(\tilde{X}_{N}\) to all of \(H^{2}_{m}\otimes\mathcal{E}_{N+1}\) linearly, we obtain a longer resolution essentially equivalent to the original one
+
+\[0\longrightarrow H^{2}_{m}\otimes\mathcal{E}_{N+1}\stackrel{{ X_{N+1}}}{{\longrightarrow}}H^{2}_{m}\otimes(\mathcal{E}_{N+1}\oplus\mathcal{E}_{N})\stackrel{{\tilde{X}_{N}}}{{\longrightarrow}}\cdots\longrightarrow\mathcal{M}\longrightarrow 0.\]
+
+Moreover, the new resolution will be inner if the original one is.
+
+The proof of the preceding theorem shows that any finite inner resolution by DA-modules is equivalent to a series of such trivial extensions of the resolution
+
+\[0\longrightarrow H^{2}_{m}\otimes\mathcal{E}\stackrel{{ X}}{{\longrightarrow}}H^{2}_{m}\otimes\mathcal{E}\longrightarrow 0,\]
+
+for some Hilbert space \(\mathcal{E}\) and \(X=I_{H^{2}_{m}\otimes\mathcal{E}}\). We will refer to such a resolution as a _trivial inner resolution_. We use that terminology to summarize this supplement to the theorem in the following statement.
+
+Corollary 4.5**.**: _All finite inner resolutions for a pure co-spherically contractive Hilbert module \(\mathcal{M}\) are trivial inner resolutions._
+
+What happens when we relax the conditions on the module maps \(\{X_{k}\}\) so that \(\mbox{ran}\,X_{k}=\mbox{ker}\,X_{{k-1}}\) for all \(k\) but do not require them to be partial isometries? In this case, non-trivial finite resolutions do exist, completely analogous to what happens for the case of the Hardy or Bergman modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) for \(m>1\). We describe a simple example.
+
+Consider the module \(\mathbb{C}_{(0,0)}\) over \(\mathbb{C}[z_{1},z_{2}]\) defined so that
+
+\[p(z_{1},z_{2})\cdot\lambda=p(0,0)\lambda,\;\;\mbox{where~{}}p\in\mathbb{C}[z_{1},z_{2}]\,\mbox{and~{}}\lambda\in\mathbb{C},\]
+
+and the resolution:
+
+\[0\longrightarrow H^{2}_{2}\stackrel{{ X_{2}}}{{\longrightarrow}}H^{2}_{2}\oplus H^{2}_{2}\stackrel{{ X_{1}}}{{\longrightarrow}}H^{2}_{2}\stackrel{{ X_{0}}}{{\longrightarrow}}\mathbb{C}_{(0,0)}\longrightarrow 0,\]
+
+where \(X_{0}f=f(0,0)\) for \(f\in H^{2}_{2}\), \(X_{1}(f_{1}\oplus f_{2})=M_{z_{1}}f_{1}+M_{z_{2}}f_{2}\) for \(f_{1}\oplus f_{2}\in H^{2}_{2}\oplus H^{2}_{2}\), and \(X_{2}f=M_{z_{2}}f\oplus(-M_{z_{1}}f)\) for \(f\in H^{2}_{2}\). One can show that this sequence, which is closely related to the Koszul complex, is exact and non-trivial; in particular, it does not split as trivial resolutions do.
+
+Another question one can ask is the relationship between the inner resolution for a pure co-spherically contractive Hilbert module and more general, _not necessarily inner_, resolutions by DA-modules. In particular, is there any relation between the minimal length of a not necessarily inner resolution and the inner resolution. Theorem 3.3 provides some information on this matter.
+
+A parallel notion of resolution for Hilbert modules was studied by Arveson [3], which is different from the one considered in this paper. For Arveson, the key issue is the behavior of the resolution at \(\bm{0}\in\mathbb{B}^{m}\) or the localization of the sequence of connecting maps at \(\bm{0}\). His main goal, which he accomplishes and is quite non trivial, is to extend an analogue of the Hilbert’s syzygy theorem. In particular, he exhibits a resolution of Hilbert modules in his class which ends in finitely many steps. The resolutions considered in ([7], [6]) and this paper are related to dilation theory although the requirement that the connecting maps are partial isometries is sometimes relaxed.
+
+## 5. Hilbert modules over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\)
+
+Although we use the following lemma only in the non-commutative case, it also holds in the commutative case as indicated.
+
+Lemma 5.1**.**: _If \(\mathcal{H}\) is a co-spherically contractive Hilbert module over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) or \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\), respectively, which is similar to \(H^{2}_{m}\otimes\mathcal{F}\), or \(F^{2}_{m}\otimes\mathcal{F}\), respectively, for some Hilbert space \(\mathcal{F}\), then \(\mathcal{H}\) is pure._
+
+Proof. We use the notation for the commutative case but the proof in both cases is the same. Let \(X:\mathcal{H}\to H^{2}_{m}\otimes\mathcal{F}\) be an invertible module map. Then \(M_{i}=X^{-1}M_{z_{i}}X\) for all \(i=1,\ldots,m\). Since \(\{P^{l}_{\mathcal{H}}(I_{\mathcal{H}})\}^{\infty}_{l=0}\) is a decreasing sequence of positive operators, it suffices to show that
+
+\[\mbox{WOT}-\mbox{lim}_{l\rightarrow\infty}P_{\mathcal{H}}^{l}(I_{\mathcal{H}})=0.\]
+
+To see that this is the case, let \(f_{1}\) be a vector in \(\mathcal{H}\) and set \(f=X^{*-1}f_{1}\). Then
+
+\[\begin{split}{\langle}\sum_{|\bm{k}|=l}M^{\bm{k}}M^{*\bm{k}}f_{1},f_{1}{\rangle}=&{\langle}\sum_{|\bm{k}|=l}X^{-1}M_{z}^{\bm{k}}XX^{*}M_{z}^{*\bm{k}}X^{*-1}f_{1},f_{1}{\rangle}={\langle}\sum_{|\bm{k}|=l}M_{z}^{\bm{k}}XX^{*}M_{z}^{*\bm{k}}f,f{\rangle}\\ &\leq\|X\|^{2}\sum_{|\bm{k}|=l}\langle M_{z}^{k}M^{*k}_{z}f,f\rangle.\end{split}\]
+
+Letting \(l\rightarrow\infty\) in the last expression, we conclude that the required limit is zero, which completes the proof.
+
+Actually, the proof shows that two similar co-spherically contractive Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{m}]\), or two similar contractive Hilbert modules over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\), are either both pure or both not pure.
+
+Theorem 5.2**.**: _Let \(\mathcal{H}\) be a pure co-spherically contractive Hilbert module over \(\mathbb{F}[Z_{1},\ldots,Z_{m}]\). Then \(\mathcal{H}\) is similar to \(F^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\) if and only if the characteristic operator \(\Theta\) of \(\mathcal{H}\) in \(\mathcal{L}(F^{2}_{m}\otimes\mathcal{E},F^{2}_{m}\otimes\mathcal{E}_{*})\), for some Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\), is left invertible; that is, if and only if there exists a multi-analytic operator \(\Psi:F^{2}_{m}\otimes{\mathcal{E}_{*}}\to F^{2}_{m}\otimes\mathcal{E}\) such that_
+
+\[\Psi\Theta=I_{F^{2}_{m}\otimes\mathcal{E}}.\]
+
+Proof. First, using Theorem 2.3 we realize the pure contractive Hilbert module \(\mathcal{H}\) as the quotient module given by its characteristic function \(\Theta\), which is an isometric multi-analytic map. That is,
+
+\[\mathcal{H}\cong\mathcal{H}_{\Theta}=(F^{2}_{m}\otimes\mathcal{E}_{*})/\Theta(F^{2}_{m}\otimes\mathcal{E}).\]
+
+Now given a module map \(X:F^{2}_{m}\otimes\mathcal{F}\rightarrow\mathcal{H}_{\Theta}\), we appeal to the noncommutative analogue of the CLT (see Theorem 6.1 in [23] or Theorem 5.1 in [24]) to obtain a multi-analytic operator \(\Phi:F^{2}_{m}\otimes\mathcal{F}\to F^{2}_{m}\otimes\mathcal{E}_{*}\) such that
+
+\[P_{\mathcal{H}_{\Theta}}\Phi=X.\]
+
+Consider, the bounded module map
+
+\[Z:(F^{2}_{m}\otimes\mathcal{F})\oplus(F^{2}_{m}\otimes\mathcal{E})\to F^{2}_{m}\otimes\mathcal{E}_{*}\]
+
+defined by
+
+\[Z(f\oplus g)=\Phi f+\Theta g,\]
+
+for all \(f\oplus g\in(F^{2}_{m}\otimes\mathcal{F})\oplus(F^{2}_{m}\otimes\mathcal{E})\). Then \(Z\) is invertible if and only if \(X\) is invertible. This follows by noting that \(X\) is invertible if and only if the range of \(Z\), which is the span of \(\mathcal{H}_{\Theta}\) and \(\mbox{ran}\,\Theta\) is \(F^{2}_{m}\otimes\mathcal{E}_{*}\), and \(X\) is one-to-one if and only if \(Z\) is.
+
+To prove the necessity part of the theorem, assume that \(X\) is invertible or, equivalently, that \(Z\) is invertible. Consequently, we can define a module idempotent \(Q\) on \(F^{2}_{m}\otimes\mathcal{E}_{*}\) such that
+
+\[Q\Theta=\Theta\]
+
+and
+
+\[\mbox{ran}\,Q=\mbox{ran}\,\Theta.\]
+
+Then the bounded module map \(\hat{Q}:F^{2}_{m}\otimes\mathcal{E}_{*}\to F^{2}_{m}\otimes\mathcal{E}\) defined by
+
+\[\hat{Q}(\Phi f+\Theta g)=g,\quad\quad\quad\quad\quad\Phi f+\Theta g\in(F^{2}_{m}\otimes\mathcal{E}_{*})\]
+
+satisfies
+
+\[Q=\Theta\hat{Q}.\]
+
+Since \(\hat{Q}\) is a module map, there exists a multi-analytic operator \(\Psi:F^{2}_{m}\otimes\mathcal{E}_{*}\to F^{2}_{m}\otimes\mathcal{E}\) such that
+
+\[\hat{Q}=\Psi.\]
+
+Hence
+
+\[\Theta=Q\Theta=\Theta\hat{Q}\Theta=\Theta\Psi\Theta.\]
+
+Since \(\Theta\) is an isometry, the necessity part follows; that is, \(\Theta\) has a left inverse.
+
+To prove the sufficiency part, let \(\Psi:F^{2}_{m}\otimes\mathcal{E}_{*}\to F^{2}_{m}\otimes\mathcal{E}\) be a multi-analytic operator such that
+
+\[\Psi\Theta=I_{F^{2}_{m}\otimes\mathcal{E}}.\]
+
+Then \(Q=\Theta\Psi\) is an idempotent on \(F^{2}_{m}\otimes\mathcal{E}_{*}\) and any \(f\) in \(F^{2}_{m}\otimes\mathcal{E}_{*}\) can be expressed as
+
+\[f=(f-\Theta\Psi f)+\Theta\Psi f,\]
+
+where \(f-\Theta\Psi f\) is in \(\mbox{ker}\,\Psi\) and \(\Theta\Psi f\) is in \(\mbox{ran}\,\Theta\). Thus,
+
+\[\mbox{ran}\,Q=\mbox{ran}\,\Theta,\quad\quad\mbox{and}\quad\quad\mbox{ker}\,\Psi=\mbox{ran}\,(I-Q).\]
+
+Since \(\mbox{ker}\,\Psi\) is a submodule of \(F^{2}_{m}\otimes\mathcal{E}_{*}\), by Theorem 2.4, the noncommutative version of the BLHT, there exists an inner multi-analytic operator \(\Phi:F^{2}_{m}\otimes\mathcal{F}\to F^{2}_{m}\otimes\mathcal{E}_{*}\) for some Hilbert space \(\mathcal{F}\) such that
+
+\[\mbox{ker~{}}\Psi=\mbox{ran~{}}(I-Q)=\Phi(F^{2}_{m}\otimes\mathcal{F}),\]
+
+Consequently,
+
+\[F^{2}_{m}\otimes\mathcal{E}_{*}=\mbox{ran~{}}\Phi\stackrel{{\bm{.}}}{{+}}\mbox{ran~{}}\Theta.\]
+
+Then one can define the invertible module map \(Z\) as in the necessity part. Setting \(X=P_{\mathcal{H}_{\Theta}}\Phi\) defines the required similarity between \(\mathcal{H}_{\Theta}\) and \(F^{2}_{m}\otimes\mathcal{F}\), which completes the proof.
+
+As mentioned in the introduction, specializing the preceding proof to the (commutative) \(m=1\) case yields a new proof of the old result on the similarity of contraction operators to unilateral shifts.
+
+The main difference in the above proof and that of Corollary 3.7 for the commutative case is that here we can assume that \(\Theta\) has no kernel and one of the complemented submodule is isomorphic to a DA-module.
+
+In the proof of Theorem 5.2, we did not use the fact that the characteristic function is an isometry but the fact that \(\mbox{ker~{}}\Theta=\{0\}\) and \(\mbox{ran~{}}\Theta\) is closed. Hence we can state a more general result in terms of a module resolution.
+
+Theorem 5.3**.**: _Let \(\mathcal{E}\) and \(\mathcal{E}_{*}\) be Hilbert spaces and \(\Theta:F^{2}_{m}\otimes\mathcal{E}\to F^{2}_{m}\otimes\mathcal{E}_{*}\) be a multi-analytic operator such that \(\mbox{ker~{}}\Theta=\{0\}\) and \(\mbox{ran~{}}\Theta\) is closed. Then the quotient space \(\mathcal{H}_{\Theta}\), given by \((F^{2}_{m}\otimes\mathcal{E}_{*})/\,\mbox{ran}\,\Theta\) is similar to \(F^{2}_{m}\otimes\mathcal{F}\) for some Hilbert space \(\mathcal{F}\) if and only if \(\Theta\Psi\Theta=\Theta\) for some multi-analytic operator \(\Psi:F^{2}_{m}\otimes\mathcal{E}_{*}\to F^{2}_{m}\otimes\mathcal{E}\)._
+
+## 6. Concluding remarks
+
+Observe that if \(\mathcal{H}\) is a Hilbert module over \(\mathbb{C}[z_{1},\ldots,z_{m}]\) (or \(A(\Omega)\), where \(\Omega\) is a bounded connected open subset of \(\mathbb{C}^{m}\)), Corollary 3.7 remains true under the assumption that the analogue of the CLT holds for the class of Hilbert modules under consideration. In particular, Corollary 3.7 can be generalized to any reproducing kernel Hilbert module where the kernel is given by a complete Nevanlinna-Pick kernel.
+
+## References
+
+* [1] W. B. Arveson, _Subalgebras of \(C^{*}\)-algebras. III. Multivariable operator theory_, Acta Math. 181 (1998), no. 2, 159–228. MR 2000e:47013.
+* [2] W. B. Arveson, _The curvature invariant of a Hilbert module over \(C[z_{1},\cdots,z_{d}]\)_, J. Reine Angew. Math. 522 (2000), 173–236. MR 1758582.
+* [3] W. B. Arveson, _The free cover of a row contraction_, Doc. Math. 9 (2004), 137–161 MR 2054985.
+* [4] J. A. Ball, T. Trent and V. Vinnikov, _Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces_, Operator theory and analysis (Amsterdam, 1997), 89–138, Oper. Theory Adv. Appl., 122, Birkhäuser, Basel, 2001. MR 2002f:47028.
+* [5] K. R. Davidson and T. Le, _Commutant Lifting for Commuting Row Contractions_, http://arxiv.org/abs/0906.4526
+* [6] R. G. Douglas and G. Misra, _On quasi-free Hilbert modules_, New York J. Math. 11 (2005), 547–561; MR 2007b:46044.
+* [7] R. G. Douglas and G. Misra, _Quasi-free resolutions of Hilbert modules_, Integral Equations Operator Theory 47 (2003), no. 4, 435–456; MR 2004i:46109.
+* [8] R. G. Douglas and V. I. Paulsen, _Hilbert Modules over Function Algebras_, Research Notes in Mathematics Series, 47, Longman, Harlow, 1989. MR 91g:46084.
+* [9] R. G. Douglas and J. Sarkar, _On unitarily equivalent submodules_, Indiana Univ. Math. J. 57 (2008), no. 6, 2729–2743. MR 2482998
+* [10] S. W. Drury, _A generalization of von Neumann’s inequality to the complex ball_, Proc. Amer. Math. Soc., 68 (1978), 300 304, MR 80c:47010.
+* [11] C. Foias and A. Frazho, _The commutant lifting approach to interpolation problems_, Operator Theory: Advances and Applications, 44. Birkhäuser Verlag, Basel, 1990. MR 92k:47033.
+* [12] D. C. V. Greene, _Free resolutions in multivariable operator theory_, J. Funct. Anal. 200 (2003), no. 2, 429–450. MR 2004c:47014
+* [13] D. C. V. Greene, S. Richter and C. Sundberg, _The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels_, J. Funct. Anal. 194 (2002), 311–331. MR 2003h:46038
+* [14] K. Guo, J. Hu and X. Xu, _Toeplitz algebras, subnormal tuples and rigidity on reproducing \(\mathbb{C}[z_{1},\ldots,z_{d}]\)-modules_, J. Funct. Anal. 210 (2004), 214–247. MR 2005a:47007.
+* [15] C. Jiang and Z. Wang, _Structure of Hilbert space operators_, World Scientific Publishing Co. Pvt. Ltd., Hackensack, NJ, 2006. MR 2008j:47001.
+* [16] S. McCullough and T. T. Trent, _Invariant subspaces and Nevanlinna-Pick kernels_, J. Funct. Anal. 178 (2000), no. 1, 226–249. MR 2002b:47006.
+* [17] V. Muller and F.-H. Vasilescu, _Standard models for some commuting multioperators_, Proc. Amer. Math. Soc. 117 (1993), 979–989.MR 93e:47016
+* [18] B. Sz.-Nagy and C. Foias, _Sur les contractions de l’espace de Hilbert. X. Contractions similaires à des transformations unitaires_, Acta Sci. Math. (Szeged) 26 (1965), 79–91. MR 34 1856.
+* [19] B. Sz.-Nagy and C. Foias, _On contractions similar to isometries and Toeplitz operators_, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 553–564. MR 58 30376.
+* [20] B. Sz.-Nagy and C. Foias, _Harmonic Analysis of Operators on Hilbert Space_, North Holland, Amsterdam, 1970. MR 43 947.
+* [21] N. Nikolski, _Operators, functions, and systems: an easy reading. Vol. 2: Model operators and systems_, Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI. MR 2003i:47001b.
+* [22] G. Popescu, _Isometric dilations for infinite sequences of noncommuting operators_, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. MR 90c:47006.
+* [23] G. Popescu, _Characteristic functions for infinite sequences of noncommuting operators_, J. Operator Theory 22 (1989), no. 1, 51–71. MR 91m:47012.
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+

arxiv_ground_truth/0907.4096.md

@@ -0,0 +1,82 @@
+# On Fixed Points of Order K of RSA
+
+_Shaohua Zhang__1,2_
+
+1 School of Mathematics, Shandong University,
+Jinan, Shandong, 250100, China
+2 The key lab of cryptography technology and information security,
+Ministry of Education, Shandong University,
+Jinan, Shandong, 250100, China
+E-mail: shaohuazhang@mail.sdu.edu.cn
+
+###### Abstract
+
+In this paper, we gave a preliminary dynamical analysis on the RSA cryptosystem and obtained a computational formulae of the number of the fixed points of \(k\) order of the RSA. Thus, the problem in [8, 9] has been solved.
+
+**Keywords:** RSA; fixed point; fixed points of order \(k\); fixed points attack; dynamical analysis
+
+**2000 MR Subject Classification:** 11A25; 11T71; 37A45
+
+## 1 Introduction
+
+Shortly after Diffie and Hellman [1] introduced the idea of public key cryptography, Rivest, Shamir and Adleman (RSA) [2] proposed such a cryptosystem. A simplified version of RSA is the following:
+
+Let \(n=pq\) be the product of two large primes of the same size. Let \(e,d\) be two integers satisfying \(ed\equiv 1(\mod\varphi(n))\). Call \(n\) the RSA modulus, \(e\) the encryption exponent, and \(d\) the decryption exponent. Let \(e\) and \(n\) be public keys, and let \(d\) be the corresponding secret key. A message is an integer \(m\in Z_{n}\). To encrypt \(m\), one computes \(m^{e}\equiv c(\mod n)\). To decrypt the ciphertext \(c\), the receiver computes \(m\equiv m^{ed}\equiv c^{d}(\mod n)\). Denote such a cryptosystem by \(RSA(n,e)\). We call \(m\) a fixed point of \(RSA(n,e)\) if \(m^{e}\equiv m(\mod n)\). And call \(m\) a fixed point of order \(k\) if \(k\) is the smallest positive integer such that \(m^{e^{k}}\equiv m(\mod n)\). Clearly, \(f:x\longrightarrow x^{e}(\mod n)\) is a dynamical system. Thus, \(k\) is exactly the period of \(m\). For more details on the arithmetic of dynamical systems, see [7].
+
+In 1979, Blakley and Borosh [3] first pointed out that there were at least 9 fixed points in \(RSA(n,e)\). For more references on fixed points, also see [4]-[6]. Denoted the set of all fixed points of order \(k\) of \(RSA(n,e)\) by \(E_{n,e,k}\) and the cardinality of the set \(S\) by \(|S|\). In [8, 9], Yu considered the general case of fixed points of order \(k\) and gave geometric mean value of \(|T_{n,e,k}|\) and pointed out that it was difficult to give a quantitative description of \(|T_{n,e,k}|\), where \(k\) is a given positive integer and
+
+\[T_{n,e,k}=\{x|\forall m<k,m\in N,x\in Z_{n}^{\ast},x^{e^{k}}\equiv x(\mod n),x^{e^{m}}\neq x(\mod n)\}.\]
+
+In this essay, we preliminarily consider this question and obtain the following results:
+
+**Theorem 1 \(|T_{n,e,k}|=\sum_{d|k}\mu(k/d)(e^{d}-1,p-1)(e^{d}-1,q-1)\)** , where \(\mu(\cdot)\) is the Möbius function.
+
+Based on this result, we get Theorem 2.
+
+**Theorem 2 \(|E_{n,e,k}|=\sum_{d|k}\mu(k/d)((e^{d}-1,p-1)+1)((e^{d}-1,q-1)+1)\)**.
+
+## 2 Proof of Main Theorems
+
+We denote the set of positive integers by \(N\) . For given positive integers \(a\) and \(b\), we write \(a|b\) if \(a\) divides \(b\) . And denote the greatest common divisor of \(a\) and \(b\) by \((a,b)\). Denote a complete set of residues modulo \(n\) by \(Z_{n}\), where \(1<n\in N\) , and a reduced set of residues modulo \(n\) is denoted by \(Z_{n}^{\ast}\). Let \(a\) be an integer relatively prime to \(n\). The order of \(a\) modulo \(n\), denoted by \(ord_{n}(a)\), which is the smallest positive integer \(d\) such that \(a^{d}\equiv 1(\mod n)\).
+
+**Lemma 1[8]** For \(1<n\in N\), \(r\in N\), let the canonical factorization of \(n\) be \(\prod\limits_{i=1}^{m}p_{i}^{a_{i}}\) and \(T_{n,r}=\{x|x^{r}\equiv 1(\mod n),1\leq x<n\}\), then \(|T_{n,r}|=\prod\limits_{i=1}^{m}(r,\varphi(p_{i}^{a_{i}}))\).
+
+**Lemma 2** For \(1<n\in N\), \(a,m,k\in N\), \(e\in Z_{\varphi(n)}^{\ast}\), if \(a^{e^{k}}\equiv a(\mod n)\) and \(k|m\), then \(a^{e^{m}}\equiv a(\mod n)\).
+
+**Proof** Let \(m=tk\). When \(t=1\), clearly \(a^{e^{k}}\equiv a^{e^{m}}\equiv a(\mod n)\). Suppose that \(a^{e^{m}}\equiv a(\mod n)\) when \(t=l\). And when \(t=l+1\), we have \(a^{e^{m}}\equiv a^{e^{lk}e^{k}}\equiv a^{e^{k}}\equiv a(\mod n)\). It immediately shows that Lemma 2 is true by induction.
+
+**Lemma 3** For \(1<n\in N\), \(a,m,k\in N\), \(e\in Z_{\varphi(n)}^{\ast}\), if \(a^{e^{m}}\equiv a(\mod n)\) and \(a\in E_{n,e,k}\), then \(k|m\).
+
+**Proof** Let \(m=kt+r\), \(t\in N\), \(0\leq r<k\). We have \(a^{e^{m}}\equiv a^{e^{kt}e^{r}}\equiv a^{e^{r}}\equiv a(\mod n)\) by Lemma 2. Since \(a\in E_{n,e,k}\), hence \(r=0\), and Lemma 3 is true.
+
+**Proof of Theorem 1** By Lemma 1 and Lemma 3, it is easy to deduce \(\sum_{d|k}|T_{n,e,d}|=\prod\limits_{i=1}^{m}(e^{k}-1,\varphi(p_{i}^{a_{i}}))\). By Möbius inversion, it immediately shows that Theorem 1 is true.
+
+**Proof of Theorem 2** By Lemma 2 and Lemma 3, analogously, using Chinese Remainder Theorem and the method of proof of Theorem 1, it is easy to deduce that Theorem 2 is true.
+
+**Corollary 1** Let \(1<n\in N\), \(r\in N\), and let the canonical factorization of \(n\) be \(\prod\limits_{i=1}^{m}p_{i}^{a_{i}}\). Then \(|\{x|ord_{n}(x)=r,1\leq x\in Z_{n}^{\ast}\}|=\sum_{d|r}(\mu(r/d)\prod\limits_{i=1}^{m}(d,\varphi(p_{i}^{a_{i}})))\).
+
+**Corollary 2** Let \(1<n\in N\), \(r\in N\), let the canonical factorization of \(n\) be \(\prod\limits_{i=1}^{m}p_{i}^{a_{i}}\), and let \(F_{n,r}=\{x|\forall k<r,k\in N,1\leq x\leq n,x^{r}\equiv x(\mod n),x^{k}\neq x(\mod n)\}\). Then \(|F_{n,r}|=\sum_{d|r}(\mu(r/d)\prod\limits_{i=1}^{m}(1+(d-1,\varphi(p_{i}^{a_{i}}))))\).
+
+## 3 Conclusion
+
+Clearly, if the factorization of \(n\) is known, then computing the number of the fixed points of order \(k\) of the RSA cryptosystem is simple and convenient by the presented formulae. This is useful to pick the encryption exponent, which is necessary to ensure the resulting RSA safe from fixed points attack. Maybe we are not afraid of a fixed point. However, the following problem should be further considered: Is there a polynomial-time algorithm for finding a fixed point \(m\), where \(m\neq 0,\pm 1\)? This problem and Factoring the RSA modulus perhaps are equivalent.
+
+**Remark:** This paper is the revision of paper [10] in the proceedings of China Crypt’2006, whose Chinese version has been accepted by _Journal of Mathematics_ (Wuhan, China).
+
+## 4 Acknowledgements
+
+I am thankful to the referees for their suggestions improving the presentation of the paper and also to my supervisor Professor Wang Xiaoyun for her valuable help and encouragement. Thank Institute for Advanced Study in Tsinghua University for providing us with excellent conditions. This work was partially supported by the National Basic Research Program (973) of China (No. 2007CB807902) and the Natural Science Foundation of Shandong Province (No. Y2008G23).
+
+## References
+
+* [1] Diffie W. and Hellman M., New directions in cryptography, _IEEE Transactions on Information Theory_, 1976, IT-22: 644-654.
+* [2] Rivest R. L., Shamir A. and Adleman L., A method for obtaining digital signatures and public key cryptosystems, _Communications of the ACM_, 1978, 21: 120-126.
+* [3] Blakley G.R. and Borosh I., Rivest-Shamir-Adleman public key cryptosystems do not always conceal messages, _Comp.& Maths. with Appls._, 1979, 5:169-178.
+* [4] Blakley Bob and Blakley G.R., Security of number theoretic public key cryptosystems against random attack , _Cryptologia_, 1978, 2(4): 306-321.
+* [5] Blakley Bob and Blakley G.R., Security of number theoretic public key cryptosystems against random attack , _Cryptologia_, 1979, 3(1): 29-42.
+* [6] Blakley Bob and Blakley G.R., Security of number theoretic public key cryptosystems against random attack , _Cryptologia_, 1979, 3(2): 105-118.
+* [7] Silverman J.H., _The arithmetic of dynamical systems_, GTM 241, Springer-Verlag, 2007.
+* [8] Yu X.Y., A note on fixed points of an RSA system, _Chinese Journal of Computers_, 2001, 24(9): 998-1001.
+* [9] Yu X.Y., A note on the RSA fixed points , _Chinese Journal of Computers_, 2002, 25(5): 497-500.
+* [10] Zhang S.H., On Fixed Points of Order k of RSA, China Crypt’2006, Science Press, 2006, 265-267. Accepted by _Journal of Mathematics_, in press.
+

arxiv_ground_truth/0907.4417.md

@@ -0,0 +1,443 @@
+# Magnetic Nanoparticle Assemblies
+
+Dimitris Kechrakos ¹
+
+Footnote 1: Contact author: Tel : +30-210-2896705, Fax : +30-210-2896713, E-mail : _dkehrakos@aspete.gr_ 
+
+###### Abstract
+
+This chapter provides an introduction to the fundamental physical ideas and models relevant to the phenomenon of magnetic hysteresis in nanoparticle assemblies. The concepts of single-domain particles and superparamagnetism are discussed. The mechanisms of magnetization by coherent rotation and the role of temperature in the gradual decay of magnetization are analyzed in the framework of simple analytical models. Modern numerical techniques (Monte Carlo simulations, Magnetization Dynamics) used to study dense nanoparticle assemblies are presented. An overview of the most common experimental techniques used to measure the magnetic hysteresis effect in nanoparticle assemblies are presented and the underlying principles are exposed.
+
+Department of Sciences, School of Pedagogical and Technological Education (ASPETE), Athens 14121, Greece
+
+_Keywords_: magnetic nanoparticles; magnetic anisotropy; dipolar interactions; magnetic hysteresis; superparamagnetism; mean field theory; Monte Carlo; magnetization dynamics
+
+###### Contents
+
+1. 1 Introduction
+2. 2 Isolated magnetic nanoparticles
+  1. 2.1 Single-Domain Particles
+  2. 2.2 Magnetization by Coherent Rotation
+  3. 2.3 Magnetic behavior at finite temperature
+  1. 2.3.1 Superparamagnetism and Blocking temperature
+  2. 2.3.2 Thermal relaxation under an applied field
+3. 3 Magnetic Measurements
+  1. 3.1 Field-cooled (FC) and Zero-Field-Cooled (ZFC) Magnetization
+  2. 3.2 Remanent magnetization and Coercive field
+4. 4 Interacting nanoparticle assemblies
+  1. 4.1 Introduction
+  2. 4.2 Mean Field models
+  3. 4.3 Numerical Techniques
+  1. 4.3.1 The Monte Carlo method
+  2. 4.3.2 The Magnetization Dynamics method
+  3. 4.3.3 Time scale of numerical methods
+  4. 4.3.4 MMC study of dipolar interacting assemblies : A case study
+5. 5 Summary
+6. 6 Future perspectives
+
+## 1 Introduction
+
+Magnetic nanoparticles (MNPs) are minute parts of magnetic materials with typical size well below \(10^{-7}m\). They are present in different materials found in nature such as rocks, living organisms, ceramics and corrosion products, but they are also artificially made and used as the active component of ferrofluids, permanent magnets, soft magnetic materials, biomedical materials and catalysts. Their diverse applications in geology, physics chemistry, biology and medicine renders the study of their properties of great importance both to science and technology.
+
+In geology, the nature and origin of magnetic phenomena related to the presence of magnetite nanoparticles in rocks is of great interest to the palaeomagnetist who searches for the geomagnetic record of rocks. The presence of magnetite particles associated with the trigeminal nerve in pigeons offers a reliable explanation to the Earth’s magnetic field detection and the consequent navigation capability. In fine arts, magnetic analysis of ancient paintings facilitates the reconstruction of the production techniques of ancient ceramics. In living organisms, the role of ferritin, a magnetic nanoparticle per se, is important among the iron storage proteins. MNPs are also used as contrast agents in Magnetic Resonance Imaging. Recent work has involved the development of bioconjugated MNPs, which facilitated specific targeting of these MRI probes to brain tumors. MNPs are also used as highly active catalysts which has long been demonstrated by the the use of finely divided metals in several reactions. Owing to their high surface-to-volume ratio MNPs of iron are more efficient at waste remediation than bulk iron.
+
+High density magnetic data storage media provide a major technological driving force for further exploration of MNPs. It is expected that if MNPs with diameter \(~{}5nm\) can be used as individually addressed magnetic bits, magnetic data storage densities of \(~{}1Tbit/in^{2}\) would be achieved, namely an order of magnitude higher than the present record (Moser 2002). MNPs have also been demonstrated to be functional elements in magneto-optical switches, sensors based on Giant Magneto-Resistance and magnetically controllable Single Electron Transistor devices.
+
+The most common preparation methods for MNPs produce assemblies with different structural and compositional characteristics that depend on the particular method adopted. Granular films, ferrofluids and cluster-assembled films are characterized as assemblies with random order in MNP locations, while ordered arrays are found in patterned media (known also as magnetic dots) and self-assembled films. The MNP preparation methods are divided to top-down and bottom-up. In top-down methods , the NPs are formed from a larger system by appropriate physical processing, such as thermal treatment, etching, etc. In bottom-up methods, the NPs are formed by an atomic nucleation process that takes place either in ultrahigh vacuum or in a liquid environment. The latter method relies on colloidal chemistry techniques and presently appears to be the most promising method for production of nanoparticles with extremely narrow size distribution. Colloidal synthesis methods combined with self-assembly methods produce MNP samples with both size uniformity and long range structural order. It is worth noticing that structural order in a MNP assembly is a decisive property for production of ultrahigh density storage media. Owing to their attractive features and their low cost, colloidal synthesis methods and self-assembly attract presently intense research activity in the field of MNP preparation (Petit 1998, Murray 2001, Willard 2004, Farrell 2005, Darling 2005).
+
+The magnetic properties of MNPs and their assemblies provide a fascinating field for basic research, which is done on two different scales, the atomic and the mesoscopic. In the atomic scale, the properties of individual MNPs are examined and they are revealed in samples with low particle concentration. In the mesoscopic scale, dense samples are examined which exhibit collective magnetic behavior arising from interparticle interactions. The study of the magnetic properties can be naturally divided in the investigation of the ground state configuration (long range order, disorder, etc) and the excitations from it. Excitations can be either weak, as for example at low temperature and weak external magnetic field, or strong, as for example, close to a thermal phase transition or under a reversing magnetic field.
+
+For individual MNPs the ground state configuration can differ remarkably from the parent bulk material in various ways. For example, owing to energy balance reasons, the abundance of magnetic domains that form in a bulk magnet can be replaced by a single domain in a MNP, which then becomes magnetically saturated even in the absence of an external magnetic field (Néel 1949). The application of an external field forces the atomic magnetic moments of a single-domain MNP to rotate coherently (Stoner 1948). Also, for temperature above a threshold, the direction of particle’s magnetization fluctuates at random, making the particle bahave as a molecule with a giant magnetic moment. The applications of this effect, known as superparamagnetism (Bean 1959), are presently a lot, ranging from geology to medicine. Finally, we should remark that the above described simplified picture of a single-domain MNP becomes invalid if one considers the crucial effect of the MNP surface. Reduced crystal symmetry and chemical disorder close to the surface can produce variations between the surface and interior magnetic structure and modify the overall response of the MNP to an applied field (Kodama 1999).
+
+When MNPs form dense assemblies, interparticle interactions produce a collective behavior, by coupling the magnetic moments of individual MNPs. This fact renders in most cases even the determination of the ground state configuration an intricate physical problem. The collective behavior of dense (interacting) assemblies is reflected also on the modified magnetic response of the assembly, compared to isolated MNPs. The most complex behavior occurs in samples with random morphology and long-range magnetostatic interactions. Various experimental measurements have been proposed to reveal the nature of the interparticle interactions, and various measuring protocols probe different aspects of the collective behavior. On the other hand, analytical models have difficulties in predicting or explaining the magnetic behavior of these interacting MNP assemblies, and most of the curret research relies on numerical simulations.
+
+In this chapter we provide an introduction to the fundamental ideas and concepts pertaining to the magnetic properties of MNP assemblies. Emphasis is given to the response of MNP assemblies to an applied magnetic field and the related issue of magnetization reversal. The chapter is organized as follows : In Section 2 we discuss the magnetic properties of individual (isolated) MNPs. Fist, the condition under which a single-domain MNP is formed is derived, and then the magnetic response under an applied field is examined. The presentation is based on a simple theoretical model (Néel 1949, Stoner 1948). In Section 3 we give a brief overview of the most common magnetic characterization techniques and explain the information extracted from each one. In Section 4 we discuss the response of a dense MNP assembly to a magnetic field, when the interparticle interactions are important and lead to a collective behavior of the MNPs. Mean-field models are presented and an introduction to modern numerical techniques (Monte Carlo, Magnetization dynamics) to tackle this problem are presented. The chapter is summarized in Section 5 and the perspectives in this field are presented in Sections 6.
+
+## 2 Isolated magnetic nanoparticles
+
+In this section we derive the criterion for formation of single-domain MNPs and examine the magnetization process at zero temperature by coherent rotation of magnetization (Stoner-Wohlfarth model). The behavior of a MNP assembly at finite temperature is discussed and the related concepts of superparamagnetism and blocking temperature are introduced. The effects of an applied dc magnetic field is examined within the simplest model assuming uniaxial anisotropy and bistability of particle moments (Néel model).
+
+### Single-Domain Particles
+
+The ground state magnetic structure of a ferromagnetic (FM) material is the outcome of the balance between three different types of energies, namely, the exchange (\(U_{ex}\)), the magnetostatic (\(U_{m}\)) and the anisotropy energy (\(U_{a}\)). The exchange interaction has its origin in the Pauli exclusion principle for electrons. Let the FM material be divided in small cubic elements each one carrying a magnetic moment \(\overrightarrow{\mu_{i}}\). The exchange interaction between the cubic elements favors parallel alignment of neighboring magnetic moments and it is written in the usual Heisenberg form as \(U_{ex}=-(A/a^{2})\sum_{ij}cos\theta_{ij}\), where \(A\) is the stiffness constant, \(a\) is the lattice constant and \(\theta_{ij}\) is the angle between moments at sites \(i\) and \(j\). The stiffness constant is related to the microscopic exchange energy \(J\) through the relation \(A=zJS^{2}/a\), where \(S\) is the atomic spin and \(z=1,2,4\) for sc, fcc and bcc lattice, respectively. The magnetostatic energy, is the sum of Coulomb energies between the magnetic moments comprising the FM material. It can be expressed as \(U_{m}=-\mu_{0}\overrightarrow{H_{d}}\cdot\overrightarrow{M}/2\), where \(H_{d}\) is the _demagnetizing_ field and \(M\) the sample magnetization. The anisotropy energy, is the energy required to orient the magnetization at an angle (\(\theta\)) relative to certain fixed axes of the system, known as the _easy_ axes. The microscopic mechanisms leading to anisotropy can be quite diverse and the most common types of anisotropy found in FMs are as follows:
+
+(i) _Crystal_ anisotropy. It arises from the combined effects of spin-orbit coupling and quenching of the orbital momentum that produce a preferred orientation of the magnetization along a symmetry axes of the underlying crystal. For a uniaxial materials (e.g. hexagonal Co) it has the form \(U_{a}=K_{1}sin^{2}\theta+K_{2}sin^{4}\theta+...\), where \(K_{1},K_{2},...\) are the anisotropy constants, and \(\theta\) the angle between the magnetization direction and the easy axis. Typical values for cobalt are \(K_{1}=4.5\times 10^{6}~{}J/m^{3}\) and \(K_{2}=1.5\times 10^{5}~{}J/m^{3}\). For cubic crystals (e.g. fcc Fe, Ni) it reads \(U_{a}=K_{1}(a_{1}^{2}a_{2}^{2}+a_{2}^{2}a_{3}^{2}+a_{3}^{2}a_{1}^{2})+K_{2}a_{1}^{2}a_{2}^{2}a_{3}^{2}+...\), where \(a_{1},a_{2},a_{3}\) are the direction cosines of the magnetization direction. Typical values for Fe are \(K_{1}=4.8\times 10^{4}~{}J/m^{3}\) and \(K_{2}=\pm 0.5\times 10^{4}~{}J/m^{3}\).
+(ii) _Stress_ anisotropy. It is produced by the presence of stress in the sample and it has a uniaxial character \(U_{a}=K_{\sigma}sin^{2}\theta\), where \(K_{\sigma}=\frac{3}{2}\lambda_{i}\sigma\), with \(\lambda_{i}\) the magnetically induced isotropic strain and \(\sigma\) the stress.
+(iii) _Surface_ anisotropy. This is caused by the presence of sample free boundaries, where the reduced symmetry and the presence of defects can induce additional anisotropy. It is important in MNPs because of the substantial surface-to-volume ratio.
+(iv) _Shape_ anisotropy. This occurs because on one hand the demagnetizing field depends on the shape of the magnetized body and takes the lowest value along the longest axis of the sample, and on the other hand, \(U_{m}\) is minimized when \(M\) is parallel to \(H_{d}\). As an example, consider a specimen in the shape of prolate spheroid with major axis \(c\) and minor axis \(a\), magnetized at an angle \(\theta\) with respect to \(c\)-axis. Then, \(U_{m}=\frac{\mu_{0}}{2}\left[N_{c}(M\cos\theta)^{2}+N_{a}(M\sin\theta)^{2}\right]=\frac{1}{2}(N_{c}-N_{a})M^{2}\sin^{2}\theta\) , where \(N_{c}\) and \(N_{a}\) are the _demagnetizing_ factors along the corresponding axes. This expression for \(U_{m}\) has the form of uniaxial anisotropy with \(K_{s}=\frac{1}{2}(N_{c}-N_{a})M^{2}\). Typical cases, are a spherical specimen with \(K_{s}=0\), an infinitely thin planar specimen with \(N_{\|}=0\) (in-plane) and \(N_{\bot}=1\), and a infinitely long (needle-shaped) specimen with \(N_{\|}=\frac{1}{2}\) (along the axis) and \(N_{\bot}=0\).
+
+In studies of the magnetic properties of MNPs, it is a common practice, to describe, within the simplest approximation, the overall effect of the various anisotropy types by an _effective_ uniaxial anisotropy term \(U_{a}=K_{eff}\sin^{2}\theta\). The constant \(K_{eff}\) accounts for the total effect of crystalline, surface and shape anisotropy.
+
+A bulk FM material is composed of many uniformly magnetized regions (_domains_). The direction of magnetization in different domains varies, and in a bulk sample it is randomly distributed leading to a non-magnetized sample even at temperatures far below the Curie point. The formation of magnetic domains in FM materials results from the competition between the exchange and the magnetostatic energy. The former favors perfect alignment of neighboring moments and the latter is reduced by breaking a uniformly magnetized body into as many as possible regions with opposite magnetization directions. The outcome of this competition is the formation of a certain number of domains in a sample with a particular orientation of the magnetization directions. A typical domain size in a bulk ferromagnet is \(1\mu m\).
+
+Neighboring magnetic domains are separated by a region where the local magnetization changes gradually direction between the two opposite sides, known as _domain wall_ (DW). Domain walls have finite width \((\delta_{w})\) determined by the balance between the exchange and anisotropy energy. As an example, consider an one-dimensional model of a DW in a uniaxial material, where a \(180^{0}\) rotation of magnetization is distributed over N sites, as shown in Fig. 1.
+
+Figure 1: One-dimensional model of a FM. (a) Long-range order. (b) An infinitely thin DW (dashed line). The increase of exchange energy at the wall is higher than the decrease of the magnetostatic energy. (c) A \(180^{0}\) domain wall spread over \(N=10\) sites. The gradual rotation of atomic moments produces a state with lower total energy compared to (b).
+
+The total energy per unit area reads
+
+\[\sigma(N)=\sigma_{ex}+\sigma_{a}=JS^{2}(\pi/N)^{2}(N/a^{2})+NaK_{1}.\] (1)
+
+Minimization with respect to \(N\) leads to
+
+\[\delta_{w}=Na=\pi(A/K_{1})^{1/2}.\] (2)
+
+For a typical exchange stiffness value (\(A\approx 10^{-11}~{}J/m\)), Eq.(2) predicts for iron \(\delta_{w}\approx 0.4~{}\mu m\) while for a magnetically harder material like cobalt, \(\delta_{w}\approx 60~{}nm\). Substituting the result of Eq.(2) into in Eq.(1) provides the areal energy density of the DW
+
+\[\sigma_{w}=2\pi(AK_{1})^{1/2}\] (3)
+
+Consider a finite sample of a FM material, with size \(d\). As the size of the sample is reduced, the number of DWs it contains decreases, because fewer regions with opposite directions of magnetization are required to reduce the magnetostatic energy. Below a critical value of the system size, the sample does not contain any DW and it is in a _single domain_ (SD) state exhibiting saturation magnetization \((M_{s})\). For a spherical particle, the critical diameter \((d_{c})\) can be estimated as follows: the SD state is stable when the energy needed to create a DW that spans the whole particle, \(U_{w}=\sigma_{w}\pi r^{2}\), is greater than the magnetostatic energy gain from the reduction to a multidomain state, which is approximately equal to the magnetostatic energy stored in a uniformly magnetized sphere, \(U_{m}=\frac{1}{3}\mu_{0}M_{s}^{2}V\), with \(M_{s}\) the saturation magnetization and \(V=\frac{4\pi}{3}r^{3}\). The condition \(U_{w}=U_{m}\) provides
+
+\[r_{c}=9\frac{(AK_{1})^{1/2}}{\mu_{0}M_{s}^{2}}\] (4)
+
+For Fe, this approximation gives \(r_{c}\approx 3~{}nm\), which is by far too small. The reason is that the DW is assumed to have the same one-dimensional structure as in the bulk material. An improved calculation that considers a three-dimensional confinement of the DW provides for the critical radius:
+
+\[r_{c}=\sqrt{\frac{9A}{\mu_{0}M_{s}^{2}}\left[ln\left(\frac{2r_{c}}{a}\right)-1\right]}\] (5)
+
+In the case of Fe, numerical solution of Eq.(5) gives \(r_{c}\approx 25~{}nm\), which is very close to more accurate micromagnetic calculations and the experimentally obtained value (Cullity 1972).
+
+### Magnetization by Coherent Rotation
+
+The magnetization \((M)\) of a bulk FM crystal that contains many magnetic domains, changes under application of an external magnetic field \((H)\), a process known as _technical magnetization_. However, the value of \(M\) is not a unique function of \(H\) and the state of the sample _prior_ to application of the field is important. This is the phenomenon of magnetic _hysteresis_, which is commonly depicted by drawing the \(M-H\) dependence under a cyclic variation of the field from a positive to a negative and back to a positive saturation value (_hysteresis loop_). Two important characteristic values of a hysteresis loop are the _remanence_ (\(M_{r}\)), namely the magnetization after removal of the saturating field, and the _coercivity_ (\(H_{c}\)), namely the field required for the magnetization to vanish. In a bulk FM crystal, the magnetization proceeds by two basic mechanisms, namely domain wall motion (weak fields) and rotation of magnetization (strong fields).
+
+In MNPs, the change of magnetization under an applied field proceeds only by rotation, because formation of DWs is energetically unfavorable. During the magnetization rotation the atomic moments of the MNP remain parallel to each other and the MNP behaves as a giant molecule carrying a magnetic moment of a few thousand Bohr magnetons (\(\mu\sim 10^{4}\mu_{B}\) for a \(5~{}nm\) diameter Fe MNP). This process of magnetization is known as _coherent_ rotation or _Stoner-Wohlfarth_ (SW) model, after the authors who introduced and solved it (Stoner 1948). We discuss it briefly next. Consider a MNP with uniaxial (effective) anisotropy \(K_{1}\) along an easy axis taken to be the \(z\)-axis (Fig. 2). For an applied field that makes an angle \(\theta_{0}\) with the easy axis, we wish to determine the equilibrium position of the magnetic moment \(\mu=M_{s}V\). Let \(\overrightarrow{\mu}\) make an angle \(\theta\) with the easy axis, then the total energy density reads
+
+\[u=-K_{1}\cos^{2}(\theta-\theta_{0})-\mu_{0}HM_{s}\cos\theta\] (6)
+
+The equilibrium condition (zero-torque) is
+
+\[\dfrac{du}{d\theta}=0\Rightarrow 2K_{1}sin(\theta-\theta_{0})\cos(\theta-\theta_{0})+\mu_{0}HM_{s}\sin\theta=0\] (7)
+
+and introducing the dimensionless quantity \(h=H/H_{a}\) with the _anisotropy_ field \(H_{a}=2K_{1}/\mu_{0}M_{s}\), Eq.(7) becomes
+
+\[\sin\left(2(\theta-\theta_{0})\right)+2h\sin\theta=0.\] (8)
+
+We define the reduced magnetization along the field \(m=\mu\cos\theta/M_{s}V=cos\theta\) and the solution of Eq.(8) is written as
+
+\[2m(1-m^{2})^{1/2}\cos 2\theta_{0}+\sin 2\theta_{0}(1-2m^{2})+2h(1-m^{2})^{1/2}=0\] (9)
+
+Figure 2: (a) Sketch of a magnetic nanoparticle with uniaxial anisotropy along the \(z\)-axis and an applied field at an angle (\(\theta_{0}\)) with respect to the easy axis. (b) Magnetization curves within the Stoner-Wohlfarth model for various field directions. The initial direction of the magnetization is taken along the field.
+
+The remanence (\(h=0\)) and coercivity (\(m=0\)) are readily obtained from Eq.(9) as
+
+\[m_{r}=\cos\theta_{0}\quad\textnormal{and}\quad h_{c}=\sin\theta_{0}\cos\theta_{0}.\] (10)
+
+For non-zero field values, Eq.(9) is solved for \(h\) as a function of \(m\) and the data are shown in Fig. 2. Consider the two extreme cases, namely for \(\theta_{0}=90^{0}\) (hard-axis magnetization) and \(\theta_{0}=0^{0}\) (easy-axis magnetization). In the former case, the magnetization shows zero coercivity and a linear field dependence. In the latter case, the magnetization remains constant until the reversing field becomes equal to the anisotropy field, and then an _irreversible_ jump of the reduced magnetization from \(m=+1\) to \(m=-1\) is seen. These extreme cases demonstrate the distinct mechanism of switching by rotation that can occur in an assembly. More generally, at an arbitrary field angle, an irreversible jump of the magnetization occurs at the so called _switching_ field (\(H_{s}\)) defined as the field value satisfying \(dm/dh\rightarrow\infty\). At \(H=H_{s}\) the local minimum of the total energy, corresponding to the higher energy state (magnetization opposite to the applied field) disappears and the system jumps to the remaining minimum that corresponds to a magnetization direction along the field (see Fig. 3). In other words, \(H_{s}\) is an instability point of the total energy and it satisfies \(du/d\theta=0\) and \(d^{2}u/d\theta^{2}=0\). In the SW model, the stability condition reads
+
+\[\dfrac{d^{2}u}{d\theta^{2}}=0\Rightarrow\cos 2(\theta-\theta_{0})\pm h\sin\theta=0\] (11)
+
+From Eqs.(8) and (11) we obtain for the switching field \(h_{s}=H_{s}/H_{a}\)
+
+\[h_{s}=(\cos^{2/3}\theta_{0}+\sin^{2/3}\theta_{0})^{-3/2}\] (12)
+
+By comparison of Eqs.(10) and (12) one finds that \(h_{c}<h_{s}\) for \(45^{0}<\theta_{0}<90^{0}\) , namely switching happens after the magnetization changes sign, while for field angles close to the easy axis, \(0^{0}<\theta_{0}<45^{0}\), the magnetization changes sign by an irreversible jump (\(h_{c}=h_{s}\)). The physical distinction between \(h_{s}\) and \(h_{c}\) can be understood by the following example. Consider a SW particle under application of a reversing field \(h=h_{c}\), which brings the particle’s moment \(\overrightarrow{\mu}\) in a direction perpendicular to the field, so that \(m=0\). Then the field is switched off adiabatically. If \(h_{c}<h_{s}\) (i.e. \(45^{0}<\theta_{0}<90^{0}\)), \(\overrightarrow{\mu}\) will return back to the positive remanence value \((m=+1)\), while if \(h_{c}=h_{s}\) (i.e. \(0^{0}<\theta_{0}<45^{0}\)), \(\overrightarrow{\mu}\) will jump to the negative remanence state \((m=-1)\).
+
+Figure 3: Dependence of total energy on the direction of the particle’s moment (see Eq.(6)), for various strengths of the applied field \((h=H/H_{a})\). The energy minimum at \(\theta=\pi\) becomes unstable at the switching field \(h_{s}=1\).
+
+The switching field of a hard (i.e. large anisotropy) magnetic material is a physical quantity with great technological interest in magnetic recording applications. In these, the information bit is stored in the direction of magnetization and the switching field is the field required to write or erase this information.
+
+Stoner and Wohlfarth (Stoner 1948) also studied an assembly of isolated MNPs with easy axes directions distributed uniformly on a sphere (_random anisotropy model_, RIM). The reported values for the remanence and coercivity are
+
+\[m_{r}=0.5\quad\textnormal{and}\quad h_{c}=0.48.\] (13)
+
+This result is particularly useful as random easy axis distribution is found in most MNP-based materials (granular films, cluster-assembled films, self-assembled arrays, etc)
+
+As a final remark, we remind that in the SW model thermal effects are ignored \((T=0)\), thus energy-minimization with respect to the magnetic moment direction is a sufficient condition to determine the field-dependent magnetization at equilibrium. The magnetic behavior of SD particles at finite temperature is discussed in the following section.
+
+### Magnetic behavior at finite temperature
+
+How do thermal fluctuations affect the average magnetization direction of an isolated MNP ? How does the presence of an applied field modify the magnetic response at finite temperature ? Is the assembly magnetization stable in time, when the MNP moment are subject to thermal fluctuations ? These points are briefly discussed next, along the lines of a model first studied by Néel (Néel 1949).
+
+#### 2.3.1 Superparamagnetism and Blocking temperature
+
+Consider an assembly of identical SD particles with uniaxial anisotropy. The energy (per particle) is \(U=-K_{1}V\cos^{2}\theta\), where \(\theta\) is the angle between the single particle magnetic moment \(\overrightarrow{\mu}\) and the easy axis. The energy barrier that must be overcome for a MNP to rotate its magnetization is \(E_{b}=K_{1}V\). As first pointed out by Néel (Néel 1949), thermal fluctuations could provide the required energy to overcome the anisotropy barrier and spontaneously (i.e. without externally applied field) reverse the magnetization of a MNP from one easy direction to the other. This phenomenon can be thought of as a Brownian motion of a particle’s magnetic moment. The assembly shows paramagnetic behavior, however it is the giant moments of the MNPs that fluctuate rather than the atomic moments of a classical bulk paramagnetic material. This magnetic behavior of the MNPs is called _superparamagnetism_ (SPM) (Bean 1959) At high enough temperature, \(k_{B}T>>K_{1}V\), the anisotropy energy can be neglected and the assembly magnetization can be described by the well known Langevin function \(M=nM_{s}\L(x)\), where \(n\) is the particle number density, and \(x=\mu_{0}\mu H/k_{B}T\). Thus, the features serving as signature of superparamagnetism are the scaling of magnetization curves with \(H/T\), as dictated by the Langevin function, and the lack of hysteresis, i.e. vanishing remanence and coercivity. Moreover, the major difference between classical paramagnetism of bulk materials and SPM is the weak fields \((H\sim 0.1~{}T)\) required to achieve saturation of a MNP assembly magnetization \(M\). This occurs because of the large particle moment \((\mu\sim 10^{4}\mu_{B})\) compared to the atomic moments \((\mu_{at}\sim\mu_{B})\).
+
+Measurement of magnetization curves at sufficiently high temperature can, in principle, be used to extract the particle moment \(\mu\). In practice, two complication arise. First, the presence of different particle sizes in any sample produces a convolution of the Langevin function with the volume distribution function. Second, interparticle interactions, modify the reversal mechanism and the SW model needs extensions, which are discussed in the Section 4.
+
+At low temperature, \(k_{B}T<<K_{1}V\), the anisotropy barriers are very rarely overcome (weak thermal fluctuations), the assembly shows hysteresis and this is called the _blocked_ state.
+
+One might now ask, whether there exists a temperature value that draws the border between the blocked and the SPM state. Following Néel’s arguments, we assume that thermal activation over the anisotropy barrier can be described within the _relaxation time_ approximation (or _Arrhenius law_) as
+
+\[\tau=\tau_{0}\exp(K_{1}V/k_{B}T),\] (14)
+
+where \(1/2\tau\) is the probability per unit time for a reversal of \(\overrightarrow{\mu}\). The intrinsic time \(\tau_{0}\) depends on the material parameters (magnetostriction constant, Young modulus, anisotropy constant and saturation magnetization). Typical values are \(\tau_{0}\sim 10^{-10}-10^{-9}s\) as obtained by Néel. To detect the superparamagnetic behavior experimentally, the MNP must be probed for a long enough period of time to perform many switching events that would produce a vanishing small time-average magnetic moment. If \(\tau_{m}\) is the measuring time-window, the condition for SPM behavior is \(\tau_{m}\gg\tau\). The strong (exponential) dependence of \(\tau\) on temperature (see Eq.(14)) permits us to define a temperature value (or more precisely, a very narrow temperature range) above which the relaxation time is so small that SPM behavior is observed. This is called the _blocking temperature_ (\(T_{b}\)) of the assembly, and is given by
+
+\[T_{b}=K_{1}V/k_{B}\ln(\tau_{m}/\tau).\] (15)
+
+For \(T<T_{b}\), the particle moments fluctuate without switching direction (on average) and the assembly is in the blocked state exhibiting hysteresis. For \(T>T_{b}\) the assembly is in the SPM state, hysteresis disappears and thermal equilibrium is established. It is remarkable, that the value of \(T_{b}\) depends on \(\tau_{m}\), which is a characteristic of the experimental technique adopted. For example, in dc susceptibility measurements \(\tau_{m}\approx 100s\), in ac susceptibility \(\tau_{m}\approx 10^{-8}-10^{4}s\), in Mössbauer spectroscopy \(\tau_{m}\approx 10^{-9}-10^{-7}s\) and in neutron spectroscopy \(\tau_{m}\approx 10^{-12}-10^{-8}s\). Therefore, if \(T_{b}\) is of interest for a particular application, the measurement technique implemented must imitate the real conditions. For example, to study the reliability of magnetic storage media, dc magnetic measurements over a wide time window (\(\tau_{m}\sim 10^{2}-10^{4}~{}s\)) should be used, while to study magnetic recording speed, ac measurements are appropriate.
+
+Brown (Brown 1963) extended the treatment of thermal activation over the anisotropy barrier, allowing also for fluctuations of \(\mu\) transverse to the easy axis, which Néel has neglected, and obtained a different expression for \(\tau_{0}\). However, the common feature of both studies is the temperature and volume dependence of \(\tau\), so the final result, Eq.(14), is referred to as the _Néel-Brown_ model.
+
+In a _polydisperse_ assembly, the distribution of particle volumes \(f(V)\), produces a corresponding distribution of blocking temperatures \(f(T_{b})\). Then, at a certain temperature \(T\) the assembly contains a mixture of blocked and SPM particles. The MNPs with volumes above a critical value \(V_{c}\), fulfill the requirement of strong thermal energy with respect to their anisotropy barrier, and are SPM, while those while those with \(V\leq V_{c}\) are blocked. From Eq.(14), the critical volume reads \(V_{c}=k_{b}T\ln(\tau_{m}/\tau_{0})/K_{1}\). As explained above for \(T_{b}\), also for \(V_{c}\) the experimental determination depends on the technique adopted. Most preparation techniques result in polydisperse samples and the problem of extracting the size distribution function from magnetic measurements, pioneered by Bean and Jacobs more than fifty years ago (Bean 1956) remains a difficult task mainly due to the complications introduced by interparticle interactions. Knobel and colleagues have recently reviewed this subject (Knobel 2008).
+
+#### 2.3.2 Thermal relaxation under an applied field
+
+Figure 4: Total energy of an isolated particle with uniaxial anisotropy subject to a negative field parallel to the easy axis with value less than the switching field \((0<H<H_{s})\). Energy barriers \((E_{b})\) and relaxation times \((\tau)\) for the forward \((+)\) and the backward \((-)\) process are not equal.
+
+Consider an assembly of \(N\) identical MNPs with uniaxial anisotropy along the \(z\)-axis and let their moments point initially along the \(+z\)-axis. Assume that a magnetic field \(H\), weaker than the switching field , which is equal to \(H_{a}\), is applied along the \(-z\)-axis. Then, the total energy per particle reads, \(U=-K_{1}V\cos^{2}(\theta-\theta_{0})+\mu_{0}HM_{s}V\cos\theta\). It exhibits two non-equivalent local minima at \(\theta=0,\pi\) with values \(U_{\pm}=-K_{1}V\pm M_{s}VH\) and a maximum at \(\theta=\pi/2\) with \(U_{max}=K_{1}V(H/H_{a})^{2}\), as shown in Fig. 4. The energy barriers and the corresponding relaxation times for the forward \((+)\) and the backward \((-)\) rotations are
+
+\[E_{b}^{\pm}(H)=K_{1}V(1\mp H/H_{a})^{2}\quad\textnormal{and}\quad\tau_{\pm}=\tau_{0}\exp({E_{b}^{\pm}/k_{B}T}).\] (16)
+
+The change of \(\tau_{0}\) due to the field is much weaker than the change of the exponential factor and as such it is neglected in the above equation.
+
+The blocking temperature, as measured within a time-window \(\tau_{m}\), is reached when the observation time equals the _forward_ relaxation time \(\tau_{+}\), because the latter corresponds to a moment flip from the initial state along \(+z\) to the opposite direction, namely a process that reduces the initial magnetization. From Eq.(16) one obtains
+
+\[T_{b}(H)=\frac{K_{1}V(1-H/H_{a})^{2}}{k_{B}\ln(\tau_{m}/\tau)}\equiv T_{b}(0)(1-H/H_{a})^{2}.\] (17)
+
+which indicates that the blocking temperature is reduced by the presence of a reverse field. By completely symmetric arguments one could show that \(T_{b}\) increases in the presence of a field with the same direction as the initial magnetization.
+
+Since thermal fluctuations act in synergy to a reverse field in switching the moment of a MNP, it is expected that the coercivity of an assembly will decay with temperature. As discussed above, for a particle with its moment along the \(+z\)-axis, a reverse field \((0<H<H_{a})\) reduces the barrier for reversal to the value \(E_{b}^{+}\) given in Eq.(16). If the field is strong enough, it will reduce the barrier to the value appropriate for superparamagnetic relaxation, namely \(k_{B}T\ln(\tau_{m}/\tau_{0})\), and the (time-average) magnetization will vanish. On the other hand, the reverse field that makes the magnetization vanish is by definition the coercive field. Therefore, the following relation holds
+
+\[K_{1}V(1-H_{c}/H_{a})^{2}=k_{B}T\ln(\tau_{m}/\tau_{0})\] (18)
+
+which, using Eq.(15), provides the temperature dependent coercivity
+
+\[H_{c}(T)=H_{a}\left[1-(T/T_{b})^{1/2}\right].\] (19)
+
+The microscopic mechanism of thermal activation of the MNP moment over the anisotropy barrier, produces a macroscopically measured time-decay of the magnetization. We derive this dependence assuming that when a moment switches direction it continues to remain along the easy axis (Néel 1949). Then, at time \(t\), \(N_{+}\) particles occupy the lower minimum at \(\theta=0\), and the rest \(N_{-}=N-N_{+}\) particles occupy the higher minimum at \(\theta=\pi\). The time-evolution of \(N_{+}\) is governed by the _rate equation_
+
+\[\dfrac{dN_{+}}{dt}=-\dfrac{N_{+}}{\tau_{+}}+\dfrac{N_{-}}{\tau_{-}}.\] (20)
+
+The magnetization per particle is given as \(M(t)\equiv(2N_{+}(t)/N-1)M_{s}\), and solution of Eq.(20) provides
+
+\[M(t)=M_{\infty}+(M_{0}-M_{\infty})\exp(-t/\tau)\] (21)
+
+with \(1/\tau=1/\tau_{+}+1/\tau_{-}\) being the reduced relaxation time and
+
+\[M_{\infty}=\dfrac{\tau_{+}-\tau}{\tau_{+}+\tau_{-}}M_{s}\quad\textnormal{and}\quad M_{0}=\left(\dfrac{2N_{+}(0)}{N_{-}(0)}-1\right)M_{s}\] (22)
+
+the time-asymptote and initial values of the particle magnetization, respectively. Eq.(21) indicates that the magnetization decays exponentially towards the equilibrium value \(M_{\infty}\), reached as \(t\rightarrow\infty\). In other words, equilibrium is reached when the population of the energy minima is proportional to the corresponding relaxation times \((N_{+}/N_{-}=\tau_{+}/\tau_{-})\), as dictated by Eq.(20). When the applied field is strong enough \((H>H_{s})\) to produce only one minimum, thermal equilibrium is always reached. Obviously, in the absence of an external field, thermal equilibrium is reached when the two equivalent minima are equally populated \((N_{+}=N_{-})\), resulting in a vanishing magnetization.
+
+Notice that in Eq.(20) we assumed bistability of the moment direction, which is a valid approximation provided the anisotropy barrier is high \((K_{1}V\gg k_{B}T)\). For lower anisotropy barriers or elevated temperature \((K_{1}V\approx k_{B}T)\), the transverse fluctuations of \(\overrightarrow{\mu}\), or, in other words, intra-valley motion around the energy minimum should be taken into account. A general treatment of thermal relaxation of SD MNPs was pioneered by Brown (Brown 1963) and extended to the case of an applied external field (Aharoni 1965, Coffey 1998, Garannin 1999).
+
+If an assembly is polydisperse, characterized by a volume distribution \(f(V)\), a distribution of blocking temperatures \(f(T_{b})\) exists. However, it remains still unclear if the mean value \(<T_{b}>\) is the appropriate blocking temperature of the assembly, which should be substituted, for example, in Eq.(19). This point is discussed further in the literature (Nunes _et al_ 2004).
+
+In a polydisperse assembly, a distribution of relaxation times \(f(\tau)\) exists, with \(f(\tau)d(\ln\tau)\) the probability of a MNP to have \(\ln\tau\) in the range \(\left(\ln\tau,~{}\ln\tau+d\ln\tau\right)\) and the normalization condition \(\int_{0}^{\infty}f(\tau)d(\ln\tau)=1\). In this case the magnetization can be obtained by a superposition of the single-particle magnetization properly weighted, as follows
+
+\[M(t)=M_{s}\int_{0}^{\infty}\left[1-\exp(-t/\tau)\right]\frac{f(\tau)}{\tau}d\tau,\] (23)
+
+where the term in brackets is the probability per unit time for a particle not to flip its moment. For a broad enough distribution, the observation time \(t\) will satisfy \(\tau_{1}\ll t\ll\tau_{2}\), where \(\tau_{1}\) and \(\tau_{2}\) are the minimum and maximum relaxation times of the assembly, respectively. Assuming a uniform distribution \(f(\tau)\), it can be shown that the magnetization exhibits a logarithmic relaxation
+
+\[M(H,t)=M(H,0)-S(H,T)\ln(t/\tau_{0})\] (24)
+
+with \(S\) the _magnetic viscosity_ of the system. Thus, polydispersity produces a much slower decay of magnetization with time.
+
+The discussion so far, refers to a field applied parallel to the easy axis. However, random anisotropy is most commonly found in MNP assemblies and the the necessity to study the effect of a tilted field with respect to the easy axis, arises. In this case, the calculation of the energy barriers and relaxation time is a much more complicated task and no analytical solution exists. Numerical studies (Pfeiffer 1990) showed that the energy barrier for an applied field at an angle \(\theta_{0}\) to the easy axis can be approximately written as
+
+\[E_{b}(\theta_{0})=K_{1}V(1-H/H_{a})^{0.86+1.14h_{s}}\] (25)
+
+where \(h_{s}\) is given by Eq.(12). In the limit of \(\theta_{0}=0\), Eq.(25) reduces to Eq.(16).
+
+The temperature dependence of the coercivity for a monodisperse assembly with random anisotropy has also been obtained numerically (Pfeiffer 1990) as
+
+\[H_{c}(T)=0.48H_{a}\left[1-(T/T_{b})^{0.77}\right],\] (26)
+
+which at \(T=0\) reduces to the SW result of Eq.(13). A detailed theoretical study of the relaxation time for a non-uniaxial applied field can be found in the review by Coffey and colleagues (Coffey 1993).
+
+As a concluding remark, the presence of polydispersity and random anisotropy makes the description of the magnetic behavior of an assembly intractable to exact analytical treatment. Instead, numerical approximations and simulation methods provide the alternative theoretical tools to study these systems. Numerical simulation approaches are introduced in Section 4.
+
+## 3 Magnetic Measurements
+
+Thermal relaxation has a dynamic character, therefore, the relation between the various relaxation times of the assembly and the measurement time is a decisive parameter for the outcome of a measurement. Additionally, if the assembly is not at equilibrium during the measurement, or if it changes its equilibrium state (for example, by adiabatic changes of the applied field) the result of the measurements depends on the measurement protocol followed. In what follows we discuss two very common types of static measurements, that reveal the temperature and field dependence of the magnetization and provide evidence for superparamagnetic relaxation. Dynamic measurements are not discussed in this article. The interested reader can find more on the physical principles behind the most common magnetic measurement techniques in the review of Dormann and colleagues (Dormann 1997).
+
+### Field-cooled (FC) and Zero-Field-Cooled (ZFC) Magnetization
+
+This is a measurement protocol adopted for investigation of the temperature dependent magnetization of an assembly and it reveals superparamagnetic behavior. It is performed in three stages. In the first, the sample is initially at a high enough temperature (\(T_{max}\)) to ensure a SPM state and it is cooled to low temperature (\(T_{min}\)) to approach its ground state. In the second stage, a weak field is applied (\(H\ll H_{sat}\)), the sample is heated up to \(T_{max}\) and the magnetization is measured as a function of temperature. This is the ZFC curve. In the third stage, the system is cooled down to \(T_{min}\), without removing the field, while the magnetization is recorded again, producing the FC curve. During cooling and heating the temperature changes at the same constant rate. A typical ZFC-FC curve is shown in Fig. 5.
+
+Figure 5: (a) Typical FC-ZFC magnetization curves. The curves join at the peak of the ZFC which corresponds to \(T_{b}\). Arrows indicate the direction the measurements are taken. For \(T>T_{b}\) the system is in thermal equilibrium and the heating process is reversible. (b) FC-ZFC curves for a dilute (non-interacting) assembly of Fe nanoparticles (\(D=3.0~{}nm,M_{s}=1720~{}emu/cc\) and \(K_{1}=2.4\times 10^{5}~{}erg/cc\)). The blocking temperature (dotted line) decreases weakly with increasing measuring field. The non-zero values of \(M_{ZFC}(T=0)\) are due to the finite value of the measuring field. Data produced by Monte Carlo simulations (Section 4.3)
+
+As the temperature rises, the blocked magnetic moments align easier along the applied field leading to an initial increase of the ZFC curve. However, as soon as thermal fluctuations push the moments over the anisotropy barrier, thermal randomization of the moments produces a drop of the curve. Therefore, the peak of ZFC curve corresponds to the blocking temperature of the assembly. Notice that above \(T_{b}\) the ZFC and FC curves coincide, because the system is in thermal equilibrium and the the heating (cooling) process is reversible. On cooling below \(T_{b}\) the moments remain partially aligned along the field, and the magnetization tends to a non-zero value. The magnetization vanishes at the ground state (\(T_{min}\)) if the measuring field is very weak, a random distribution of the easy axes exists and the assembly is non-interacting (dilute). Deviations from any of the above conditions produce a non-zero value for \(M_{ZFC}(T=0)\). For isolated MNPs, the ZFC-FC curves are only weakly sensitive to the value of the applied field, provided that it is weak \((H\ll H_{s})\).
+
+### Remanent magnetization and Coercive field
+
+Remanent magnetization at a certain field, \(M_{r}(H)\), is measured after switching off the previously applied field \(H\). In an assembly of MNPs, remanence arises because the moments of some particles, which have rotated under an applied field and to do so they have overcome an energy barrier, cannot rotate back to their original direction after removal of the field. In a polydisperse assembly, at finite temperature \(T\), only the blocked MNPs, namely those with \(T_{b}<T\) contribute to the remanence. Therefore, \(M_{r}/M_{s}=\int_{E_{b,c}}^{\infty}f(E_{b})dE_{b}\) where \(E_{b,c}=K_{1}V_{c}\) is the critical barrier for SPM relaxation at temperature \(T\). Taking into account that \(T_{b}\sim V\) (see Eq.(15)), we deduce that
+
+\[dM_{r}(T)/dT=f(T_{b})\] (27)
+
+namely, the slope of \(M_{r}(T)\) provides the barrier (or blocking temperature) distribution function of the assembly. There are three different measurement protocols for the remanent magnetization, as first suggested by Wohlfarth (Wohlfarth 1958) :
+
+(i) _Thermoremanence_\(TRM(H,T)\), measured at the end of a FC process with field \(H\) from \(T_{max}\) down to the measuring temperature \(T\).
+(ii) _Isothermal Remanence_\(IRM(H,T)\), measured at the end of ZFC process from \(T_{max}\) down to the measuring temperature \(T\), at which a field \(H\) is applied and then removed.
+(iii) _DC Demagnetization remanence_\(DcD(H,T)\). First, a ZFC process from \(T_{max}\) down to the measuring temperature \(T\) is performed. Second, the sample is brought to _saturation_ remanence \(IRM(\infty,T)\). Third, a _reverse_ field \(H\) is applied and then removed to leave the sample at the \(DcD(H,T)\) remanence.
+
+Wohlfarth pointed out that for isolated MNPs the different remanent magnetizations are related as \(DcD(H)=IRM(\infty)-2\cdot IRM(H)\). More interestingly, the deviations from this equality, defined as
+
+\[\Delta M(H)=DcD(H)-\left[IRM(\infty)-2\cdot IRM(H)\right]\] (28)
+
+quantify the character and strength of interparticle interactions and are obtained experimentally (O’Grady _et al_ 1993). Positive \(\Delta M\) values imply interactions with magnetizing character, and negative values indicate demagnetizing interactions. We should say that this is only a phenomenological characterization of the interactions, because Eq.(28) does not provide any information about their microscopic origin. However, Eq.(28) has been proved a standard tool for quantification of interparticle interactions in complex MNP assemblies such as those used in modern industry of magnetic recording media (granular films, particulate media). Interparticle interactions are discussed in the Section 4.
+
+## 4 Interacting nanoparticle assemblies
+
+### Introduction
+
+The magnetic interactions that are present in bulk magnetic materials pertain to MNP assemblies and they preserve their physical origin and characteristics. In particular, (direct) exchange between atomic moments separated by a few lattice constants can couple ferromagnetically or antiferromagnetically two MNPs via their surface atoms. Indirect exchange or Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction exists between MNPs hosted in a metallic matrix, which provides free electrons required to mediate the interaction between the atomic moments of the MNPs. Finally, magnetostatic interactions, which are of minor importance in bulk magnets due to their weakness, become the dominant interactions in MNP assemblies with _well separated_ MNPs. This situation occurs for two reasons. First, the exchange interactions have a very short range (up to \(\sim 5~{}\)Å), the RKKY interactions have an oscillating FM/AFM character with a period of a few Å, which renders to zero their average effect on the MNP volume, so both have a weak effect in interparticle coupling. On the other hand, magnetostatic interactions, in the lowest approximation, namely the dipolar contribution, are proportional to the magnitude of the coupled magnetic moments, which for SD particles has an enormously large value compared to the atomic moments (\(\mu_{MNP}\sim 10^{3}\mu_{B}\sim 10^{3}\mu_{atom}\)).
+
+Further on we discuss the effects of magnetostatic (dipolar) interactions on the magnetic properties of MNP assemblies and their interplay with single-particle anisotropy. The complexity of this problem arises from the _long-range_ (\(\sim 1/d^{3}\), \(d\) being the interparticle distance) and _anisotropic_ character of the dipolar interactions, namely the dependence of interaction energy on the orientation of the moments relative to the bond joining the particle centers (Fig. 6).
+
+Figure 6: Ground state configuration of magnetic nanoparticles with elliptic shape coupled by magnetostatic (dipolar) forces. The easy axis coincides with the long axis of the ellipse (shape anisotropy). Dipolar coupling induces antiferromagnetic ordering when the moments are forced (by anisotropy) to remain normal to the bond, as in (a). FM ordering (nose-to-tail) is favored when the easy axes are parallel to the bond, as in (b). In an assembly with random anisotropy, as in (c), the moments are aligned along the local easy axes. Dipolar interactions have a complex effect, leading to misalignment of the moments with respect to the local easy axis.
+
+Understanding and controlling the effects of dipole-dipole interactions (DDI) in MNP assemblies is of paramount importance to modern technology of magnetic recording media for two opposite reasons. First, DDI couple the MNPs of an assembly. The ultimate goal in magnetic recording applications is to address each MNP individually and treat it as a magnetic bit. In this case, DDI have a parasitic role and one wishes to estimate and reduce their impact in the magnetic properties of an assembly. On the contrary, magnetic logic devices, have been proposed and built that exploit the magnetostatic coupling between ordered MNP arrays (linear or planar) to transfer a magnetic bit (usually a flipped moment) between two distant points in the array (Cowburn 2006). In this case, DDI are of central importance and the goal is to enhance and tailor their effects.
+
+Over the last two decades, many research groups have prepared and measured MNP assemblies in various forms (granular films, ferrofluids, cluster assembled films, self-assembled nanoparticles, lithographic arrays of magnetic dots) and studied the intrinsic factors (host and particle material, particle size, particle density) and the extrinsic factors (temperature, field, measurement protocol) that control the magnetic behavior. In many of these studies the presence of magnetostatic interactions has been confirmed. Among the above mentioned systems, the self-assembled MNPs prepared by a synthetic route offer the advantage of containing well separated MNPs with a very narrow size distribution (\(\sigma_{V}\sim 5-10\%\)), so they are ideal systems to study DDI effects. Experimental observations on self-assembled MNPs that have been attributed to DDI include, reduction of the remanence at low temperature (Held 2001), increase of the blocking temperature (Murray 2001), increase of the barrier distribution width (Woods 2001), deviations of the zero-field cooled magnetization curves from the Curie behavior (Puntes 2001), and difference between the in-plane and out-of-plane remanence (Russier 2000). Long-range ferromagnetic order in linear chains (Russier 2003), and hexagonal arrays (Puntes 2004, Yamamoto 2008) of dipolar coupled single-domain magnetic nanoparticles has been demonstrated, supporting the existence of a _dipolar superferromagnetic_ ground state, characterized by ferromagnetic long-range order of the particle moments.
+
+Investigations of the static and dynamic magnetic properties of dipolar interacting nanoparticle assemblies brought up fundamental issues related to the existence of a ground state which shares common features with _spin glasses_, such as slow relaxation, memory and ageing effects (Sasaki 2005). The latter are magnetic systems characterized by disorder and competing interactions that produce an energy landscape with many local minima, considered responsible for the occurrence of these effects. Dipolar interparticle interactions in dense and random nanoparticle assemblies are believed to cause a _spin-glass-like_ behavior (Dormann 1997).
+
+Theoretical models have been developed in an effort to explain these observations and related previous ones in assemblies with randomly located MNPs (granular films, cluster-assembled films). On a microscopic level, the presence of DDI between MNPs modifies the magnetization switching mechanism, which for an isolated MNP obeys the Néel-Arrhenius model. When anisotropic MNPs are dipolar coupled, the reversal mechanism is determined by the interplay between the single-particle anisotropy energy (\(E_{a}\sim K_{1}V\)) and the dipolar interaction energy (\(E_{d}\sim\mu_{i}\mu_{j}/r_{ij}^{3}\)). For weak interactions (\(E_{d}\ll E_{a}\)), the moments reverse _independently_ by thermal activation over energy barriers, which are however modified due to DDI. This limiting case is treated within a mean-field approximation and is discussed in Section 4.2. For strong interactions (\(E_{d}\gg E_{a}\)), the single-particle reversal is no longer valid. Reversal of one particle can excite the reversal of others, and the assembly behaves in a collective manner. Many-body energy barriers exist in the system, with values that depend on the configuration of all moments. Their evaluation becomes a formidable task and numerical simulations offer in this case an indispensable tool. Numerical methods are briefly discussed in Section 4.3.
+
+For a detailed review on the magnetic properties of dipolar interacting MNP assemblies the reader is referred to the relevant literature (Dormann 1997, Farrell _et al_ 2005, Knobel _et al_ 2008, Kechrakos and Trohidou 2008). The role of magnetostatic interactions in patterned magnetic media has been reviewed by Martín _et al_ (Martín 2003)
+
+### Mean Field models
+
+In an early attempt to include the effect of interparticle interactions in the thermal relaxation of MNPs , Shtrikman and Wohlfarth (Shtrikman 1981) assumed that the single-particle anisotropy barrier of a MNP is increased by the Zeeman energy due to the _interaction field_\(H_{int}\) produced by the moments of neighboring particles. In this model, the Néel relaxation time is obtained from Eq.(16) with the applied field \(H\) replaced by the interaction field \(H_{int}\). The mean-field approximations consists in replacing \(H_{int}\) by its thermal average value, which in Néel’s model is
+
+\[\overline{H_{int}}=H_{int}\tanh(\mu_{0}\mu H_{int}/k_{B}T)\approx\mu_{0}\mu\overline{H_{int}^{2}}/k_{B}T\] (29)
+
+the latter approximation being valid for weak interaction fields. Substitution of Eq.(29) into Eq.(16) gives
+
+\[\tau=\tau_{0}\exp\left[\frac{K_{1}V+\mu_{0}^{2}\mu^{2}\overline{H_{int}^{2}}/k_{B}T}{k_{B}T}\right].\] (30)
+
+Using the approximation \(1+x\approx 1/(1-x)\) we write Eq.(30) in the form
+
+\[\tau\approx\tau_{0}\exp\left[\frac{K_{1}V}{k_{B}(T-T_{0})}\right]\] (31)
+
+with \(T_{0}=\mu_{0}^{2}\mu^{2}\overline{H_{int}^{2}}/k_{B}K_{1}V\). Eq.(31), also known as the _Vogel-Fulcher law_, indicates that the relaxation time of an assembly of interacting MNPs is the same as that of the isolated MNPs at a _lower_ temperature.
+
+In the Shtrikman-Wohlfarth model, the temperature \(T_{0}\), or equivalently the thermal average \(\overline{H_{int}^{2}}\), is not related to the microscopic parameters of the assembly, i.e. particle location, and is treated as a phenomenological parameter, fitted to experimental data. Dormann and colleagues (Dormann 1988, Dormann 1997) developed a statistical model for the average barrier in a dipolar interacting assembly which quantifies the interaction field and provides for the single-particle energy barrier
+
+\[E_{b}=K_{1}V+n_{1}a_{1}M_{s}^{2}V\L(a_{1}\mu^{2}/Vk_{B}T)\] (32)
+
+with \(n_{1}\) the number of nearest neighbors of a particle, \(a_{1}=x_{v}/\sqrt{2}\), \(x_{v}\) the volume concentration of the particles and \(\L(\cdot)\) the Langevin function. Eq.(32) indicates that the anisotropy barrier is increased due to DDI, thus the model of Dormann _et al_ predicts an increase of the blocking temperature due to DDI. This model behavior has been observed in almost all types of MNP assemblies, with a few exceptions (Hansen and Mørup 1998).
+
+More recently, Allia _et al_ (Allia 2001) used also a phenomenological approach to describe a superparamagnetic assembly with weak DDI. Namely, an assembly in a regime that the remanence and coercivity vanish, but the field-dependent magnetization varies with concentration of MNPs indicating the presence of DDI. The authors (Allia 2001) suggested that the dipolar field changes at a high rate and in random direction and therefore acts similar to the thermal field. The effect is accounted for by an apparent increase of the system temperature. The magnetization at temperature \(T\) is given by \(M=M_{s}\L\left[\mu H/k_{B}(T+T^{*})\right]\), with \(T^{*}\) related to the average dipolar energy via \(k_{B}T^{*}=n_{1}\mu^{2}/\overline{d^{3}}\) and obtained by a fitting procedure. This model interpreted successfully the magnetization behavior of Co nanoparticles in Cu matrix and established the existence of the interacting superparamagnet regime (Allia 2001).
+
+### Numerical Techniques
+
+The mean-field models have the advantage of providing analytical expressions suitable for extracting system parameters from the experimental data by a fitting process. However, they are not applicable to strongly dipolar systems and they do not account for collective effects. Numerical techniques on the other hand, have the major advantage that they treat rigorously the local and temporal statistical fluctuations of the macroscopic quantities characterizing the MNP assembly and provide an efficient interpolation scheme between the weak and the strong interaction regimes. We discuss briefly two most common numerical approaches, the Monte Carlo (MC) method and the Magnetization Dynamics (MD) method.
+
+#### 4.3.1 The Monte Carlo method
+
+Different algorithms that mimic thermal fluctuations of the degrees of freedom of a physical system by means of (pseudo)random numbers go under the umbrella of Monte Carlo techniques. In the case of MNPs, two widely used algorithms are the Metropolis Monte Carlo (MMC) and the Kinetic Monte Carlo (KMC). The former is appropriate for a description of the equilibrium behavior of an assembly, while the latter also accounts, within a certain time scale, for the transition to equilibrium. Both algorithms provide thermal averages of macroscopic quantities of interest in the canonical ensemble, i.e. at constant temperature. To do so a sampling of the phase space is performed, however the sampling procedures differ, as outlined below.
+
+The MMC algorithm samples the phase space, visiting preferentially states close to the equilibrium states (_Importance Sampling_). This is achieved when subsequently visited states form a Markov chain, meaning that the probability of visiting the next state depends only on the last visited one. To do so, one chooses the transition from state \(s\) to \(s^{\prime}\) to occur with certainty, if it reduces the total energy (\(E_{s^{\prime}}\leq E_{s}\)) and with a finite probability \(p(s\to s^{\prime})=\exp(-\frac{E_{s^{\prime}}-E_{s}}{kT})\), if it increases the total energy (\(E_{s^{\prime}}>E_{s}\)). Thus the system is allowed to climb-up energy barriers and slide-down toward energy minima until it reaches eventually the global minimum.
+
+In KMC the system jumps from a state \(s\) at a local minimum to a new state \(s^{\prime}\) being also a local minimum by overcoming a barrier \(E_{b}\). The jump is performed within a predefined time step \(\Delta t\) with probability \(p(\Delta t)=1-\exp(-\Delta t/\tau)\) where \(\tau\) is the corresponding relaxation time with Arrhenius behavior, \(\tau=\tau_{0}\exp(E_{b}/kT)\).
+
+In both algorithms, interparticle interactions are included by replacing the applied field \(H\) with the _total_ field \(H_{i}=H+\sum_{j(\neq i)}H_{int,ij}\), which includes the contribution from the _interaction_ field \(H_{int,ij}\). In contrast to mean-field theories, in MC and MD (see next section) techniques the interaction field is treated exactly, meaning that its value depends on the configuration of all the moments of the assembly and it changes at each time-step.
+
+An important distinction between the MC algorithms is that KMC simulates the relaxation of the system in physical time, while time quantification of the MMC time step is possible only in the absence of interparticle interactions (Nowak 2000, Chubykalo 2003). However, a serious difficulty in KMC arises from the calculation of the local energy barrier required to obtain the transition probability. In an interacting system the barrier depends on _all_ degrees of freedom and its calculation is a formidable task (Chubykalo 2004, Jensen 2006), usually performed in an approximate manner (Pfeiffer 1990, Chantrell 2001). Furthermore, the KMC assumes that the system evolves through thermally activated jumps over energy barriers, an approximation that becomes invalid at elevated temperatures \((kT\sim E_{b})\), or when collective behavior becomes important, as, for example, in strongly interacting MNPs. Collective effects are better described within the MMC algorithm.
+
+For a detailed description and technical implementation of MC algorithms the interested reader could refer to the book by Landau and Binder (Landau 2000)
+
+#### 4.3.2 The Magnetization Dynamics method
+
+In this method, the equations of motion for the magnetic moments are integrated in time and time averages of the macroscopic magnetization are recorded. At zero temperature, the time-evolution of a magnetic moment \(\mu_{i}\) under a total field \(H_{i}\) is described by the Landau-Lifshitz-Gilbert (LLG) equation
+
+\[\frac{d\overrightarrow{\mu}_{i}}{dt}=-A(\overrightarrow{\mu}\times\overrightarrow{H}_{i})-B_{i}\overrightarrow{\mu}_{i}\times(\overrightarrow{\mu}_{i}\times\overrightarrow{H}_{i})\] (33)
+
+with \(A\equiv\gamma/(1+\alpha^{2})\), \(B_{i}\equiv\alpha\gamma/(1+\alpha^{2})\mu_{i}\), \(\gamma\) the gyromagnetic ratio, and \(\alpha\) a dimensionless damping parameter. The first term on the r.h.s. of Eq.(33) is the torque term leading to precession around the field axis and the second one is a phenomenological damping torque that tends to align the precessing moment with the field \(H_{i}\).
+
+The dynamics at finite temperature are described by introduction of an additional field (\(H_{f,i}\)) term in Eq.(33) with stochastic character. \(H_{f}\) is assumed to have zero time-average (_white noise_) and its values at different sites \(i,j\) or different instants \(t,t^{\prime}\) are uncorrelated. The LLG equation augmented by the thermal field term is commonly referred to as the _Langevin_ or _stochastic LLG_ equation.
+
+#### 4.3.3 Time scale of numerical methods
+
+The MC and MD techniques are complementary since they describe thermal relaxation of magnetic properties in different time scales. In the MD method, the characteristic time is a fraction (\(\sim 10^{-2}\)) of the precessional (Larmor) period \((\sim 10^{-10}s)\), which implies that simulation times up to \(\sim 1~{}ns\) are presently attainable. Thus, MD is the appropriate scheme to investigate fast-relaxation phenomena as, for example, the reversal path of magnetization under an applied short field pulse (Berkov 2002, Suess 2002).
+
+In KMC the characteristic time is the single-particle relaxation time (see Eq.(14)), which is much larger than \(\tau_{0}\sim 10^{-10}s\) in the temperature range of interest \((kT\ll E_{b})\), a fact that makes the method suitable to treat slow-relaxation problems, as, for example the thermal decay of magnetization in permanent magnets, a phenomenon that evolves within days or years (Van de Veerdonk 2002).
+
+Finally, when static magnetic properties are concerned, the system is at a stable (or metastable) state and the MMC is a powerful and sufficient scheme to describe, for example, long range order at the ground state or collective behavior at finite temperature (Kechrakos 1998, Jensen 2003).
+
+#### 4.3.4 MMC study of dipolar interacting assemblies : A case study
+
+In this section we show typical results from MMC simulations of the magnetic properties of dipolar interacting MNP assemblies (Kechrakos 1998, Kechrakos 2002). Our system contains \(N\) identical SD NPs with diameter \(D\) and uniaxial anisotropy in a random direction. The MNPs are located randomly in space or on the vertices of a hexagonal lattice. The former is an appropriate model for granular samples, and the latter for self-assembled MNPs. The total energy of the system is
+
+\[E=g\sum_{ij}\frac{\widehat{S}_{i}\cdot\widehat{S}_{j}-3(\widehat{S}_{i}\cdot\widehat{R}_{ij})(\widehat{S}_{i}\cdot\widehat{R}_{ij})}{R_{ij}^{3}}-k\sum_{i}(\widehat{S}_{i}\cdot\widehat{e_{i}})^{2}-h\sum_{i}(\widehat{S}_{i}\cdot\widehat{H})\] (34)
+
+where \(\widehat{S}_{i}\) is the magnetic moment direction (spin) of the \(i\)-th particle, \(\widehat{e_{i}}\) is the easy-axis direction, and \({R}_{ij}\) is the center-to-center distance between particles \(i\) and \(j\). Hats indicate unit vectors. The energy parameters entering Eq.(34) are: (i) the dipolar energy \(g\equiv\mu_{0}^{2}\mu^{2}/4\pi d^{3}\), with \(\mu=M_{s}V\) the particle moment and \(d\) the minimum interparticle distance. (ii) the anisotropy energy \(k\equiv K_{1}V\), and (iii) the Zeeman energy \(h\equiv\mu_{0}\mu H\) due to the applied dc field \(H\). The energy parameters (\(g,k,h\)) entering Eq.(34), the thermal energy \(k_{B}T\), and the treatment history of the sample determine the micromagnetic configuration at a certain temperature and field. The freedom to choose an arbitrary energy scale makes the numerical results applicable to a class of materials with the same parameter ratios rather than to a specific material. The crucial parameter that determines the transition from single-particle to collective behavior is the ratio of the dipolar to the anisotropy energy (\(g/k\)).
+
+Figure 7: Dependence of the saturation isothermal remanence of a random assembly (a) on volume fraction of MNPs, at very low temperature (t/k=0.001) and, (b) on temperature, for fixed volume fraction. The particles have random anisotropy. The data are obtained by MMC simulations.
+
+We show in Fig. 7 the concentration and temperature dependence of the remanence magnetization of a random assembly. Notice in Fig. 7a, that weak DDI produce an increase of the remanence with concentration, while strong DDI have the opposite effect. Remarkably, the presence of free sample boundaries, can reverse the increasing trend of the remanence, due to the presence of a demagnetizing field. When DDI are much stronger than single-particle anisotropy (\(g/k\sim 10\)), the remanence value is sensitive to the morphology of the assembly, as the peak around the percolation threshold indicates. This behavior is explained by the anisotropic character of DDI (see Fig. 6). In Fig. 7b, DDI interactions are shown to produce a much slower temperature decay, producing finite remanence values above the blocking temperature of the isolated MNPs. This result supports the predictions of the mean-field theory about the increase of the measured blocking temperature in dipolar interacting systems (Dormann 1988).
+
+Figure 8: ZFC curves and blocking temperature for an ordered (hexagonal) assembly of identical MNPs with diameter \(D\) and center-to-center distance \(d\). (a) Evolution of ZFC curves with decreasing \(d\) values. (b) Linear scaling of \(T_{b}\) with inverse cube of \(d\). Data obtained by MMC simulations.
+
+In chemically-prepared, self-assembled MNPs the possibility to control the interparticle separation by variation of the surfactant (Willard 2004) offers the possibility to study the dependence of \(T_{b}\) on interparticle spacing while preserving the geometrical arrangement of the assembly (hexagonal). In Fig. 8 we show results for the ZFC magnetization (\(M_{ZFC}\)) and the blocking temperature as obtained from the peak of the \(M_{ZFC}(T)\) curve for a hexagonal array of dipolar interacting MNPs with random anisotropy. Parameters corresponding to Co nanoparticles are used (Kechrakos 2002). The characteristic dependence of \(T_{b}\) on the inverse cube of interparticle spacing can be used as a proof of the dominant character of DDI in an assembly. Notice also that \(M_{ZFC}(T\approx 0)\) assumes a positive value that increases with \(d\) values. This feature arises from the gradual formation of a long range ferromagnetic ground state, due to DDI.
+
+More examples of MC or MD simulations and comparison to experiments on MNP assemblies can be found in the relevant literature (Vedmedenko 2007, Kechrakos and Trohidou 2008).
+
+## 5 Summary
+
+We have discussed the main theoretical concepts that pertain to the magnetization properties of isolated (non-interacting) nanoparticles and their assemblies. We estimated the critical radius for formation of single domain particles and studied the zero-temperature magnetization reversal mechanism of coherent rotation (Stoner-Wohlfarth model), the thermally activated reversal (Néel-Brown model) and the related phenomenon of superparamagnetism occurring above the blocking temperature. Complications arising from size polydispersity, distribution of easy axes directions, and applied field on the relaxation time for magnetization reversal were discussed. Two standard experimental techniques for (static) magnetic measurements, namely the field and temperature dependence of magnetization were outlined. Finally, the subject of interparticle dipolar interactions was introduced along with the most common theoretical techniques used to analyze interacting systems. Examples from Monte Carlo studies of MNP assemblies were given.
+
+## 6 Future perspectives
+
+The dynamic behavior of MNPs in the presence of interparticle interactions is expected to remain a topic of intense scientific and technological research in the coming years. The research effort is expected to focus on both the atomic scale properties of individual magnetic nanoparticles and on the mesoscopic properties of nanoparticle assemblies.
+
+On the _atomic_ scale, future goals will include : (i) Reduction of the magnetic particle size without violating thermal stability (_superparamagnetic limit_) at room temperature . The technological benefit from progress in this direction will be the development of magnetic data-storage media with higher areal density. Given that the SPM effect is not observed below a certain size of a MNP, due to disorder effects on the particle surface, the search for new high-anisotropy materials is required. Composite nanoparticles with a core-shell morphology (Skumryev 2003) constitute an interesting perspective.
+(ii) Understanding and control of surface effects. With reduction of particle size the contribution from surface moments become of increasing importance. The chemical structure of the surface (disorder, defects) controls the magnitude and type of the surface anisotropy, which is usually much (up to \(\sim 10\) times) larger than the core anisotropy. Synthetic methods can offer indispensable routes to surface structure modification. _Ab-initio_ electronic structure calculations are a valuable tool to predict the surface anisotropy values and modeling of MNPs as multi-spin system will reveal complex magnetization reversal mechanisms beyond the Stoner-Wohlfarth model (Kachkachi 2000). Experiments on individual nanoparticles (Wernsdorfer 2000) offer a unique test of the above theories.
+
+The future task on the _mesoscopic_ scale will be to understand and control collective magnetic behavior in ordered nanostructures (self-assembled MNPs and magnetic patterned media). Ordered nanostructures include chemically prepared self-assembled MNPs and lithographically prepared magnetic patterned media. Chemical synthesis of MNPs and self-assembly (bottom-up approach) is a very promising and cost-effective method to produce ordered MNP arrays (Willard 2004). However, deeper understanding and improvement of the self-assembly process is required in order to achieve larger (beyond \(1mm^{2}\)) sample area with structural coherence. There is still a remaining problem as nanoparticles self-assemble into hexagonal arrays that are incompatible with the square arrangements required in industrial applications. A resolution to this problem could be the recently demonstrated templated assembly (Cheng 2004). Lithographic patterning (top-down approach) offers better control over the geometrical aspects of the assembly but cannot yet produce nanostructures with size below \(\sim 100nm\) (Martin 2003). Increase of lithographic resolution is demanded in order to achieve patterned media with smaller (below \(0.1\mu m\)) characteristic size. On the measurements side, improvement of existing techniques to probe mesoscopic magnetic order and excitations is demanded. Recent examples are the observation of mesoscopic sale magnetic order in self-assembled Co nanoparticles by an indirect method (small-angle neutron scattering)(Sachan 2008) and by direct methods such as magnetic force microscopy (Puntes 2004) and electron holography (Yamamoto 2008). From the point of view of basic physics, ordered nanostructures constitute model systems to study collective magnetic behavior driven by magnetostatic interactions, because the size, the shape and the spatial arrangement of the magnetic nanostructures is well controlled. Known phenomena are to be demonstrated on the mesoscopic scale and new ones possibly to be discovered. As a recent example, we refer to the observation of magnetic frustration is magnetostatically coupled magnetic microrods (Wang 2006), a phenomenon previously met in bulk magnetic random alloys (spin glasses).
+
+Finally, progress in numerical modeling will provide methods for bridging the atomic scale and the mesoscopic scale simulations. Such multi-scale simulations point to the future of theoretical investigations in the field of relaxation in magnetic nanoparticles and have only recently started to appear (Yanes 2007, Kazantseva 2008).
+
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+

arxiv_ground_truth/0907.5464.md

@@ -0,0 +1,436 @@
+# Robust Downlink Beamforming in Multiuser MISO Cognitive Radio Networks
+
+ Ebrahim A. Gharavol1, Ying-Chang Liang2, and Koenraad Mouthaan1
+
+ 1 Department of Electrical and Computer Engineering, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077
+Email: {Ebrahim,k.mouthaan}@nus.edu.sg
+
+ 2 Institute of Infocomm Research, Agency for Science, Technology and Research (A*STAR), 1 Fusionpolis Way, Singapore 138632
+Email: ycliang@i2r.a-star.edu.sg
+
+###### Abstract
+
+This paper studies the problem of robust downlink beamforming design in a multiuser Multi-Input Single-Output (MISO) Cognitive Radio Network (CR-Net) in which multiple Primary Users (PUs) coexist with multiple Secondary Users (SUs). Unlike conventional designs in CR-Nets, in this paper it is assumed that the Channel State Information (CSI) for all relevant channels is imperfectly known, and the imperfectness of the CSI is modeled using an Euclidean ball-shaped uncertainty set. Our design objective is to minimize the transmit power of the SU-Transmitter (SU-Tx) while simultaneously targeting a lower bound on the received Signal-to-Interference-plus-Noise-Ratio (SINR) for the SU’s, and imposing an upper limit on the Interference-Power (IP) at the PUs. The design parameters at the SU-Tx are the beamforming weights, i.e. the precoder matrix. The proposed methodology is based on a worst case design scenario through which the performance metrics of the design are immune to variations in the channels. We propose three approaches based on convex programming for which efficient numerical solutions exist. Finally, simulation results are provided to validate the robustness of the proposed methods.
+
+Index Terms: Robust beamforming, cognitive radio network, multi-user MISO communication, worst case design, imperfect CSI
+
+## I Introduction
+
+A Cognitive Radio Network (CR-Net) [1], [2] is an intelligent solution to the spectrum scarcity problem. In a CR-Net, the Secondary Users (SUs) are allowed to operate within the service range of the Primary Users (PUs), though the PUs have higher priority in utilizing the spectrum. There are two types of CR-Nets: opportunistic CR-Nets for which the SUs sense the spectrum and try to utilize the unused channels when they are not occupied by PUs; and concurrent CR-Nets in which SUs are allowed to use the spectrum even when PUs are active, provided that the amount of interference power to each PU is kept below a certain threshold [2]. Hereafter, we will focus on concurrent CR-Nets.
+
+Fig. 1-a illustrates the downlink scenario of multiuser multiple-input single-output (MISO) CR-Net with \(K\) SUs coexist with \(L\) PUs. There are two sets of relevant channel state information (CSI) which play important roles in the system design: one set describes the channels between SU-Transmitter (SU-Tx) and SU-Receivers (SU-Rx’s) while the other set describes the channels between SU-Tx and PU-Receivers (PU-Rx’s). For simplicity, we term the first set of CSI as SU-link CSI and the second set as PU-link CSI. When PUs are inactive, the system becomes conventional multiuser MISO system, and SU-link CSI is needed for transmission design. This knowledge is usually acquired through transmitting pilot symbols from SU-Tx to SU-RXs, and feeding back the estimated CSI from SU-Rxs to SU-Tx. In practice, however, because of the time variant nature of wireless channels, it is not possible to acquire the CSI perfectly, either due to channel estimation error and/or feedback error. On the other hand, when PUs are active, PU-link CSI is further needed at SU-Tx for the purpose of controlling interferences at the PU-Rx’s. This CSI knowledge has to be acquired by SU-Tx through environmental learning [3], which again will introduce errors in CSI. In this paper we consider the transmit design for a multiuser MISO CR-Net with uncertain CSI in both SU-link and PU-link.
+
+Previously in conventional radio network design, ad-hoc methods, such as diagonal loading [4], were exploited in the design procedure of robust beamforming systems. Quite recently these designs are based on well-reputed mathematical methodologies, such as the systematical worst case designs [5]-[8]. These methods deal with a Minimum Variance Distortionless Response (MVDR) problem in the signal processing domain and show that the problem may be recast as a Second-Order Cone Program (SOCP) [9]. Also, it was shown that this worst case design scenario is equivalent to an adaptive diagonal loading [5]. One of the first worst case designs was published by Bengtsson and Otterstten [10]-[11]. They showed that the robust maximization of SINR would lead to a Semi-Definite Program (SDP) [9], after a simple Semidefinite Relaxation (SDR). Sharma, _et al.,_[12] developed a model to cover the Positive Semi-definiteness (PSD) of the channel covariance matrix. They proposed two SDPs, a conventional SDP and a SDP based on an iterative algorithm. Also Mutapcic _et al.,_[13] proposed a new tractable method to solve the robust downlink beamforming. Their method is based on the cutting set algorithm which is also an iterative method. Also [14]-[16] are targeting the robust design of a beamforming system using the worst case scenarios for Quality-of-Service (QoS) constraints.
+
+Quite a few works are published on the robust design for CR-Nets [18], [19], [20] and [21]. Zhang _et al._[18], [19] have studied such a CR-Net from an information theoretic perspective. The CR-Net considered in [18], [19] consists of one PU-Rx and one SU-Rx, and the SU-link CSI is assumed to be perfectly known, but the PU-link CSI has uncertainty. A duality theory was developed to cope with the CSI imperfectness. Additionally, the authors proposed an analytic solution for this case. Also, Zhi _et al._[20] designed a robust beamformer for a CR-Net, where the system setup is the same as in [19], however there may be some uncertainty in both the channel covariance matrix as well as the antenna manifold. Finally, Cumanan _et al._[21] considered a CR-Net having multiple PUs and only one SU. In this work, both channels are assumed to be imperfect. They also used the worst case design method to come up with a convex problem that can be solved efficiently.
+
+In this paper, we consider a downlink system of a CR-Net with multiple PUs and multiple SUs whose relevant CSI is imperfectly known. The imperfectness of the CSI is modeled using an Euclidean ball. Our design objective is to minimize the transmit power of the SU-Tx while simultaneously targeting a lower bound on the received Signal-to-Interference-plus-Noise-Ratio (SINR) for the SUs, and imposing an upper limit on the Interference-Power (IP) at the PUs. The design parameters at the SU-Tx are the beamforming weights, i.e. the precoder matrix. The proposed methodology is based on a worst case design scenario through which the performance metrics of the design are immune to variations in the channels. We propose three approaches based on convex programming for which efficient numerical solutions exist. In the first approach, the worst case SINR is derived through using loose upper and lower bounds on the terms appearing in the numerator and denominator of the SINR. Working in this line, SDP is developed which provides us the robust beamforming coefficients. In the second approach, the minimum SINR is found through minimizing its numerator while maximizing its denominator. Different from the first method, we chose exact upper and lower bounds on the previously mentioned terms. This approach does not lead to a SDP, but the resulting problem is still convex and may be solved efficiently. Finally, in our third approach, we find the exact minimum of SINR directly, and this method is also a general convex optimization problem.
+
+The rest of the paper is organized as follows. In Section II, the model of our studied system is described. In Section III, the robust design of a multiuser MISO CR-Net with multiple SUs and multiple PUs is considered. In Section IV, we show that the resulting optimization problem, using loose upper and lower bounds, is a SDP. In Sections V and VI we propose two more general problem formulations based on stricter bound and exact bound on the minimum value of SINR, respectively. In Section VII the simulation results that demonstrate the robustness of the proposed schemes are presented. Finally, in Section VIII we conclude the paper.
+
+**Notations:** Matrices and vectors are typefaced using slanted bold uppercase and lowercase letters, respectively. Conjugate and conjugate transpose of the matrix \(\mathbf{A}\) are denoted as the \(\mathbf{A}^{\dagger}\) and \(\mathbf{A}^{*}\), respectively. The trace of a matrix is annotated using \(\mathsf{Tr}\left[\cdot\right]\). Positive semi-definiteness of the matrix \(\mathbf{A}\) is depicted using \(\mathbf{A}\succeq 0\). The symbol “\(\triangleq\)” means “defined as”. \(\mathbb{C}^{m\times n}\) is used to describe the complex space of \(m\times n\) matrices. A Zero-Mean Circularly Symmetric Complex Gaussian (ZMCSCG) random variable with the variance of \(\sigma^{2}\) is denoted using \(\mathcal{CN}(0,\sigma^{2})\). For a vector like \(\mathbf{x}\), \(\|\mathbf{x}\|\) is the Euclidean norm while the norm of a matrix like \(\|\mathbf{A}\|\) is the Frobenius norm. To show the differentiation of a function, \(f\), with respect to some of its parameters, \(\mathbf{a}\), \(\nabla_{\mathbf{a}}f(\cdot)\) is used. Finally, mathematical expectation is described as \(\mathsf{E}\left\{\cdot\right\}\).
+
+## II System Model
+
+Fig. 1 shows the downlink scenario of a multiuser MISO CR-Net coexisting with a Primary-Radio Network (PR-Net) having \(L\) PUs each equipped with a single antenna. The SU-Tx equipped with \(N\) antennas transmits independent symbols, \(s_{k}\), to \(K\) different single antenna SUs, \(\{s_{k}\in\mathbb{C}\}_{k=1}^{K}\). It is assumed that the transmitted symbols are all Gaussian, zero-mean and independent, i.e., \(s_{k}\sim\mathcal{CN}(0,1)\). Each symbol is precoded by a weight vector, \(\{\mathbf{w}_{k}\in\mathbb{C}^{N\times 1}\}_{k=1}^{K}\), resulting in a vector signal, \(\{\mathbf{s}_{k}=\mathbf{w}_{k}s_{k}\}_{k=1}^{K}\), for each one. It is known that \(\mathbf{s}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{Q}_{k})\), where \(\mathbf{Q}_{k}\) is the covariance matrix of \(\mathbf{s}_{k}\); \(\mathbf{Q}_{k}=\mathsf{E}\{\mathbf{s}_{k}\mathbf{s}_{k}^{\dagger}\}=\mathbf{w}_{k}\mathbf{w}_{k}^{\dagger}\succeq 0\) and \(\mathbf{0}\) is the zero vector. The channel from SU-Tx to each SU-Rx is determined using a complex-valued vector, \(\{\mathbf{h}_{k}\in\mathbb{C}^{N\times 1}\}_{k=1}^{K}\) which is not perfectly known and there is some kind of uncertainty in channel gains. This uncertainty is described using an uncertainty set, \(\mathcal{H}_{k}\) which is defined as an Euclidean ball
+
+\[\mathcal{H}_{k}=\{\mathbf{h}|\|\mathbf{h}-\mathbf{\tilde{h}}_{k}\|\leq\delta_{k}\}.\] (1)
+
+In this definition, the ball is centered around the actual value of the channel vector, \(\mathbf{\tilde{h}}_{k}\), and the radius of the ball is determined by \(\delta_{k}\) which is a positive constant. Using this notion, the channel is modeled as
+
+\[\mathbf{h}_{k}=\mathbf{\tilde{h}}_{k}+\mathbf{a}_{k},\] (2)
+
+where \(\mathbf{a}_{k}\) is a norm-bounded uncertainty vector, \(\|\mathbf{a}_{k}\|\leq\delta_{k}\).
+
+The SU-Tx combines the signals and transmits the combination, \(\mathbf{x}\),
+
+\[\mathbf{x}=\sum_{k=1}^{K}\mathbf{s}_{k}=\mathbf{Ws},\] (3)
+
+where \(\mathbf{s}=[s_{1},\cdots,s_{K}]^{T}\in\mathbb{C}^{K\times 1}\) contains the transmitted symbols and as we know, \(\mathbf{s}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{K})\), also \(\mathbf{W}=[\mathbf{w}_{1},\cdots,\mathbf{w}_{K}]\in\mathbb{C}^{N\times K}\), is called the precoding matrix. For \(\mathbf{x}\) we know that \(\mathbf{x}\sim\mathcal{CN}(\mathbf{0},\mathbf{Q})\), where \(\mathbf{Q}=\mathsf{E}\{\mathbf{x}\mathbf{x}^{\dagger}\}=\mathbf{WW}^{\dagger}\succeq 0\). The design objective is to determine this precoding matrix \(\mathbf{W}\) based on certain criteria that we will discuss in the next few paragraphs.
+
+The channel from the SU-Tx to a PU-Rx is also defined using a complex valued vector, i.e., \(\{\mathbf{g}_{\ell}\in\mathbb{C}^{N\times 1}\}_{\ell=1}^{L}\). Here it is assumed that the CSI for these users is also uncertain. We use the same notation to describe the uncertainty for these channels. The uncertainty is defined using a set, \(\mathcal{G}_{\ell}\), which is
+
+\[\mathcal{G}_{\ell}=\{\mathbf{g}|\|\mathbf{g}-\mathbf{\tilde{g}}_{\ell}\|\leq\eta_{\ell}\}.\] (4)
+
+Equivalently, we may write
+
+\[\mathbf{g}_{\ell}=\mathbf{\tilde{g}}_{\ell}+\mathbf{b}_{\ell},\] (5)
+
+where \(\mathbf{b}_{\ell}\) is a norm-bonded uncertain vector, \(\|\mathbf{b}_{\ell}\|\leq\eta_{\ell}\) and \(\mathbf{\tilde{g}}_{\ell}\) is the actual value of the channel.
+
+Figure 1: A Typical Multiuser MISO CR-Net System with Uncertain CSI
+
+## III Problem Formulation
+
+For the system depicted in Fig.1 the total transmitted power, \(\mathtt{TxP}\), is given by
+
+\[\mathtt{TxP} \triangleq\mathsf{E}\left\{\|\mathbf{x}\|^{2}\right\}\]
+\[=\sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}.\] (6)
+
+The received signal at the \(k\)th SU is
+
+\[y_{k} =\mathbf{h}_{k}^{\dagger}\mathbf{x}+n_{k}\]
+\[=\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}s_{k}+\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{h}_{k}^{\dagger}\mathbf{w}_{i}s_{i}+n_{k}.\] (7)
+
+The right-hand side of (7) has three terms. The first term is the received signal of the intended message, while the second and the third terms show the interference from other messages and noise, which is white and Gaussian, i.e. \(n_{k}\sim\mathcal{CN}(0,\sigma_{n}^{2})\), respectively. The average received power for \(k\)th SU, \(\mathtt{S}_{k}\), from the intended message is
+
+\[\mathtt{S}_{k} \triangleq\mathsf{E}\left\{|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}s_{k}|^{2}\right\}\]
+\[=|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2}.\] (8)
+
+Similarly, it is easy to show that the received interference power, \(\mathtt{IP}_{k}\), is given by
+
+\[\mathtt{IP}_{k}\triangleq\mathsf{E}\left\{\left|\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{h}_{k}^{\dagger}\mathbf{w}_{i}s_{i}\right|^{2}\right\}=\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}|\mathbf{w}_{i}^{\dagger}\mathbf{h}_{k}|^{2}.\] (9)
+
+Using (8) and (9), the SINR of \(k\)th SU-Rx, \(\mathtt{SINR}_{k}\), is given by
+
+\[\mathtt{SINR}_{k}\triangleq\frac{\mathtt{S}_{k}}{\sigma_{n}^{2}+\mathtt{IP}_{k}}=\frac{|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2}}{\sigma_{n}^{2}+\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}|\mathbf{w}_{i}^{\dagger}\mathbf{h}_{k}|^{2}}.\] (10)
+
+In robust design problems relating to SINR, expressions with the form of \(|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2}\) are frequently used. We can write
+
+\[|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2} =\mathbf{w}_{k}^{\dagger}(\mathbf{\tilde{h}}_{k}+\mathbf{a}_{k})(\mathbf{\tilde{h}}_{k}+\mathbf{a}_{k})^{\dagger}\mathbf{w}_{k}\]
+\[=\mathbf{w}_{k}^{\dagger}(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{w}_{k},\] (11)
+
+where \(\mathbf{\tilde{H}}_{k}=\mathbf{\tilde{h}}_{k}\mathbf{\tilde{h}}_{k}^{\dagger}\) is a constant matrix and \(\mathbf{\Delta}_{k}\) is given by
+
+\[\mathbf{\Delta}_{k}=\mathbf{\tilde{h}}_{k}\mathbf{a}_{k}^{\dagger}+\mathbf{a}_{k}\mathbf{\tilde{h}}_{k}^{\dagger}+\mathbf{a}_{k}\mathbf{a}_{k}^{\dagger}.\] (12)
+
+Note that \(\mathbf{\Delta}_{k}\) is a norm bounded matrix, \(\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}\). It is straightforward to find the following relation:
+
+\[\epsilon_{k}\geq\|\mathbf{\Delta}_{k}\| =\|\mathbf{\tilde{h}}_{k}\mathbf{a}_{k}^{\dagger}+\mathbf{a}_{k}\mathbf{\tilde{h}}_{k}^{\dagger}+\mathbf{a}_{k}\mathbf{a}_{k}^{\dagger}\|\]
+\[\leq\|\mathbf{\tilde{h}}_{k}\mathbf{a}_{k}^{\dagger}\|+\|\mathbf{a}_{k}\mathbf{\tilde{h}}_{k}^{\dagger}\|+\|\mathbf{a}_{k}\mathbf{a}_{k}^{\dagger}\|\]
+\[\leq\|\mathbf{\tilde{h}}_{k}\|\ \|\mathbf{a}_{k}^{\dagger}\|+\|\mathbf{a}_{k}\|\ \|\mathbf{\tilde{h}}_{k}^{\dagger}\|+\|\mathbf{a}_{k}\|^{2}\]
+\[=\delta_{k}^{2}+2\delta_{k}\|\mathbf{\tilde{h}}_{k}\|.\] (13)
+
+Using (13) it is possible to choose \(\epsilon_{k}=\delta_{k}^{2}+2\delta_{k}\|\mathbf{\tilde{h}}_{k}\|\). We may use the identity of \(\mathbf{x}^{\dagger}\mathbf{Ax}=\mathsf{Tr}\left[\mathbf{Axx}^{\dagger}\right]\), to further simplify this expression, which gives
+
+\[|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2}=\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{w}_{k}\mathbf{w}_{k}^{\dagger}\right].\] (14)
+
+We also adopt the notation of \(\mathbf{W}_{k}=\mathbf{w}_{k}\mathbf{w}^{\dagger}_{k}\) in our design formulation. With this, we find that
+
+\[|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2}=\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{k}\right].\] (15)
+
+It is noted that, from now on, similar expressions will appropriately be used for the other terms of \(\mathtt{SINR}_{k}\).
+
+Also, the received signal at the \(\ell\)th PU is
+
+\[z_{\ell} =\mathbf{g}_{\ell}^{\dagger}\mathbf{x}+\nu_{\ell}\]
+\[=\sum_{k=1}^{K}\mathbf{g}_{\ell}\mathbf{w}_{k}s_{k}+\nu_{\ell},\] (16)
+
+and the interference power, \(\mathtt{IP}_{\ell}\), to this PU-Rx would be
+
+\[\mathtt{IP}_{\ell}\triangleq\sum_{k=1}^{K}|\mathbf{w}_{k}^{\dagger}\mathbf{g}_{\ell}|^{2}.\] (17)
+
+Again using similar formulation as \(|\mathbf{w}_{k}^{\dagger}\mathbf{h}_{k}|^{2}\), we get
+
+\[|\mathbf{w}_{k}^{\dagger}\mathbf{g}_{\ell}|^{2} =\mathbf{w}_{k}^{\dagger}(\mathbf{\tilde{G}}_{\ell}+\mathbf{\Lambda}_{\ell})\mathbf{w}_{k}\]
+\[=\mathsf{Tr}\left[\left(\mathbf{\tilde{G}}_{\ell}+\mathbf{\Lambda}_{\ell}\right)\mathbf{W}_{k}\right],\] (18)
+
+where \(\mathbf{\tilde{G}}_{\ell}\) is a constant matrix, \(\mathbf{\tilde{G}}_{\ell}=\mathbf{\tilde{g}}_{\ell}\mathbf{\tilde{g}}_{\ell}^{\dagger}\) and \(\mathbf{\Lambda}_{\ell}\) is the norm bounded uncertainty matrix, \(\|\mathbf{\Lambda}_{\ell}\|\leq\xi_{\ell}\). Similarly we know that \(\xi_{\ell}=\eta_{\ell}^{2}+2\eta_{\ell}\|\mathbf{\tilde{g}}_{\ell}\|\).
+
+Our design objective is to minimize the transmitted power, \(\mathtt{TxP}\), while guaranteeing that the SINR at SU-Rx for all the channel realizations is higher than the QoS-constrained threshold, \(\{\mathtt{SINR}_{k}\geq\gamma_{i}\}_{k=1}^{K}\), and simultaneously IP at PU-Rx is less than the PR-net–imposed threshold, \(\{\mathtt{IP}_{\ell}\leq\kappa_{\ell}\}_{\ell=1}^{L}\), respectively. Mathematically, this problem can be described as
+
+\[\mathop{\mbox{Minimize}}_{\{\mathbf{W}_{k}\}_{k=1}^{K}} \quad\mathtt{TxP}\] (19)
+Subject to \[\quad\mathop{\mathtt{SINR}_{k}}_{\forall\mathbf{h}_{k}\in\mathcal{H}_{k}}\geq\gamma_{k},\quad k=1,\cdots,K\]
+\[\quad\mathop{\mathtt{IP}_{\ell}}_{\forall\mathbf{g}_{\ell}\in\mathcal{G}_{\ell}}\leq\kappa_{\ell},\quad\ell=1,\cdots,L.\]
+
+The above problem is a problem with an infinite number of constraints. To deal with such a problem one well-known method is to find the minimum and maximum values of \(\mathtt{SINR}_{k}\) and \(\mathtt{IP}_{\ell}\), respectively, related to those realizations of the channels which are claimed as the “worst ones”. The worst channel realizations for SINR and IP would lead to the minimum and maximum value of SINR and IP, respectively. In that case, the problem will guarantee that the smallest possible SINR and largest possible IP also satisfy the constraints. Using this worst case design methodology, we could recast (19) to a simpler problem set as follows:
+
+\[\mathop{\mbox{Minimize}}_{\{\mathbf{W}_{k}\}_{k=1}^{K}} \quad\mathtt{TxP}\] (20)
+Subject to \[\quad\min_{\mathbf{h}_{k}\in\mathcal{H}_{k}}\ \mathtt{SINR}_{k}\geq\gamma_{k},\quad k=1,\cdots,K\]
+\[\quad\max_{\mathbf{g}_{\ell}\in\mathcal{G}_{\ell}}\ \mathtt{IP}_{\ell}\leq\kappa_{\ell},\quad\ell=1,\cdots,L.\]
+
+Adopting the previously mentioned notations, we rewrite it as
+
+\[\mathop{\mbox{Minimize}}_{\begin{subarray}{c}\{\mathbf{W}_{k}\}_{k=1}^{K}\end{subarray}} \quad\sum_{k=1}^{K}\mathsf{Tr}\left[\mathbf{W}_{k}\right]\]
+Subject to \[\quad\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\ \frac{\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{k}\right]}{\sigma_{n}^{2}+\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{i}\right]}\geq\gamma_{k},\quad k=1,\cdots,K\] (21a)
+\[\quad\max_{\|\mathbf{\Lambda}_{\ell}\|\leq\eta_{\ell}}\ \sum_{k=1}^{K}\mathsf{Tr}\left[\left(\mathbf{\tilde{G}}_{\ell}+\mathbf{\Lambda}_{\ell}\right)\mathbf{W}_{k}\right]\leq\kappa_{\ell},\quad\ell=1,\cdots,L.\] (21b)
+
+In the next sections, we will solve the robust problem of (21) and will show that this problem can be recast as a series of simple optimization problems.
+
+## IV Loosely Bounded Robust Solution
+
+In this section we will deal with the problem of (21). In [10] and [11], it is suggested to minimize the SINR through minimizing the numerator while maximizing its denominator. So (21a) is equivalent to
+
+\[\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\max_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]\geq\gamma_{k}\sigma_{n}^{2}.\] (22)
+
+As it is known, this method is a conservative way to find the minimum of the SINR.
+
+### _Minimization of SINR_
+
+To minimize the numerator,
+
+\[\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\ \mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{k}\right],\] (23)
+
+we adopt a loose lower bound, proposed by [10], [11]. Using this lower bound, we have
+
+\[\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\ \mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{k}\right]=\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}-\epsilon_{k}\mathbf{I}_{N}\right)\mathbf{W}_{k}\right],\] (24)
+
+and to maximize the denominator, the following term should be maximized
+
+\[\max_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\ \mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{i}\right].\] (25)
+
+Using a similar approximation, we have
+
+\[\max_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\ \mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{i}\right]=\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\epsilon_{k}\mathbf{I}_{N}\right)\mathbf{W}_{i}\right].\] (26)
+
+Using these results, the problem of SINR minimization (21a) is recast as
+
+\[\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}-\epsilon_{k}\mathbf{I}_{N}\right)\mathbf{W}_{k}\right]-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\epsilon_{k}\mathbf{I}_{N}\right)\mathbf{W}_{i}\right]\geq\sigma_{n}^{2}\gamma_{k},\ k=1,\cdots,K,\] (27)
+
+and by regrouping the left hand side of this equation we find
+
+\[\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right)\right]-\epsilon_{k}\mathsf{Tr}\left[\mathbf{W}_{k}+\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right]\geq\sigma_{n}^{2}\gamma_{k},\ k=1,\cdots,K.\] (28)
+
+### _The Whole Conventional Program_
+
+Using the same methodology as before, IP maximization (21b) leads to the following problem
+
+\[\sum_{k=1}^{K}\mathsf{Tr}\left[\left(\mathbf{\tilde{G}}_{\ell}+\xi_{\ell}\mathbf{I}_{N}\right)\mathbf{W}_{k}\right]\leq\kappa_{\ell},\quad\ell=1,\cdots,L.\] (29)
+
+Then the whole conventional program targeting to solve the robust downlink optimization in MISO CR-Nets becomes
+
+\[\mathop{\mbox{Minimize}}_{\begin{subarray}{c}\{\mathbf{W}_{k}\}_{k=1}^{K}\end{subarray}} \quad\sum_{k=1}^{K}\mathsf{Tr}\left[\mathbf{W}_{k}\right]\]
+Subject to \[\quad\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right)\right]-\epsilon_{k}\mathsf{Tr}\left[\mathbf{W}_{k}+\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right]\geq\sigma_{n}^{2}\gamma_{k},\ k=1,\cdots,K\] (30a)
+\[\quad\mathsf{Tr}\left[\left(\mathbf{\tilde{G}}_{\ell}+\xi_{\ell}\mathbf{I}_{N}\right)\sum_{k=1}^{K}\mathbf{W}_{k}\right]\leq\kappa_{\ell},\quad\ell=1,\cdots,L,\] (30b)
+\[\qquad\mathbf{W}_{k}=\mathbf{W}_{k}^{\dagger},\quad k=1,\cdots,K,\] (30c)
+\[\qquad\mathbf{W}_{k}\succeq 0,\quad k=1,\cdots,K.\] (30d)
+
+Please note the fact that the last two constraints are inherent in the structure of the problem formulation. Also note that to come up with a convex problem formulation, a non-convex constraint, \(\mathsf{rank}\{\mathbf{W}_{k}\}=1\), is eliminated [5], [10], [12]. This final form of the problem is an SDP and can be solved using efficient numerical methods [22]. Finally it should be noted that unlike [12] the beamforming weights are not exactly the principal eigenvector¹ of the matrix solution. To get the beamforming weights, the eigen decomposition of the \(\mathbf{W}_{k}\) is used. In this decomposition, \(\mathbf{W}_{k}\) may be decomposed to a series of rank one matrices, i.e.,
+
+Footnote 1: The principal eigenvector of a rank one matrix is the eigenvector corresponding to the only non-zero eigenvalue.
+
+\[\mathbf{W}_{k}=\sum_{n=1}^{N}\lambda_{n,k}\ \mathbf{e}_{n,k}\ \mathbf{e}_{n,k}^{\dagger},\] (31)
+
+where in this expansion, \(\lambda_{n,k}\) denotes the \(n\)th eigenvalue and \(\mathbf{e}_{n,k}\) is its respective eigenvector. The solution matrix of \(\mathbf{W}_{k}\) itself is a rank one matrix, then all the eigenvalues are equal to zero except one, let’s say \(\lambda_{N,k}\). Therefore the above mentioned equation may be written as
+
+\[\mathbf{W}_{k} =\lambda_{N,k}\ \mathbf{e}_{N,k}\ \mathbf{e}_{N,k}^{\dagger}\]
+\[=(\sqrt{\lambda_{N,k}}\ \mathbf{e}_{N,k})(\sqrt{\lambda_{N,k}}\ \mathbf{e}_{N,k})^{\dagger}\]
+\[=\mathbf{w}_{k}\mathbf{w}_{k}^{\dagger},\] (32)
+
+where \(\mathbf{w}_{k}=\sqrt{\lambda_{N,k}}\ \mathbf{e}_{N,k}\). The next two sections deal with the same problem but in different ways. It should be noted that the next two problems are not SDP and are generally convex problems, but, for these problems we use the same formula to acquire the beamforming weights from the solution matrix.
+
+## V Strictly Bounded Robust Solution
+
+In the previous section, the minimum of SINR was found using loose upper and lower bounds for its constituent terms. In this section, we try to minimize the SINR using the same method: we minimize the numerator and maximizing the denominator. But here, we try to find the exact maximum and the exact minimum for each term respectively:
+
+\[\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\max_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]\geq\gamma_{k}\sigma_{n}^{2}.\] (33)
+
+Our main tool, is the Lagrangian Multiplier method.
+
+### _Minimization of SINR_
+
+We start with the first minimization problem.
+
+_Proposition 1:_ For the terms \(\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k}\right)\mathbf{W}_{k}\right]\), using a norm-bounded variable \(\mathbf{\Delta}_{k}\), \(\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}\), the minimizer and maximizer would be
+
+\[\mathbf{\Delta}_{k}^{min}=-\epsilon_{k}\frac{\mathbf{W}_{k}^{\dagger}}{\|\mathbf{W}_{k}\|},\] (34)
+
+and
+
+\[\mathbf{\Delta}_{k}^{max}=\epsilon_{k}\frac{\mathbf{W}_{k}^{\dagger}}{\|\mathbf{W}_{k}\|},\] (35)
+
+respectively.
+
+_Proof:_ Please refer to Appendix A.
+
+Using the above results, we have
+
+\[\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]=\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}-\epsilon_{k}\frac{\mathbf{W}_{k}^{\dagger}}{\|\mathbf{W}_{k}\|}\right)\mathbf{W}_{k}\right]=\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\mathbf{W}_{k}\right]-\epsilon_{k}\|\mathbf{W}_{k}\|,\] (36)
+\[\max_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]=\mathsf{Tr}\left[\left(\mathbf{\tilde{H}}_{k}+\epsilon_{k}\frac{\mathbf{W}_{i}^{\dagger}}{\|\mathbf{W}_{i}\|}\right)\mathbf{W}_{i}\right]=\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\mathbf{W}_{i}\right]+\epsilon_{k}\|\mathbf{W}_{i}\|.\] (37)
+
+So we may rewrite (33) as
+
+\[\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\mathbf{W}_{k}\right]-\epsilon_{k}\|\mathbf{W}_{k}\|-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{k}\left(\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\mathbf{W}_{i}\right]+\epsilon_{k}\|\mathbf{W}_{i}\|\right)\] (38)
+\[\ =\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}\mathbf{W}_{i}\right)\right]-\epsilon_{k}\left(\|\mathbf{W}_{k}\|+\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}\|\mathbf{W}_{i}\|\right)\geq\gamma_{k}\sigma_{n}^{2}.\] (39)
+
+### _The Whole Program_
+
+Similarly, the IP constraints may be written as:
+
+\[\max_{\|\mathbf{\Lambda}_{\ell}\|\leq\xi_{k}} \sum_{k=1}^{K}\mathsf{Tr}\left[(\mathbf{\tilde{G}}_{\ell}+\Lambda_{\ell})\mathbf{W}_{k}\right]=\sum_{k=1}^{K}\left(\mathsf{Tr}\left[\mathbf{\tilde{G}}_{\ell}\mathbf{W}_{k}\right]+\xi_{\ell}\|\mathbf{W}_{k}\|\right)\leq\kappa_{\ell}.\] (40)
+
+Finally, the whole program is
+
+\[\mathop{\mbox{Minimize}}_{\begin{subarray}{c}\{\mathbf{W}_{k}\}_{k=1}^{K}\end{subarray}} \quad\sum_{k=1}^{K}\mathsf{Tr}\left[\mathbf{W}_{k}\right]\]
+Subject to \[\quad\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right)\right]-\epsilon_{k}\left(\|\mathbf{W}_{k}\|+\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\|\mathbf{W}_{i}\|\right)\geq\sigma_{n}^{2}\gamma_{k},\]
+\[\qquad\qquad\qquad k=1,\cdots,K\] (41a)
+\[\quad\sum_{k=1}^{K}\left(\mathsf{Tr}\left[\mathbf{\tilde{G}}_{\ell}\mathbf{W}_{k}\right]+\xi_{\ell}\|\mathbf{W}_{k}\|\right)\leq\kappa_{\ell},\quad\ell=1,\cdots,L\] (41b)
+\[\quad\mathbf{W}_{k}=\mathbf{W}_{k}^{\dagger},\quad k=1,\cdots,K\] (41c)
+\[\quad\mathbf{W}_{k}\succeq 0,\quad k=1,\cdots,K.\] (41d)
+
+Although this final problem is not an SDP, it is in fact convex, and this problem can be solved using standard numerical optimization packages, like CVX [22].
+
+## VI Exact Robust Solution
+
+In the last two sections we dealt with the problem of minimizing the SINR using a conservative method. In this section we find the exact solution instead. We start again with the problem of (21), but with a simple alteration. This problem is stated as:
+
+\[\mathop{\mbox{Minimize}}_{\{\mathbf{W}_{k}\}_{k=1}^{K}} \quad\sum_{k=1}^{K}\mathsf{Tr}\left[\mathbf{W}_{k}\right]\]
+Subject to \[\ \min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\left(\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]\right)\geq\sigma_{n}^{2}\gamma_{k},\ \ k=1,\cdots,K;\] (42a)
+\[\ \max_{\|\mathbf{\Lambda}_{\ell}\|\leq\xi_{\ell}}\sum_{k=1}^{K}\mathsf{Tr}\left[(\mathbf{\tilde{G}}_{\ell}+\mathbf{\Lambda}_{\ell})\mathbf{W}_{k}\right]\leq\kappa_{\ell},\quad\ell=1,\cdots,L.\] (42b)
+
+In the above problem, we try to minimize the SINR directly and without using conservative assumptions. First we have the following proposition:
+
+_Proposition 2:_ The minimizer of (42a) has the form of
+
+\[\mathbf{\Delta}_{k}^{min}=-\epsilon_{k}\frac{\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right)^{\dagger}}{\|\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\|}.\] (43)
+
+_Proof:_ Please refer to Appendix B.
+
+Using this proposition we find
+
+\[\min_{\|\mathbf{\Delta}_{k}\|\leq\epsilon_{k}}\left(\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]\right) =\]
+\[\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i})\right]-\epsilon_{k}\ \|\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\| \geq\sigma_{n}^{2}\gamma_{k}.\] (44)
+
+Finally we come up with the final and general problem:
+
+\[\mathop{\mbox{Minimize}}_{\{\mathbf{W}_{k}\}_{k=1}^{K}} \quad\sum_{k=1}^{K}\mathsf{Tr}\left[\mathbf{W}_{k}\right]\]
+Subject to \[\qquad\mathsf{Tr}\left[\mathbf{\tilde{H}}_{k}(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i})\right]-\epsilon_{k}\ \|\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\|\geq\sigma_{n}^{2}\gamma_{k},\quad k=1,\cdots,K;\] (45a)
+\[\qquad\sum_{k=1}^{K}\left(\mathsf{Tr}\left[\mathbf{\tilde{G}}_{\ell}\mathbf{W}_{k}\right]+\xi_{\ell}\|\mathbf{W}_{k}\|\right)\leq\kappa_{\ell},\quad\ell=1,\cdots,L;\] (45b)
+\[\qquad\mathbf{W}_{k}=\mathbf{W}_{k}^{\dagger},\quad k=1,\cdots,K\] (45c)
+\[\qquad\mathbf{W}_{k}\succeq 0,\quad k=1,\cdots,K.\] (45d)
+
+The above problem is the most general form of the original problem. It should be noted that the beamforming weights are also the principal eigenvector of the solutions of this problem. Also it should be mentioned that the IP part of these two last problems, (41) and (45), are the same, i.e., these two problems have the same performance in IPs.
+
+## VII Simulation Results and Discussions
+
+To validate our developed methods, a set of simulations were conducted. It is assumed that the BS is equipped with a Uniform Linear Array (ULA) having 8 elements with a spacing of half wave length. A set of \(K=3\) SU-Rx’s are served and the CR-Net should protect a set of \(L=2\) PU-Rx’s. The SU-Rx’s are located in the directions of \(\theta_{1}=20^{\circ}\), \(\theta_{2}=35^{\circ}\) and \(\theta_{3}=50^{\circ}\) relative to the antenna boreside, respectively. The PU-Rx’s are also located at the directions of \(\phi_{1}=80^{\circ}\) and \(\phi_{2}=85^{\circ}\), respectively. It is assumed that the change in Direction of Arrival (DoA) of input waves to the SU-Tx may be changed up to \(\pm 5^{\circ}\) arbitrarily. The noise power is assumed to be \(\sigma_{n}^{2}=0.01\), and constant for all of the users. Also, a constant SINR level of 10dB is targeted for all the SUs, while the constant interference threshold of 0.01 is used to protect the PUs. The channel model for PUs and SUs is assumed to be in line with a simple model of
+
+\[[\mathbf{h}_{k}(\theta_{k})]_{i} =e^{j\pi(i-1)\cos(\theta_{k})},\quad i,k=1,\cdots,K,\] (46)
+\[[\mathbf{g}_{\ell}(\phi_{\ell})]_{i} =e^{j\pi(i-1)\cos(\phi_{\ell})},\quad i,\ell=1,\cdots,L.\] (47)
+
+The uncertainty sets are characterized with \(\epsilon_{k}=\eta_{\ell}=0.05\). We have used the CVX Software Package [22] to solve the proposed problems numerically.
+
+In Fig. 2, the array gains toward each user using different weight vectors are depicted. In this figure and the subsequent figures, LCBS, SCBS and ExCS denote Loosely Bounded Convex Solution, Strictly Bounded Convex Solution, and Exact Convex Solution, respectively. In this figure, the vertical solid lines show the DoA corresponding to different SU-Rx’s, while the vertical dashed lines show the DoA of PU-Rx’s. From this figure, it is clear that the proposed method can transmit the desired data to the SU-Rx’s while it is protecting the PU-Rx’s. It is also apparent that all these approaches produce similar results.
+
+Figure 2: Array Gain for Different Users
+
+In Fig. 3, we have plotted the histogram of normalized constraints for both SU-Rx’s and PU-Rx’s. The normalized constraints of SUs, \(C_{k}^{\mathrm{(sinr)}}\), defined as \({\mathtt{SINR}_{k}}/{\gamma_{k}}\), is equivalent to
+
+\[C_{k}^{\mathrm{(sinr)}}=\frac{1}{\sigma_{n}^{2}\gamma_{k}}\mathbf{w}_{k}^{\dagger}\mathbf{H}_{k}\mathbf{w}_{k}-\frac{1}{\sigma_{n}^{2}}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{w}_{i}^{\dagger}\mathbf{H}_{k}\mathbf{w}_{i},\]
+
+where \(\mathbf{H}_{k}=\mathbf{h}_{k}\mathbf{h}_{k}^{\dagger}\), and the normalized PU constraint, \(C_{k}^{\mathrm{(ip)}}\), is defined as
+
+\[C_{\ell}^{\mathrm{(ip)}}=\frac{1}{\kappa_{\ell}}\sum_{k=1}^{K}\mathbf{g}^{\dagger}_{\ell}\mathbf{H}_{k}\mathbf{g}_{\ell}.\]
+
+Unlike the normalized SINR constraints for a SU-Rx, when a normalized IP constraint is less than one, this constraint is considered to be satisfied. In Fig. 3-a and Fig. 3-c, the normalized SINR histograms for two different scenarios are depicted. In the first one, the uncertainty sets are chosen to be \(\epsilon_{k}=\xi_{\ell}=0.05\) while for the second scenario, the uncertainty measures are four times more than the first one, i.e. \(\epsilon_{k}=\xi_{\ell}=0.20\). It is apparent that in this scenario the gap between the ultimate value of normalized SINR, i.e. 1, and the actual values is wider than in the first scenario, having smaller uncertainty sets. In both cases, as expected, ExCS is outperforming the other two schemes due to its exact bounds on SINR. In Fig. 3-b and Fig. 3-d the normalized IP constraints for PU-Rx’s are depicted. As can be seen, there is little difference between the proposed methods in terms of their IP constraints, because of the similar structure of these constraints.
+
+Figure 3: Normalized SINR Constraints for Different Methods for SU#1 and PU#1
+
+As it can be seen, the robust design is immune for the variation of channel, whereas the non-robust model fails in such situations. Also it is apparent that for the robust case, the variation of normalized constraints is much less than the variation of normalized constraints for the non-robust case. It is also clear that SBCS and ExCS are more efficient in terms of handling the SINR. The ExCS model not only can satisfy all the constraints, but also is a parsimonious model in terms of SINR. This is because of the fact that this method uses exact minimum of SINR. Also it should be noted that the IP variations in SBCS and ExCS are the same. It is because of the identical form of the equation which describes the IP constraints in these two methods. Additionally, it should be mentioned that they are slightly better than the LCBS.
+
+Also in Fig. 4 we have plotted the normalized total transmit power versus the SINR thresholds for different amounts of allowed normalized IP. The normalized total transmit power is the ratio of total transmit power to the noise power and the normalized IP is defined using the same manner. Both quantities are dimension-less and for better clarity are displayed in dBs. As expected, ExCS is better than the other two methods. In Fig. 4-a, it is clear that for a relative IP level of \(-\)4 dB, ExCS transmits the lowest amount of power while SBCS requires to transmit a modest amount of power relative to LBCS, and finally LBCS requires to transmit the largest amount of power. In this figure, it is also observed that for a relative IP level of 0 dB, the proposed ExCS is the best scheme to use to transmit power. In Fig. 4-b, we have plotted the same graph but in the higher SINR values. In this range of SINR thresholds, all of the optimization problems with a relative IP level of \(-\)4 dB would be infeasible, so the graph is only provided for the relative IP level of 0 dB. It is clear that in such scenarios, ExCS performs the best, although it should be noted that the performance of SBCS and ExCS are very close to each other.
+
+Figure 4: The Total Tx Power vs. SINR Thresholds
+
+## VIII Conclusion
+
+The problem of robust downlink beamforming design in multiuser MISO cognitive radio networks is studied. Particularly, a set up of \(K\) SU-Rx’s and \(L\) PU-Rx’s, all equipped with a single antenna is considered, and the SU-Tx has \(N\) transmit antennas. It is assumed that the relevant CSI is not known perfectly for both sets of users. The uncertainty in the CSI is modeled using an Euclidean ball notation. Three different approaches, namely LBCS, SBCS and ExCS, are presented which can be implemented efficiently. The first solution is a SDP, while the later two solutions are the convex optimization problems. Various simulation results are presented to evaluate the robustness of proposed methods.
+
+## Appendix A Proof of Proposition 1
+
+The Lagrangian function, using an arbitrary positive multiplier, \(\lambda\geq 0\), is
+
+\[L(\mathbf{\Delta}_{k},\lambda) =\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]+\lambda(\|\mathbf{\Delta}_{k}\|^{2}-\epsilon_{k}^{2})\]
+\[=\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]+\lambda(\mathsf{Tr}\left[\mathbf{\Delta}_{k}\mathbf{\Delta}_{k}^{\dagger}\right]-\epsilon_{k}^{2}).\] (48)
+
+By differentiating [23] of this Lagrangian function with respect to \(\mathbf{\Delta}_{k}^{*}\) and equating it to zero we will find the optimizer \(\mathbf{\Delta}_{k}\),
+
+\[\nabla_{\mathbf{\Delta}_{k}^{*}}L(\mathbf{\Delta}_{k},\lambda)=\mathbf{W}_{k}^{\dagger}+\lambda\mathbf{\Delta}_{k}=0,\] (49)
+
+which is annotated by \(\mathbf{\Delta}_{k}^{opt}\),
+
+\[\mathbf{\Delta}_{k}^{opt}=-\frac{1}{\lambda}\mathbf{W}_{k}^{\dagger}.\] (50)
+
+To eliminate the role of arbitrary parameter of \(\lambda\), again, we differentiate the Lagrangian function with respect to this unknown parameter and then equate it to zero
+
+\[\nabla_{\lambda}L(\mathbf{\Delta}_{k},\lambda)=0,\] (51)
+
+to get the optimizer \(\lambda\), annotated as \(\lambda^{opt}\),
+
+\[\lambda^{opt}=\frac{1}{\epsilon_{k}}\|\mathbf{W}_{k}^{\dagger}\|.\] (52)
+
+By combining these results, finally, we come up with
+
+\[\mathbf{\Delta}_{k}^{opt}=-\epsilon_{k}\frac{\mathbf{W}_{k}^{\dagger}}{\|\mathbf{W}_{k}\|}.\] (53)
+
+To test if this solution belongs to a minimum, we should observe that the second derivative at the optimizer points should have a non-negative value:
+
+\[\nabla^{2}_{\mathbf{\Delta}_{k}^{*}}L(\mathbf{\Delta}_{k}^{opt},\lambda^{opt})=\lambda^{opt}\geq 0.\] (54)
+
+To find the maximum of such a term, again, using a positive arbitrary Lagrangian multiplier, we build a Lagrangian function.
+
+\[L(\mathbf{\Delta}_{k},\lambda) =\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]-\lambda(\|\mathbf{\Delta}_{k}\|^{2}-\epsilon_{k}^{2})\]
+\[=\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]-\lambda(\mathsf{Tr}\left[\mathbf{\Delta}_{k}\mathbf{\Delta}_{k}^{\dagger}\right]-\epsilon_{k}^{2}).\] (55)
+
+By differentiating it with respect to \(\mathbf{\Delta}_{k}\) and equating it to zero
+
+\[\nabla_{\mathbf{\Delta}_{k}^{*}}L(\mathbf{\Delta}_{k},\lambda)=\mathbf{W}_{i}^{\dagger}-\lambda\mathbf{\Delta}_{k}=0,\] (56)
+
+we will get
+
+\[\mathbf{\Delta}_{k}^{opt}=\frac{1}{\lambda}\mathbf{W}_{i}^{\dagger}.\] (57)
+
+Again, by differentiating the Lagrangian function with respect to \(\lambda\) and equating it to zero,
+
+\[\nabla_{\lambda}L(\mathbf{\Delta}_{k},\lambda)=0,\] (58)
+
+we are able to get the optimizer.
+
+\[\lambda^{opt} =\frac{1}{\epsilon_{k}}\|\mathbf{W}_{i}\|,\] (59)
+\[\mathbf{\Delta}_{k}^{opt} =\epsilon_{k}\frac{\mathbf{W}_{i}^{\dagger}}{\|\mathbf{W}_{i}\|}.\] (60)
+
+To prove if this solution belongs to a maximum, we should observe that:
+
+\[\nabla^{2}_{\mathbf{\Delta}_{k}^{*}}L(\mathbf{\Delta}_{k}^{opt},\lambda^{opt})=-\lambda^{opt}\leq 0.\] (61)
+
+## Appendix B Proof of Proposition 2
+
+The Lagrangian multiplier is adopted again:
+
+\[L(\mathbf{\Delta}_{k},\lambda) =\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{k}\right]-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathsf{Tr}\left[(\mathbf{\tilde{H}}_{k}+\mathbf{\Delta}_{k})\mathbf{W}_{i}\right]+\lambda(\mathsf{Tr}\left[\mathbf{\Delta}_{k}\mathbf{\Delta}_{k}^{\dagger}\right]-\epsilon_{k}^{2}).\] (62)
+
+By differentiating this function and equating it with zero,
+
+\[\nabla_{\mathbf{\Delta}_{k}^{*}}L(\mathbf{\Delta}_{k},\lambda)=\mathbf{W}_{k}^{\dagger}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}^{\dagger}+\lambda\mathbf{\Delta}_{k}=0,\] (63)
+
+we will come up with
+
+\[\mathbf{\Delta}_{k}^{opt}=-\frac{1}{\lambda}\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right)^{\dagger},\] (64)
+
+and to eliminate the \(\lambda\),
+
+\[\nabla_{\lambda}L(\mathbf{\Delta}_{k},\lambda)=\|\mathbf{\Delta}_{k}\|-\epsilon_{k}=0,\] (65)
+
+we will get
+
+\[\lambda^{opt} =\frac{1}{\epsilon_{k}}\|\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\|,\] (66)
+\[\mathbf{\Delta}_{k}^{opt} =-\epsilon_{k}\frac{\left(\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\right)^{\dagger}}{\|\mathbf{W}_{k}-\gamma_{k}\sum_{\begin{subarray}{c}i=1\\ i\neq k\end{subarray}}^{K}\mathbf{W}_{i}\|}.\] (67)
+
+The second order differential test to prove that this solution belongs to a minimum, in this case, is also straight forward and is not included here.
+
+## References
+
+* [1] J. Mitola and G. Q. Maguire, “Cognitive radios: making software radios more personal,” _IEEE Personal communications,_ vol. 6, pp. 13-18, 1999.
+* [2] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” _IEEE Journal on Selected Areas in Communications,_ vol. 23, no. 2, pp. 201 - 220, 2005.
+* [3] F. Gao, R. Zhang, Y.-C. Liang and X. Wang, “Multi-antenna cognitive radio systems: Environmental learning and channel training,” _Proceedings of IEEE ICASSP 2009_, Taipei, April 2009.
+* [4] D. P. Bertsekas, _Nonlinear Programming,_ Athena Scientific, 1995.
+* [5] S. A. Vorobyov, A. B. Gershman and Z. -Q. Luo, “Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem,” _IEEE Transactions On Signal Processing,_ vol. 51, no. 2, pp. 313-324, 2003.
+* [6] S. A. Vorobyov, A. B. Gershman, Z. -Q. Luo and N. Ma, “Adaptive beamforming with joint robustness against mismatched signal steering vector and interference nonstationarity,” _IEEE Signal Processing Letters,_ vol. 11, no. 2, pp. 108-111, 2004.
+* [7] S. A. Vorobyov, H. Chen and A. B. Gershman, “On the relationship between robust minimum variance beamformers with probabilistic and worst-case distortionless response constraints,” _IEEE Transactions On Signal Processing,_ vol. 56, no. 11, pp. 5719-5724, 2008.
+* [8] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beamforming,” _IEEE Transactions On Signal Processing,_ vol. 53, no. 5, pp., 1684-1696, 2005.
+* [9] S. Boyd and L. Vandenberghe, _Convex Optimization,_ Cambridge University Press, 2004.
+* [10] M. Bengtsson and B. Ottersten, “Optimal downlink beamforming using semidefinite optimization,” _Proceedings of 37th Annual Allerton Conference on Communication, Control, and Computing,_ pp. 987-996, Sep. 1999.
+* [11] M. Bengtsson and B. Ottersten, “Optimum and suboptimum transmit beamforming,” Chapter 18 of _Handbook of antennas in wireless communications_ By Lal Chand Godara, CRC Press, 2001.
+* [12] V. Sharma, I. Wajid, A. B. Gershman, H. Chen and S. Lambotharan, “Robust downlink beamforming using positive semidefinite covariance constraints,” _Proceedings of 2008 International ITG Workshop on Smart Antenna (WSA 2008),_ pp. 36-41, Feb. 2008.
+* [13] A. Mutapcic, S. -J. Kim and S. Boyd, “A tractable method for robust downlink beamforming in wireless communications,” _Proceedings of 2007 Asilomar Conference on Signals, Systems, and Computers (ACSSC 2007),_ pp. 1224-1228, Nov. 2007.
+* [14] N. Vucic and H. Boche, “Robust QoS-constrained optimization of downlink multiuser MISO systems,” _IEEE Transactions On Signal Processing,_ vol. 57, no. 2, pp. 714-725, 2009.
+* [15] M. B. Shenouda and T. N. Davidson, “Convex conic formulations of robust downlink precoder design with quality of service constraints,” _IEEE Journal of Selected Topics in Signal Processing,_ vol. 1, no. 4, pp. 714-724, 2007.
+* [16] M. B. Shenouda and T. N. Davidson, “Non-linear and linear broadcasting with QoS requirements: tractable approaches for bounded channel uncertainties,” Available online at http://arxiv.org/abs/0712.1659
+* [17] S. -J. Kim, A. Magnani, A. Mutapcic, S. P. Boyd and Z. -Q. Luo, “Robust beamforming via worst-case SINR maximization” _IEEE Transactions On Signal Processing,_ vol. 56, no. 4, pp. 1539-1574, 2008.
+* [18] L. Zhang, Y. -C. Liang and Y. Xin, “Robust cognitive beamforming with partial channel state information,” _Proceedings of 42nd Annual Conference on Information Sciences and Systems (CISS 2008),_ pp. 890-895, Mar. 2008.
+* [19] L. Zhang, Y. -C. Liang, Y. Xin, and V. Poor, “Robust cognitive beamforming with partial channel state information,” _to appear in IEEE Transactions of Wireless Communications,_ Aug. 2009.
+* [20] W. Zhi, Y. -C. Liang and M. Y. W. Chia, “Robust transmit beamforming in cognitive radio networks”, _Proceedings of 2008 International Conference on Communications Systems (ICCS 2008),_ pp. 232-236, Nov. 2008.
+* [21] C. Cumanan, R. Krishna, V. Sharma and S. Lambotharan, “A Robust beamforming based interference control technique and its performance for cognitive radios,” _Proceedings of 2008 International Symposium on Communication and Information Technologies (ISCIT 2008),_ pp. 9-13, Oct. 2008.
+* [22] M. Grant, S. Boyd and Y. Ye, “CVX: Matlab software for disciplined convex programming, ver. 1.1,” Available at www.stanford.edu./\(\sim\)boyd/cvx, Nov. 2007.
+* [23] A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: techniques and key results,” _IEEE Transactions on Signal Processing,_ vol. 55, no. 6, pp. 2740-2746, Jun. 2007.
+

arxiv_ground_truth/0908.3175.md

@@ -0,0 +1,53 @@
+# The Impact of Stellar Populations on the Dynamics of Merger Remnants
+
+B. Rothberg, J. Fischer
+
+###### Abstract
+
+Many studies and simulations suggest gas-rich mergers do not contribute significantly to the overall star-formation rate and total mass function of galaxies. The velocity dispersions (\(\sigma\)) of Luminous & Ultraluminous Infrared Galaxies measured using the 1.62 or 2.29\(\mu\)m CO bandheads imply they will form m \(<\) m_∗_ ellipticals. Yet, \(\sigma\)’s obtained with the Calcium II triplet (CaT) at 0.85\(\mu\)m suggest all types of mergers will form m \(>\) m_∗_ ellipticals. Presented here are recent results, based on high-resolution imaging and multi-wavelength spectroscopy, which demonstrate the dominance of a nuclear disk of Red Supergiants (RSG) or Asymptotic Giant Branch (AGB) stars in the near-infrared bands, where dust obscuration does not sufficiently block their signatures. The presence of these stars severely biases the dynamical mass. At _I_-band, where dust can sufficiently block RSG or AGB stars, LIRGs populate the Fundamental Plane over a large dynamic range and are virtually indistinguishable from elliptical galaxies.
+
+_Naval Research Laboratory, Remote Sensing Division, Code 7211, 4555 Overlook Ave SW, Washington D.C. 20375_
+
+## 1 Introduction
+
+The “Toomre Hypothesis” (Toomre & Toomre 1972; Toomre 1977), proposes that the merger of two-gas rich spiral galaxies will form an elliptical galaxy, often with a final stellar mass larger than the sum of the progenitors. In the local universe, Luminous and Ultraluminous Infrared Galaxies are ideal candidates for forming massive elliptical galaxies (Kormendy & Sanders 1992). These are objects with _LIR_\(>\) 10¹¹ L_⊙_ between 8-1000 \(\mu\)m(Sanders & Mirabel 1996), contain vast quantities of molecular gas, and show strong evidence of recent or ongoing merging activity. Radio recombination line observations of the nearest ULIRG Arp 220 imply a formation rate of 10³ M_⊙_ yr_-1_(Anantharamaiah et al. 2000), while CO interferometric data indicate that 0.15-0.46 of the dynamical mass of this system is gaseous (Downes & Solomon 1998; Greve et al. 2009). The star-formation rates and vast quantities of gas in LIRGs/ULIRGs could add a significant stellar component to the total mass of the merger.
+However, a number of studies, all using infrared CO bandheads to measure central velocity dispersions (\(\sigma\)_∘_), have shown that LIRGs have masses consistent with low-moderate luminosity elliptical galaxies (_L_\(\sim\) 0.03-0.15 _L∗_) (Shier & Fischer 1998) and ULIRGs have masses consistent with _L_\(\leq\)_L∗_ (e.g. Genzel et al. (2001)). These results have raised significant doubts as to whether gas-rich mergers contribute significantly to the formation of elliptical galaxies.
+Yet, \(\sigma\)_∘_ measured from the Calcium II Triplet absorption lines (CaT), suggest gas-rich mergers, including LIRGs, have masses which span nearly the entire mass range of elliptical galaxies (Lake & Dressler 1986; Rothberg & Joseph 2006) (hereafter RJ06). This difference in \(\sigma\), or \(\sigma\)-mismatch, is counter-intuitive. Namely, that LIRGs/ULIRGs, which are undergoing intense star-formation and possess large quantities of dust, should show smaller \(\sigma\)_∘_ at _longer_ wavelengths. The use of infrared stellar lines to measure \(\sigma\)_∘_ was initially motivated by the need to pierce the veil of extinction in starburst galaxies and measure their “true” dynamical masses. However, the results presented here show that IR-luminous mergers are Janus-like, that is, they reveal two different dynamical faces depending on the wavelength observed. The obscuring characteristics of dust in the optical in IR-luminous galaxies behaves in a manner beneficial for determining the true mass of merger remnants.
+
+## 2 Sample Selection & Observations
+
+The dynamical properties of a sample of 14 advanced (single-nuclei) merger remnants are compared with a sample of 23 elliptical galaxies. The merger remnants are a subsample of the 51 merger remnants discussed in detail in Rothberg & Joseph (2004) (hereafter RJ04). The photometric data for the merger remnants include _F814W_ (\(\sim\)_I_-band) imaging from the Wide-Field Planetary Camera 2 (_WFPC2_) or the Advanced Camera for Surveys Wide-field Camera (_ACS/WFC_) on _HST_ and _K_-band imaging from Quick Infrared Camera (QUIRC) on the University of Hawaii 2.2m telescope. The kinematic data for the merger remnants include CaT observations from ESI on Keck-2, and CO observations from either NIRSPEC on Keck-2 or GNIRS on Gemini South (Program GS-2007A-Q-17, P.I. Rothberg). Additional kinematic and photometric data were obtained from the literature for several merger remnants and the comparison sample of ellipticals.
+
+## 3 The Fundamental Plane and Stellar Populations
+
+Figure 1.: Shown are the _I_-band (_left_) and _K_-band (_right_) Fundamental Planes from Scodeggio et al. (1997) and Pahre et al. (1998) respectively. Overplotted in both panels are LIRG merger remnants (open circles), non-LIRG merger remnants (filled circles) and elliptical galaxies (open diamonds). All 6 LIRGs, 20/23 ellipticals, and 3/8 non-LIRG merger remnants are overplotted on the _I_-band FP.
+
+A similar \(\sigma\)-mismatch was reported for a sample of 25 nearby early-type (predominantly S0) galaxies by Silge & Gebhardt (2003). They also found that \(\sigma\)_∘,optical_\(>\)\(\sigma\)_∘,CO_, and suggested that the kinematics of the cold stellar component in S0 galaxies was obscured by dust, and detectable only in the IR while the kinematics of the hot spheroid dominated optical wavelengths. However, it remained unclear whether bonafide ellipticals produced the same discrepancy. As noted in Rothberg (2009), no discernible difference between \(\sigma\)_∘,optical_ and \(\sigma\)_∘,CO_ was found for the comparison sample of 23 elliptical galaxies. On the other hand, LIRGs showed a large discrepancy, as first noted in RJ06. The Fundamental Plane (FP) is a two-dimensional plane embedded in the three-dimensional parameter space of \(\sigma\)_∘_, the half-light (effective) radius (_Reff_), and the surface brightness within the effective radius (\(<\)\(\mu\)\(>\)eff). _All_ elliptical galaxies lie on the Fundamental Plane. The LIRG/ULIRG studies noted earlier found that while these mergers were overly luminous in the infrared, they would eventually evolve onto the FP, but the small \(\sigma\)_∘,CO_ meant they could not be the progenitors of ellipticals with _L_\(>\)_L∗_. RJ06, however, found that the observed range of \(\sigma\)_∘,CaT_ meant that gas-rich mergers could populate nearly all of the mass-range of ellipticals, including _L_\(>\)_L∗_. Figure 1 is a two panel figure which shows the _I_-band and _K_-band FPs, with LIRG merger remnants (open circles), non-LIRG merger remnants (filled circles) and ellipticals (open diamonds).
+As expected, the ellipticals show little difference between the _I_ and _K_-band FPs. The difference in the location of (primarily) the LIRGs in the _I_-band and _K_-band is striking. Figure 1 explains the apparent contradictory results between earlier LIRG/ULIRG studies, which used “pure” _H_ or _K_-band FPs and those from RJ06, which used a “hybrid” FP (CaT \(\sigma\)_∘_ and _K_-band photometry). In the _I_-band, the dynamical properties of LIRGs are indistinguishable from ellipticals. Figure 2 shows a comparison between the _Mdyn/L_ and _Mdyn_ in the _I_-band (_left_) and _K_-band (_right_) for the merger remnants and elliptical galaxies (same symbols as Figure 1). Overplotted is the evolution of _M/L_ over time for a burst population from Maraston (2005) (hereafter M05). Figure 2 shows that in the _I_-band, the measured dynamical masses and stellar ages of the LIRGs are nearly the same as elliptical galaxies. However, the _K_-band measurements imply LIRGs have smaller _Mdyn_ and young ages.
+The _K_-band is dominated by the presence of young stars. Numerical simulations have long predicted that gaseous dissipation in the merging event funnels the gas into the barycenter of the merger (e.g. Barnes & Hernquist (1991); Barnes (2002)). This forms a rotating gaseous disk in the central 1-2 kpc of the merger, which then undergoes a strong starburst, forming a rotating disk of young stars. One observational signature of this starburst is the presence of “excess light” in the surface brightness profiles of mergers (e.g. Mihos & Hernquist (1994)), first detected in the _K_-band by RJ04. The \(\sigma\)-mismatch detected in IR-luminous galaxies is another observational signature of these rotating central starbursts. Dust associated with these nuclear starbursts blocks most of their light at \(\lambda\)\(<\) 1\(\mu\)m, while allowing the random motions of the nearly virialized older stars to dominate the \(\sigma\)_∘_ measurement at _I_-band. This functions in a similar manner to an occulting mask in a coronograph. At _H_ and _K_-band, the RSG and AGB stars can account for 60-90\(\%\) of the light (M05), therefore the disk kinematics overwhelms the \(\sigma\)_∘,CO_ measurement.
+
+Figure 2.: Two panel figure showing pure _I_-band (_left_) and pure _K_-band (_right_) _M_/_L_ vs. _Mdyn_. The overplotted vector (solid line) in each panel is the evolution of _M_/_L_ for a single-burst stellar population with solar metallicity and a Salpeter IMF as computed from Maraston (2005). The dotted vertical line in each panel indicates _m∗_.
+
+## References
+
+* Anantharamaiah et al. (2000) Anantharamaiah, K. R., Viallefond, F., Mohan, N. R., Goss, W. M., & Zhao, J. H. 2000, ApJ, 537, 613
+* Barnes & Hernquist (1991) Barnes, J. E. & Hernquist, L. E. 1991, ApJ, 370, L65
+* Barnes (2002) Barnes, J. E. 2002, MNRAS, 333, 481
+* Downes & Solomon (1998) Downes, D. & Solomon, P. M. 1998, ApJ, 507, 615
+* Genzel et al. (2001) Genzel, R., Tacconi, L. J., Rigopoulou, D., Lutz, D., & Tecza, M. 2001, ApJ, 563, 527
+* Greve et al. (2009) Greve, T. R., Papadopoulos, P. P., Gao, Y., & Radford, S. J. E. 2009, ApJ, 692, 1432
+* Kormendy & Sanders (1992) Kormendy, J. & Sanders, D. B. 1992, ApJ, 390, L53
+* Lake & Dressler (1986) Lake, G. & Dressler, A. 1986, ApJ, 310, 605
+* Maraston (2005) Maraston, C. 2005, MNRAS, 362, 799 (M05)
+* Mihos & Hernquist (1994) Mihos, J. C. & Hernquist, L. 1994, ApJ, 437, L47
+* Pahre et al. (1998) Pahre, M. A., Djorgovski, S. G., & de Carvalho, R. R. 1998b, AJ, 116, 1591
+* Rothberg & Joseph (2004) Rothberg, B. & Joseph, R. D. 2004, AJ, 128, 2098 (RJ04)
+* Rothberg & Joseph (2006) —. 2006, AJ, 131, 185 (RJ06)
+* Rothberg (2009) Rothberg, B. 2009, in Galaxy Evolution: Emerging Insights and Future Challenges
+* Sanders & Mirabel (1996) Sanders, D. B. & Mirabel, I. F. 1996, ARA&A, 34, 749
+* Scodeggio et al. (1997) Scodeggio, M., Giovanelli, R., & Haynes, M. P. 1997, AJ, 113, 101
+* Shier & Fischer (1998) Shier, L. M. & Fischer, J. 1998, ApJ, 497, 163
+* Silge & Gebhardt (2003) Silge, J. D. & Gebhardt, K. 2003, AJ, 125, 2809
+* Toomre (1977) Toomre, A. 1977, in Evolution of Galaxies and Stellar Populations, p. 401
+* Toomre & Toomre (1972) Toomre, A. & Toomre, J. 1972, ApJ, 178, 623
+

arxiv_ground_truth/0909.4218.md

@@ -0,0 +1,83 @@
+# Zooming on the Quantum Critical Point in Nd-LSCO
+
+Olivier Cyr-Choinière
+
+R. Daou
+
+J. Chang
+
+Francis Laliberté
+
+Nicolas Doiron-Leyraud
+
+David LeBoeuf
+
+Y. J. Jo
+
+L. Balicas
+
+J.-Q. Yan
+
+J.-G. Cheng
+
+J.-S. Zhou
+
+J. B. Goodenough
+
+Louis Taillefer
+
+Département de physique and RQMP, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada.
+Present address: Dresden High Magnetic Field Laboratory, 01328 Dresden, Germany.
+National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310-3706, USA.
+Ames Laboratory, Ames, Iowa 50011, USA
+Texas Materials Institute, University of Texas at Austin, Austin, Texas 78712, USA.
+Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada.
+E-mail: louis.taillefer@physique.usherbrooke.ca
+
+###### Abstract
+
+Recent studies of the high-\(T_{\rm c}\) superconductor La_1.6-x_Nd_0.4_Sr_x_CuO₄ (Nd-LSCO) have found a linear-\(T\) in-plane resistivity \(\rho_{\rm ab}\) and a logarithmic temperature dependence of the thermopower \(S\,/\,T\) at a hole doping \(p=0.24\), and a Fermi-surface reconstruction just below \(p=0.24\)[1, 2]. These are typical signatures of a quantum critical point (QCP). Here we report data on the \(c\)-axis resistivity \(\rho_{\rm c}(T)\) of Nd-LSCO measured as a function of temperature near this QCP, in a magnetic field large enough to entirely suppress superconductivity. Like \(\rho_{\rm ab}\), \(\rho_{\rm c}\) shows an upturn at low temperature, a signature of Fermi surface reconstruction caused by stripe order. Tracking the height of the upturn as it decreases with doping enables us to pin down the precise location of the QCP where stripe order ends, at \(p^{\star}=0.235\pm 0.005\).We propose that the temperature \(T_{\rm\rho}\) below which the upturn begins marks the onset of the pseudogap phase, found to be roughly twice as high as the stripe ordering temperature in this material.
+
+keywords: cuprate superconductors , stripe order , quantum critical point , pseudogap phase , Nd-LSCO , c-axis resistivity 74.25.Fy , 74.72.Dn , 75.30.Kz †
+
+Footnote †: journal: Physica C
+
+One of the central questions of high-\(T_{\rm c}\) superconductivity is the nature of the pseudogap phase. Recent quantum oscillation studies [3] favour a scenario of competing order, as they reveal that the large hole-like Fermi surface of overdoped cuprates [4] transforms into small electron-like pockets in the pseudogap phase [5]. This shows that there is some “hidden” order in the pseudogap phase which breaks translational symmetry and thus causes a reconstruction of the Fermi surface [6]. In some cuprates, such as La_1.6-x_Nd_0.4_Sr_x_CuO₄ (Nd-LSCO), there is clear evidence for charge / spin order, better known as “stripe order”, setting in at low temperature (see Ref. [7] and references therein), and the pseudogap phase may be a precursor to that stripe phase.
+
+The presence of an order in the phase diagram involves the presence of a quantum critical point (QCP) at a critical doping \(p^{\star}\) where the ordering temperature goes to zero. In Nd-LSCO at a hole-doping of \(p=0.24\), the in-plane resistivity \(\rho_{\rm ab}\) is linear down to the lowest temperature [1], and the thermopower \(S\) has a \(T\ln(1/T)\) dependence over a decade of temperature [2]. These are typical signatures of a quantum phase transition for a metal with two-dimensional antiferromagnetic fluctuations [8, 9]. They show that the QCP is close to but slightly below that doping, _i.e._\(p^{\star}<0.24\).
+
+In this Letter, we present measurements of the \(c\)-axis resistivity \(\rho_{\rm c}\) in Nd-LSCO as a function of temperature \(T\) in the vicinity of the QCP. These out-of-plane measurements reveal the same behavior as found in the in-plane data, namely a linear-\(T\) resistivity down to a temperature \(T_{\rm\rho}\) below which \(\rho(T)\) starts to deviate upwards [1]. In-plane data at \(p=0.20\) and \(p=0.24\) show that \(T_{\rm\rho}\) goes to zero between these two dopings (see Fig. 1). Here we use \(c\)-axis samples at intermediate dopings to pin down with greater accuracy the critical doping \(p^{\star}\) where \(T_{\rm\rho}\to 0\).
+
+The four samples of Nd-LSCO used in this study were grown at the University of Texas, as described elsewhere [1]. They have a doping of \(p=0.20,0.22,0.23\) and \(0.24\), respectively. The resistivity \(\rho_{\rm c}\) was measured at the National High Magnetic Field Laboratory in Tallahassee, in steady magnetic fields up to \(45\) T.
+
+Figure 1: (Color online) Temperature-doping phase diagram of Nd-LSCO. The temperature \(T_{\rm\rho}\) is defined as the temperature below which the normal-state resistivity \(\rho(T)\) deviates from its linear-\(T\) behavior (blue squares for in-plane, red circles for \(c\)-axis). Data for \(p=0.15\) is from [10]; \(\rho_{\rm ab}\) data for \(p=0.20\) and \(p=0.24\) are from [1]. At \(p=0.24\), \(T_{\rho}=0\) because there is no deviation from linearity down to the lowest temperature [1]. We also show \(T_{\rm charge}\), the onset temperature for charge order, detected either via nuclear quadrupole resonance (NQR; green up triangles) or via X-ray and neutron diffraction (black diamonds) [7]. The onset temperature for spin-stripe order, \(T_{\rm spin}\), is detected by neutron diffraction (black down triangles; [10]). We can then see that \(T_{\rm\rho}\approx 2T_{\rm charge}\). The superconducting transition temperature \(T_{\rm c}\) is shown as open black circles [1, 10]. All lines are a guide to the eye.
+
+Figure 2: (Color online) \(c\)-axis resistivity \(\rho_{\rm c}\) of Nd-LSCO at \(p=0.20,0.22,0.23\) and \(0.24\) (top to bottom) as a function of temperature \(T\), in a magnetic field of \(45\) T (applied along the \(c\)-axis). The lines are a linear fit of the data between 60 and 80 K for each curve separately. We define as \(\Delta\rho_{\rm c}(T)\) the difference between data and fit, plotted in the inset. In the limit \(T\to 0\), this gives us \(\Delta\rho_{\rm c}(0)\), shown by the double-headed (red) arrow. This quantity is plotted in Fig. 3. Inset: \(\Delta\rho_{\rm c}\) vs \(T\) for the four dopings. The onset of the upturn at a temperature \(T_{\rm\rho}\) is detected as an upward deviation from zero. Arrows mark \(T_{\rm\rho}=60\pm 10,50\pm 10\) and \(40\pm 10\) K for \(p=0.20\), \(0.22\) and \(0.23\) respectively. At \(p=0.24\), \(T_{\rm\rho}\) = 0.
+
+Figure 3: (Color online) Height of the upturn in \(\rho_{\rm c}(T)\) as a function of doping \(p\), defined as the \(T\to 0\) limit of \(\Delta\rho_{\rm c}(T)\), plotted in Fig. 2. A linear extrapolation (dashed line) of the \(\Delta\rho_{\rm c}(0)\) data at \(p=0.20\), 0.22 and 0.23 yields \(p^{\star}=0.235\pm 0.005\) as the critical doping beyond which \(\rho_{\rm c}(T)\) show no upturn. This is the QCP below which Fermi-surface reconstruction occurs.
+
+Our \(\rho_{\rm c}\) data in a field of \(45\) T is presented in Fig. 2. (Note that there is negligible magneto-resistance in all cases.) A linear fit to the data below 80 K is shown as a solid line. In the inset, we plot \(\Delta\rho_{\rm c}\), the difference between data and fit. At \(p=0.24\), we see that \(\rho_{\rm c}(T)\) remains linear down to the lowest temperature, as previously reported [1]. At lower doping, an upturn is observed below a temperature \(T_{\rm\rho}\) (see arrows in inset) which is plotted vs doping in Fig. 1 (red circles). The values of \(T_{\rm\rho}\) obtained from \(\rho_{\rm c}\) are seen to agree well with the overall doping dependence of \(T_{\rm\rho}\) obtained from \(\rho_{\rm ab}\) (reproduced from ref. [1]).
+
+Another way to describe the evolution of the \(c\)-axis resistivity data is to plot the magnitude of the upturn as a function of doping, defined as \(\Delta\rho_{\rm c}(0)\), the difference between data and fit in the limit of \(T\to 0\) (red double-headed arrow in Fig. 2). Fig. 3 shows \(\Delta\rho_{\rm c}(0)\) as a function of doping. It is clear that the height of the upturn goes down as the doping is increased, extrapolating to zero at \(p=0.235\pm 0.005\). This accurately locates the quantum critical point below which Fermi-surface reconstruction begins. We infer that this is where translational symmetry is broken at \(T=0\).
+
+As noted previously from in-plane data [1, 11], the upturn begins at a temperature significantly above the ordering temperature for stripe order, at \(T_{\rm charge}\) (see Fig. 1), with \(T_{\rho}\approx 2T_{\rm charge}\). This suggests a two-step transformation of the electronic behaviour [11]: a first transformation at high temperature, detected in the resistivity and the quasiparticle Nernst signal below \(T_{\nu}\)[12], with \(T_{\rm\rho}\simeq T_{\nu}\), and a second transformation at the stripe ordering temperature, detected in the Hall [1] and Seebeck coefficients [2].
+
+We propose that the temperature \(T_{\rm\rho}\simeq T_{\nu}\) is in fact the pseudogap temperature \(T^{\star}\). The pseudogap phase would then most likely be a fluctuating precursor of the spin/charge density wave (stripe) order observed at lower temperature (below \(T_{\rm charge}\)). This QCP may be a generic feature of hole-doped cuprates. Indeed, recent measurements of \(c\)-axis resistivity in overdoped Bi-2212 crystals show upturns below a temperature \(T_{\rho}\) which also goes to zero at \(p\simeq 0.24\)[13]. Moreover, a reconstruction of the Fermi surface by stripe-like order may also be a more general occurrence, given the very similar anomalies observed in YBCO in both the Hall [11] and Seebeck [14] coefficients.
+
+## References
+
+* [1] R. Daou _et al._, Nature Phys. **5**, 31 (2009).
+* [2] R. Daou _et al._, Phys. Rev. B **79**, 180505(R) (2009).
+* [3] N. Doiron-Leyraud _et al._, Nature **447**, 565 (2007).
+* [4] B. Vignolle _et al._, Nature **455**, 952 (2008).
+* [5] D. LeBoeuf _et al._, Nature **450**, 533 (2007).
+* [6] S. Chakravarty, Science **319**, 735 (2008).
+* [7] A. W. Hunt _et al._, Phys. Rev. B **64**, 134525 (2001).
+* [8] H. v. Löhneysen _et al._, Rev. Mod. Phys. **79**, 1015 (2007).
+* [9] I. Paul and G. Kotliar, Phys. Rev. B **64**, 184414 (2001).
+* [10] N. Ichikawa _et al._, Phys. Rev. Lett. **85**, 1738 (2000).
+* [11] L. Taillefer, J. Phys.: Condens. Matter **21**, 16 (2009)
+* [12] O. Cyr-Choinière _et al._, Nature **458**, 743 (2009).
+* [13] K. Murata _et al._, to appear in M2S-IX Proceedings.
+* [14] J. Chang _et al._, arXiv:0907.5039.
+

arxiv_ground_truth/0910.1571.md

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+# Identities of the Function \(f(x,y)=x^{2}+y^{3}\)
+
+Roger Tian
+
+(July 5, 2024)
+
+###### Abstract
+
+Harvey Friedman asked in \(1986\) whether the function \(f(x,y)=x^{2}+y^{3}\) on the real plane \(\mathbb{R}^{2}\) satisfies any identities; examples of identities are commutativity and associativity. To solve this problem of Friedman, we must either find a nontrivial identity involving expressions formed by recursively applying \(f\) to a set of variables \(\{x_{1},x_{2},\ldots,x_{n}\}\) that holds in the real numbers or to prove that no such identities hold. In this paper, we will solve certain special cases of Friedman’s problem and explore the connection between this problem and certain Diophantine equations.
+
+**Acknowledgements**
+
+I would like to thank my thesis advisor, George Bergman, for introducing Lemma 2 to me, shortening the proofs of several of the results below, pointing out areas of the paper that needed clarification, and giving advice on how to better organize this thesis.
+
+Notice that, given any identity in any number of variables, one can get an identity in one variable \(x\) by replacing all the variables of the given identity by \(x\). The single variable case of Friedman’s problem, whether or not there exists a nontrivial identity that holds in the real numbers involving expressions formed by recursively applying \(f\) to the variable set \(\{x\}\), may be easier to treat than the general problem of multiple variables.
+
+_A priori_, proving that no nontrivial identity of one variable holds does not completely solve the general problem, because two expressions, if equal as polynomials, that have the same “structure” regarding the composition of \(f\)’s (ignoring the variables involved) lead to the trivial identity when all the variables are replaced by \(x\). For instance, \(f(f(x,y),f(y,x))=f(f(y,x),f(x,y))\) as polynomials implies \(f(f(x,x),f(x,x))=f(f(x,x),f(x,x))\) as polynomials. However, proving that no nontrivial identity of one variable holds would tell us that two expressions can be equal as polynomials only if they have the same structure. For instance, since \(f(x,f(x,x))\) and \(f(f(x,f(x,x)),x)\) are not equal as polynomials, we know that \(f(x,f(y,z))\) and \(f(f(x,f(y,z)),y)\) cannot be equal as polynomials. We will use this observation to prove in Lemma 2 that a nontrivial multiple-variable identity holds only if a nontrivial 1-variable identity holds.
+
+We will follow the convention that \(0\notin\mathbb{N}\).
+
+_Notation 1__._: Suppose \(G(x_{1},x_{2},\ldots,x_{n})\) is an expression formed by recursively applying \(f\) to the variable set \(\{x_{1},x_{2},\ldots,x_{n}\}\). We shall call an occurrence of a variable in \(G(x_{1},x_{2},\ldots,x_{n})\) a **variable position**. Supposing that \(G(x_{1},x_{2},\ldots,x_{n})\) contains \(l\) variable positions where \(l\in\mathbb{N}\), we will proceed from left to right and label these successive variable positions as \(v_{1}\), \(v_{2}\), …, \(v_{l}\), and for all \(i=1,2,\ldots,l\) we will denote by \(\bar{v_{i}}\) the variable in \(\{x_{1},x_{2},\ldots,x_{n}\}\) occurring in the variable position \(v_{i}\). We define the **depth** of a variable position \(v\) occurring in an \(f\)-expression \(f(A,B)\) to be one more than its depth in \(A\) or \(B\) (whichever \(v\) occurs in), where we start by defining the depth of the bare expression \(x_{i}\) where \(i\in\{1,2,\ldots,n\}\) to be \(0\); we will denote the depth of \(v\) by \(0pt(v)\). For example, the variable positions \(v_{1}\), \(v_{2}\), \(v_{3}\) of \(f(x,f(y,x))\) hold the variables \(\bar{v_{1}}=x\), \(\bar{v_{2}}=y\), and \(\bar{v_{3}}=x\), while we have \(0pt(v_{1})=1\), \(0pt(v_{2})=2\) and \(0pt(v_{3})=2\). We can associate to \(G(x_{1},x_{2},\ldots,x_{n})\) the \(l\)-tuple \(((\bar{v_{1}},0pt(v_{1})),(\bar{v_{2}},0pt(v_{2})),\ldots,(\bar{v_{l}},0pt(v_{l})))\). It is clear that \(G(x_{1},x_{2},\ldots,x_{n})\) completely determines \(((\bar{v_{1}},0pt(v_{1})),(\bar{v_{2}},0pt(v_{2})),\ldots,(\bar{v_{l}},0pt(v_{l})))\).
+
+The following result was pointed out to the author by George Bergman, and its proof follows the ideas outlined by Bergman. This lemma establishes that, to answer Friedman’s problem in the negative, it suffices to prove that no nontrivial identity of one variable holds.
+
+**Lemma 2****.**: _Suppose that \(f\) satisfies no nontrivial 1-variable identities in \(\mathbb{R}\). Let \(n\) be a positive integer and let \(\{x_{1},x_{2},\ldots,x_{n}\}\) be a set of variables. Then \(f\) satisfies no nontrivial identities involving the variables \(x_{1}\), \(x_{2}\), …, \(x_{n}\) in \(\mathbb{R}\)._
+
+Proof.: Suppose that two distinct expressions \(G(x_{1},x_{2},\ldots,x_{n})\), \(H(x_{1},x_{2},\ldots,x_{n})\) formed by recursively applying \(f\) to \(\{x_{1},x_{2},\ldots,x_{n}\}\) are equal as polynomials. It follows that \(G(x,x,\ldots,x)\) and \(H(x,x,\ldots,x)\) are equal as polynomials. Then, by the assumption of no nontrivial 1-variable identities, \(G(x,x,\ldots,x)\) and \(H(x,x,\ldots,x)\) must be the same expression, so \(G(x_{1},x_{2},\ldots,x_{n})\) and \(H(x_{1},x_{2},\ldots,x_{n})\) must have the same number of variable positions. We will label the variable positions of \(G(x_{1},x_{2},\ldots,x_{n})\) by \(v_{1}\), \(v_{2}\), …, \(v_{l}\) and the variable positions of \(H(x_{1},x_{2},\ldots,x_{n})\) by \(v_{1}^{\prime}\), \(v_{2}^{\prime}\), …, \(v_{l}^{\prime}\), where \(l\) is some positive integer. Now, \(G(x_{1},x_{2},\ldots,x_{n})\) determines the \(l\)-tuple
+
+\[((\bar{v_{1}},0pt(v_{1})),(\bar{v_{2}},0pt(v_{2})),\ldots,(\bar{v_{l}},0pt(v_{l})))\]
+
+and \(H(x_{1},x_{2},\ldots,x_{n})\) determines the \(l\)-tuple
+
+\[((\bar{v_{1}^{\prime}},0pt(v_{1}^{\prime})),(\bar{v_{2}^{\prime}},0pt(v_{2}^{\prime})),\ldots,(\bar{v_{l}^{\prime}},0pt(v_{l}^{\prime}))).\]
+
+We know that for each \(i=1,2,\ldots,l\) we have \(0pt(v_{i})=0pt(v_{i}^{\prime})\), i.e. corresponding variable positions have the same depth. Let \(j\) be the smallest positive integer such that \(\bar{v_{j}}=x_{k}\neq x_{m}=\bar{v_{j}^{\prime}}\) where \(k\neq m\). Now replace \(x_{k}\) in \(G(x_{1},x_{2},\ldots,x_{n})\), \(H(x_{1},x_{2},\ldots,x_{n})\) by \(f(x,x)\) and replace \(x_{i}\) in \(G(x_{1},x_{2},\ldots,x_{n})\), \(H(x_{1},x_{2},\ldots,x_{n})\) by \(x\) for each \(i\neq k\), and we obtain a 1-variable identity. Then there exists at least one \(p\geq j\) in \(\mathbb{N}\) such that the \(p\)th variable position of the 1-variable expression resulting from \(G(x_{1},x_{2},\ldots,x_{n})\) has a depth one greater than that of the \(p\)th variable position of the 1-variable expression resulting from \(H(x_{1},x_{2},\ldots,x_{n})\). Therefore, the two 1-variable expressions in the identity are distinct, which is a contradiction. ∎
+
+The proof of Lemma 2 leaves open a more difficult question, as the statement of Lemma 2 is weaker than what we state in the following
+
+**Conjecture 3****.**: _Suppose that \(G(x)\) is an expression formed by recursively applying \(f\) to the variable set \(\{x\}\) and that \(G(x)\) has the variable positions \(v_{1}\), \(v_{2}\), …, \(v_{l}\) for some positive integer \(l\). Let \(n\) be a positive integer and \(\{x_{1},x_{2},\ldots,x_{n}\}\) be a set of variables. Let \(G(x_{1},x_{2},\ldots,x_{n})\) be an n-variable expression obtained by letting \(\bar{v_{i}}\in\{x_{1},x_{2},\ldots,x_{n}\}\) for all \(i=1,2,\ldots,l\). If \(G(x)\) cannot occur as either side of a nontrivial 1-variable identity, then \(G(x_{1},x_{2},\ldots,x_{n})\) cannot occur as either side of a nontrivial \(n\)-variable identity._
+
+Below, we prove some results on the single variable case of Friedman’s problem. They show that certain classes of expressions cannot occur as either side of a nontrivial identity.
+
+**Definition 4****.**: _If \(p(x)=\sum_{k=m}^{n}{a_{k}x^{k}}\) is a polynomial where \(m\leq n\) are nonnegative integers and where \(a_{m}\), \(a_{n}\) are nonzero, then \(m\) will be called the **order** of \(p(x)\), and \(n\) will be called the **degree** of \(p(x)\)._
+
+In what follows, by an \(f\)**-expression** we will, unless otherwise specified, always mean a symbolic expression in \(f\) and \(x\) that is formed by recursively applying \(f\) to the variable set \(\{x\}\); we also consider \(x\) itself an \(f\)-expression. We will denote the set of all \(f\)-expressions by \(\mathrm{term}(f;x)\). Let \(e:\,\,\)\(\mathrm{term}(f;x)\)\(\,\longrightarrow\,\mathbb{Z}[x]\) be the evaluation map that assigns to each \(f\)-expression its corresponding polynomial in \(\mathbb{Z}[x]\). We say that \(e(A)\) is the **polynomial induced by** the \(f\)-expression \(A\). For example, \(e(f(x,f(x,x)))=x^{2}+(x^{2}+x^{3})^{3}\). If \(A\) and \(B\) are two \(f\)-expressions and \(e(A)=e(B)\), then we say that \(A\) and \(B\) are \(e\)**-equivalent**. We shall call an \(f\)-expression \(e\)**-isolated** if it is not \(e\)-equivalent to any other \(f\)-expression, i.e. it cannot occur as either side of a nontrivial identity. For example, \(f(x,x)\) is \(e\)-isolated because, as it is not hard to see, \(e(f(A_{1},A_{2}))=e(f(x,x))\), where \(A_{1}\), \(A_{2}\) are \(f\)-expressions, implies \(A_{1}=x\) and \(A_{2}=x\).
+
+_Notation 5__._: Let \(A\in\mathrm{term}(f;x)\). For brevity, we will denote the degree of \(e(A)\) by \(\mathrm{dege}(A)\) and the order of \(e(A)\) by \(\mathrm{orde}(A)\). For the degree and order of any polynomial \(p(x)\) that is not written as the induced polynomial of some \(B\in\mathrm{term}(f;x)\), we retain the standard notation \(\deg(p(x))\) and \(\mathrm{ord}(p(x))\) respectively. For example, we would denote the degree of \(p(x)=x^{2}+x^{3}\) as \(\deg(p(x))\) if we did not explicitly state or did not know beforehand that \(p(x)=e(f(x,x))\).
+
+In the next three propositions, let \(A,B\in\,\mathrm{term}(f;x)\).
+
+**Proposition 6****.**: _If \(e(f(C_{1},C_{2}))=e(f(x,B))\) where \(C_{1}\), \(C_{2}\) are \(f\)-expressions, then we must have \(C_{1}=x\) and \(e(C_{2})=e(B)\)._
+
+Proof.: We have \(e(f(x,B))=x^{2}+e(B)^{3}=e(C_{1})^{2}+e(C_{2})^{3}\). Notice that \(3\,\mathrm{orde}(C_{2})\geq 3\), so \(x^{2}\) must arise in the expansion of \(e(C_{1})^{2}\). This forces \(C_{1}=x\), because otherwise every term that arises in the expansion of \(e(C_{1})^{2}\) will have powers at least as high as \(4\). Then cancellation gives us \(e(C_{2})^{3}=e(B)^{3}\), which forces \(e(C_{2})=e(B)\). ∎
+
+**Proposition 7****.**: _If \(e(f(C_{1},C_{2}))=e(f(A,x))\) where \(C_{1}\), \(C_{2}\) are \(f\)-expressions, then we must have \(e(C_{1})=e(A)\) and \(C_{2}=x\)._
+
+Proof.: We have \(e(f(A,x))=e(A)^{2}+x^{3}=e(C_{1})^{2}+e(C_{2})^{3}\). If \(C_{1}=x\), then \(e(C_{1})^{2}=x^{2}\). If \(C_{1}\neq x\), then \(\mathrm{orde}(C_{1})\geq 2\), so \(e(C_{1})^{2}\) has order at least 4. Thus, \(x^{3}\) must arise in the expansion of \(e(C_{2})^{3}\). Therefore, we have \(C_{2}=x\) and it follows from cancellation that \(e(C_{1})^{2}=e(A)^{2}\), so \(e(C_{1})=e(A)\). ∎
+
+_Remark 8__._: Actually, the arguments in the proofs of the previous two propositions also apply if \(f(x,B)\) in Proposition 6 and \(f(A,x)\) in Proposition 7 are instead multiple-variable \(f\)-expressions. For example, the same arguments can be applied, repeatedly, to show that \(f\)-expressions such as \(f(f(x,f(y,z)),y)\) are \(e\)-isolated. In effect, this settles a special case of Conjecture 3.
+
+**Proposition 9****.**: _If \(e(f(C_{1},C_{2}))=e(f(f(x,x),B))\) where \(C_{1}\), \(C_{2}\) are \(f\)-expressions, then we must have \(C_{1}=f(x,x)\) and \(e(C_{2})=e(B)\)._
+
+Proof.: We have \(e(C_{1})^{2}+e(C_{2})^{3}=e(f(x,x))^{2}+e(B)^{3}=(x^{2}+x^{3})^{2}+e(B)^{3}=x^{4}+2x^{5}+x^{6}+e(B)^{3}\). If \(B=x\), then the conclusion follows by Proposition 7. Suppose \(B\) is not \(x\). Notice that the terms \(x^{4}\) and \(2x^{5}\) must arise in the expansion of \(e(C_{1})^{2}\) because \(3\,\mathrm{orde}(C_{2})\geq 6\) as \(C_{2}\neq x\). Since \(x^{4}=x^{2}\cdot x^{2}\), we must have \(e(C_{1})=e(f(x,C_{3}))=x^{2}+e(C_{3})^{3}\) where \(C_{3}\) is another \(f\)-expression. Considering \((x^{2}+e(C_{3})^{3})^{2}\) and the fact that \(x^{5}=x^{2}\cdot x^{3}\) show that the expansion of \(e(C_{3})^{3}\) contains the term \(x^{3}\), so \(e(C_{3})^{3}=x^{3}\) and thus \(C_{3}=x\). Since \(e(C_{1})=x^{2}+x^{3}\), it follows again by cancellation that \(e(C_{2})^{3}=e(B)^{3}\) and so \(e(C_{2})=e(B)\). Since \(f(x,x)\) is \(e\)-isolated, we have \(C_{1}=f(x,x)\). ∎
+
+As can be seen, the above three propositions were established with an argument that works “outside-in” in the sense that it depends only on the \(x\) and \(f(x,x)\) that are being appended to \(A\), \(B\) by \(f\), while \(A\) and \(B\) can be completely arbitrary. This argument is difficult to apply for \(f\)-expressions such as \(f(C,B)\), where \(B\) is arbitrary and \(C\) is an \(f\)-expression such that \(\mathrm{dege}(C)>\mathrm{dege}(f(x,x))\). Below we will introduce an argument that works, in some sense, “inside-out.”
+
+Let \(f(A,B)\) be an \(f\)-expression. We will be examining the leading terms of the polynomial \(e(f(A,B))\) by looking at the subexpressions from which they arise. For instance, \(e(f(x,f(x,x)))=x^{2}+e(f(x,x))^{3}=x^{2}+(x^{2}+x^{3})^{3}\) and we see that the term with the degree of the polynomial arises from the \(f(x,x)\) by the product \((x^{3})^{3}=x^{9}\). The next lemma will show that the fact that this highest degree term arises from a subexpression \(f(x,x)\) is very generally true.
+
+**Lemma 10****.**: _Every summand contributing to the highest degree term of \(e(f(A,B))\) must arise from an occurrence of \(f(x,x)\) contained in \(f(A,B)\), on expanding \(e(f(A,B))\) in powers of \(x\)._
+
+Proof.: Consider the polynomial expansion of \(e(f(A,B))\). Suppose \(\mathrm{dege}(f(A,B))=2^{m}3^{n}\) for some \(m,n\in\mathbb{N}\cup\{0\}\). Then the highest degree term of \(e(f(A,B))\) is \(px^{2^{m}3^{n}}\) for some \(p\in\mathbb{N}\). Let \(x_{d}\) denote an occurrence of \(x\) in \(f(A,B)\) such that at least one of the \(p\) copies of \(x^{2^{m}3^{n}}\) (we will denote this copy by \([x^{2^{m}3^{n}}]_{\alpha}\)) in the expansion of \(e(f(A,B))\) contains at least one factor of this occurrence of \(x\), i.e. \([x^{2^{m}3^{n}}]_{\alpha}=x_{d}^{2}\cdot x^{2^{m}3^{n}-2}\) or \([x^{2^{m}3^{n}}]_{\alpha}=x_{d}^{3}\cdot x^{2^{m}3^{n}-3}\). Suppose that \(x_{d}\) is contained in a subexpression \(f(x_{d},C)\) or \(f(D,x_{d})\) where \(C\neq x\) and \(D\neq x\). Considering the product \(e(C)^{3}\cdot x^{2^{m}3^{n}-2}\) that arises from \(f(x_{d},C)\) and the product \(e(D)^{2}\cdot x^{2^{m}3^{n}-3}\) that arises from \(f(D,x_{d})\), we see that in the expansion of \(e(f(A,B))\) any occurrence of \(x\) contained in \(C\) or in \(D\) will lead to a term with a power higher than \(2^{m}3^{n}\). This is a contradiction, so \(x_{d}\) must be contained in an \(f(x,x_{d})\) in \(f(A,B)\). ∎
+
+We will call an occurrence of \(f(x,x)\) in the \(f\)-expression \(f(A,B)\) a **core** of \(f(A,B)\) if this occurrence of \(f(x,x)\) gives rise to a summand contributing to the highest degree term of \(e(f(A,B))\). Whenever an occurrence \(f(x,x)^{\prime}\) of \(f(x,x)\) in \(f(A,B)\) is a core of \(f(A,B)\), it is clear that \(e(f(x,x)^{\prime})^{\frac{1}{3}\mathrm{dege}(f(A,B))}=(x^{2}+x^{3})^{\frac{1}{3}\mathrm{dege}(f(A,B))}\) must be a term of \(e(f(A,B))\).
+
+We define inductively what it means to **develop** an \(f\)-expression about a core:
+
+1. 1.Start with \(f(x,x)\) and label it a core of the \(f\)-expression to be developed. Then \(f(x,x)\) is the \(f\)-expression at the first stage of the development.
+2. 2.Let \(A\) be the \(f\)-expression at the \(n\)th stage of the development where \(n\geq 1\). Then the \(f\)-expression at the \((n+1)\)st stage of the development is either \(f(A,C)\) where \(C\) is an \(f\)-expression such that \(3\,\mathrm{dege}(C)\leq 2\,\mathrm{dege}(A)\) or \(f(C,A)\) where \(C\) is an \(f\)-expression such that \(2\,\mathrm{dege}(C)\leq 3\,\mathrm{dege}(A)\).
+
+Any \(f\)-expression can be developed inductively in the above manner, though the development may not be unique. For example, we can develop \(f(x,f(x,x))\) only by the sequence of steps \(f(x,x)\), \(f(x,f(x,x))\) while we can develop \(f(f(x,f(x,x)),f(f(x,x),x))\) by the sequence \(f(x,x)\), \(f(x,f(x,x))\), \(f(f(x,f(x,x)),f(f(x,x),x))\) or by the sequence \(f(x,x)\), \(f(f(x,x),x)\), \(f(f(x,f(x,x)),f(f(x,x),x))\). However, every development of an \(f\)-expression whose induced polynomial has degree \(2^{m}3^{n}\) must consist of \(m+n\) stages.
+
+Suppose that \(C_{1}\) and \(C_{2}\) are distinct cores of the \(f\)-expression \(A\). Then we can find a subexpression \(f(D_{1},D_{2})\) of \(A\) such that either \(D_{1}\) contains \(C_{1}\) and \(D_{2}\) contains \(C_{2}\) or vice versa. Since \(C_{1}\), \(C_{2}\) are both cores, we must have \(D_{1}\neq D_{2}\) and \(2\,\mathrm{dege}(D_{1})=3\,\mathrm{dege}(D_{2})\). Since \(f(D_{1},D_{2})\) is the \(f\)-expression at the \(n\)th stage of the development of \(A\) about \(C_{1}\) and the \(f\)-expression at the \(n\)th stage of the development of \(A\) about \(C_{2}\) for some \(n\in\mathbb{N}\), we see that the development of \(A\) about \(C_{1}\) and the development of \(A\) about \(C_{2}\) differ at the \((n-1)\)st stage, i.e. these two developments are distinct. This analysis shows that an \(f\)-expression has a unique core whenever it has a unique development. Therefore, an \(f\)-expression has a unique development if and only if this \(f\)-expression has a unique core. Note that a “development” is defined as a property of an \(f\)-expression, not of a polynomial. So far as we know, for a given \(f\)-expression, the properties of having a unique core and of being \(e\)-isolated are independent of one another.
+
+It is easy to see that an \(f\)-expression corresponding to a non-monic polynomial does not have a unique core. However, the number of cores of an \(f\)-expression do not necessarily equal the leading coefficient of the induced polynomial; consider \(f(x,f(f(x,f(x,x)),f(f(x,x),x)))\), which has two cores while the polynomial it induces has a leading coefficient of 8. Moreover, it is easy to prove by induction the following
+
+**Lemma 11****.**: _Suppose \(A\in\mathrm{term}(f;x)\) and \(e(A)\) is non-monic. Then we have \(\mathrm{dege}(A)=2^{p}3^{q}\) where \(p\geq 1\) and \(q\geq 2\)._
+
+Proof.: Let \(A=f(B,C)\). If \(2\,\mathrm{dege}(B)=3\,\mathrm{dege}(C)\), then neither \(B\) nor \(C\) is \(x\), so the value of \(\mathrm{dege}(f(B,C))=2\,\mathrm{dege}(B)=3\,\mathrm{dege}(C)\) will be divisible by \(2\) and \(3^{2}\), the latter because \(\mathrm{dege}(C)\) is divisible by 3. If \(2\,\mathrm{dege}(B)\neq 3\,\mathrm{dege}(C)\), then whichever of \(B\) or \(C\) contributes the higher degree term must be non-monic, and in that case we may assume inductively that either \(\mathrm{dege}(B)\) or \(\mathrm{dege}(C)\) satisfies the conclusion of the lemma. ∎
+
+We shall call an \(f\)-expression \(f(A,B)\)**disjoint** if \(2\,\mathrm{dege}(A)<3\,\mathrm{orde}(B)\) or if \(3\,\mathrm{dege}(B)<2\,\mathrm{orde}(A)\). \(f(A,B)\) is called **hereditarily disjoint** if it is disjoint at every stage of its development about some core. We also consider \(x\) to be (vacuously) hereditarily disjoint.
+
+**Proposition 12****.**: _An \(f\)-expression \(A\) is hereditarily disjoint if and only if it is either_
+
+1. 1.\(x\)__
+2. 2.\(f(x,U)\) _for_ \(U\) _a hereditarily disjoint_ \(f\)_-expression_
+3. 3.\(f(U,x)\) _for_ \(U\) _a hereditarily disjoint_ \(f\)_-expression with_ \(\mathrm{orde}(U)\geq 2\)__
+4. 4.\(f(f(x,x),U)\) _for_ \(U\) _a hereditarily disjoint_ \(f\)_-expression with_ \(\mathrm{orde}(U)\geq 3\)__
+
+_In these four cases, \(\mathrm{orde}(A)=1\), \(2\), \(3\), \(4\) respectively, and it is easy to see that \(A\) has a unique core in all cases except \(A=x\)._
+
+Proof.: If \(A\) belongs to one of the above four cases, then \(A\) is hereditarily disjoint by definition. Now assume that \(A\) is hereditarily disjoint, we will show that \(A\) belongs to one of the above four cases. Suppose that \(A\neq x\) and that for each core there are \(n\) stages in the development of \(A\) about that core. Suppose that for all \(i\leq n-1\) the \(f\)-expression at the \(i\)th stage of the development of \(A\) about each of its cores belongs to one of the above four cases. Then either \(A=f(B,C)\) where \(3\,\mathrm{orde}(C)>2\,\mathrm{dege}(B)\) or \(A=f(C,B)\) where \(2\,\mathrm{orde}(C)>3\,\mathrm{dege}(B)\). By the inductive hypothesis, we have \(\mathrm{orde}(C)\leq 4\), which forces \(\mathrm{dege}(B)<6\) for the case \(A=f(B,C)\) and \(\mathrm{dege}(B)<3\) for the case \(A=f(C,B)\). Thus, \(A=f(B,C)\) implies \(B=f(x,x)\) or \(B=x\), and \(A=f(C,B)\) implies \(B=x\). This completes the induction. ∎
+
+_Notation 13__._: Let \(A,B\in\mathrm{term}(f;x)\). Whenever we denote \(A\) by \(f(\ldots B\ldots)\), we mean that \(B\) is a subexpression of \(A\) and \(B\) contains a core of \(A\).
+
+Suppose \(A:=f(\ldots f(C,B)\ldots)\) is an \(f\)-expression of degree \(2^{p}3^{q}\) where \(B\) contains a core of \(A\). Suppose \(3\,\mathrm{dege}(B)=2^{m}3^{n}\) and \(2\,\mathrm{dege}(C)=2^{i}3^{j}\). Define the **degree-gap between \(C\) and \(B\)** to be the positive integer
+
+\[\mathrm{dgap}(C,B):=3\,\mathrm{dege}(B)-2\,\mathrm{dege}(C)=2^{m}3^{n}-2^{i}3^{j}.\] (1)
+
+In the \(\mathrm{dgap}(-,-)\) notation we use, we will ignore the order of \(B\) and \(C\). In other words, we could also have written (1) as “\(\mathrm{dgap}(B,C):=3\,\mathrm{dege}(B)-2\,\mathrm{dege}(C)=2^{m}3^{n}-2^{i}3^{j}\)” (as we have already specified that \(B\) contains a core of \(A\)). Now consider the expansion of \((e(C)^{2}+e(B)^{3})^{2^{p-m}3^{q-n}}\) and notice that the **highest degree monomial which can contain a factor coming from \(C\) in the expansion of \(e(A)\)** is
+
+\[\mathrm{maxt}_{A}(C)=x^{2^{i}3^{j}}\cdot(x^{2^{m}3^{n}})^{2^{p-m}3^{q-n}-1}=x^{2^{i}3^{j}+(2^{m}3^{n})(2^{p-m}3^{q-n}-1)};\] (2)
+
+here we are ignoring the coefficient of \(x^{2^{i}3^{j}+(2^{m}3^{n})(2^{p-m}3^{q-n}-1)}\), as it is irrelevant at this point. The definition in (2) is relative to \(A\) given, but we will abbreviate \(\mathrm{maxt}_{A}(C)\) to \(\mathrm{maxt}(C)\) where there is no danger of confusion. Of course, \(B\) gives rise to the highest degree monomial \(x^{2^{p}3^{q}}=x^{\mathrm{dege}(A)}\) of \(A\). Notice that
+
+\[\mathrm{dege}(A)-\deg(\mathrm{maxt}(C)) =\deg((e(B)^{3})^{2^{p-m}3^{q-n}})-\deg(\mathrm{maxt}(C))\]
+\[=2^{p}3^{q}-(2^{i}3^{j}+(2^{m}3^{n})(2^{p-m}3^{q-n}-1))\]
+\[=2^{m}3^{n}-2^{i}3^{j}\]
+\[=\mathrm{dgap}(C,B),\]
+
+so the degree-gap between \(C\) and \(B\) is **preserved in the expansion of \(e(A)\)**. This (and its analogue in the next paragraph) will be an important fact in Lemma 17 and Proposition 22, where we will prove that an \(f\)-expression is \(e\)-isolated by considering all possible developments that lead to an \(f\)-expression \(e\)-equivalent to the given one.
+
+Similarly, in the opposite case, where \(B:=f(\ldots f(A,D)\ldots)\) is an \(f\)-expression of degree \(2^{p}3^{q}\), \(A\) contains a core of \(B\), \(2\,\mathrm{dege}(A)=2^{m}3^{n}\), and \(3\,\mathrm{dege}(D)=2^{i}3^{j}\), we can define the **degree-gap between \(A\) and \(D\)** to be the positive integer
+
+\[\mathrm{dgap}(A,D):=2\,\mathrm{dege}(A)-3\,\mathrm{dege}(D)\] (3)
+
+and notice that the **highest degree monomial which can contain a factor coming from \(D\) in the expansion of \(e(B)\)** is
+
+\[\mathrm{maxt}_{B}(D)=x^{2^{i}3^{j}+(2^{m}3^{n})(2^{p-m}3^{q-n}-1)}.\] (4)
+
+Again, we will ignore the order of \(A\) and \(D\) in the \(\mathrm{dgap}(-,-)\) notation we use, and we will abbreviate \(\mathrm{maxt}_{B}(D)\) to \(\mathrm{maxt}(D)\) where there is no danger of confusion. As before, we can observe that
+
+\[\mathrm{dege}(B)-\deg(\mathrm{maxt}(D))=\mathrm{dgap}(A,D).\]
+
+For an \(f\)-expression \(A:=f(\ldots B\ldots)\) where \(\deg(e(B))=2^{m}3^{n-1}\), we say that \(B\) is \(e\)**-isolated with respect to**\(A\) if, for every development of every \(f\)-expression \(e\)-equivalent to \(A\), we obtain the \(f\)-expression \(B\) (not merely some \(f\)-expression \(e\)-equivalent to \(B\)) at the \((m+(n-1))\)st stage of the development. Note that if \(B\) is \(e\)-isolated with respect to \(A\), then \(B\) must be \(e\)-isolated. The converse is not true, because even though \(f(x,f(x,x))\) is \(e\)-isolated, it is not \(e\)-isolated with respect to \(f(f(x,f(x,x)),f(f(x,x),x))\). Thus, \(B\) being \(e\)-isolated with respect to \(A\) is a stronger statement than \(B\) being \(e\)-isolated. Note also that Lemma 10 is equivalent to the statement that \(f(x,x)\) is \(e\)-isolated with respect to every \(f\)-expression other than \(x\).
+
+**Definition 14****.**: _Let \(A\in\mathrm{term}(f;x)\) and let \(D_{1}(A)\), \(D_{2}(A)\) denote two developments of \(A\), not necessarily distinct. We shall say that \(D_{1}(A)\), \(D_{2}(A)\)**agree** at the \(n\)th stage if there exists \(\hat{A}\in\mathrm{term}(f;x)\) such that \(\hat{A}\) is the \(f\)-expression at the \(n\)th stage of both \(D_{1}(A)\) and \(D_{2}(A)\)._
+
+_Notation 15__._: Given \(A\in\mathrm{term}(f;x)\), we will write \(A^{[n]}\) for the \(f\)-expression at the \(n\)th stage of the development of \(A\), provided that all developments of \(A\) agree at the \(n\)th stage. Note that this notation refers only to developments of \(A\), and not to developments of \(f\)-expressions \(e\)-equivalent to \(A\), in contrast to the definition of “\(e\)-isolated with respect to \(A\)” in the paragraph preceding Definition 14.
+
+**Definition 16****.**: _Let \(p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{1}x+a_{0}\) and \(q(x)=b_{n}x^{n}+b_{n-1}x^{n-1}+\ldots+b_{1}x+b_{0}\) be two polynomials with nonnegative coefficients. Suppose \(p(x)\neq q(x)\), and let \(m\) be the greatest integer such that \(a_{m}\neq b_{m}\). Then we say that \(p(x)\) is **lexicographically greater than**\(q(x)\), denoted by \(p(x)>_{L}q(x)\), if \(a_{m}>b_{m}\)._
+
+**Lemma 17****.**: _Let \(A=f(\ldots f(x^{\prime},B)\ldots)\) be an \(f\)-expression where \(x^{\prime}:=x\) for the purpose of distinguishing it from the other occurrences of the variable \(x\) in \(A\), \(\mathrm{dege}(A)=2^{p}3^{q}\), and \(B\) contains a core of \(A\). Suppose that for every \(f\)-expression \(C\) in the ellipses \((\ldots)\) of \(A\) we have \(\deg(\mathrm{maxt}(C))\leq\deg(\mathrm{maxt}(x^{\prime}))\). Suppose there exists either an \(f\)-expression \(\bar{A}=f(\ldots f(U,B)\ldots)\) such that \(\mathrm{dege}(\bar{A})=2^{p}3^{q}\), \(U\neq x\), and \(B\) contains a core of \(\bar{A}\) or an \(f\)-expression \(\hat{A}=f(\ldots f(B,V)\ldots)\) such that \(\mathrm{dege}(\hat{A})=2^{p}3^{q}\) and \(B\) contains a core of \(\hat{A}\). Then \(e(A)<_{L}e(\bar{A})\) if \(\bar{A}\) exists and \(e(A)<_{L}e(\hat{A})\) if \(\hat{A}\) exists._
+
+Proof.: By our assumption, the subexpressions in the ellipses of \(A\) give rise to terms with powers no higher than that of \(\mathrm{maxt}(x^{\prime})\). Suppose \(e(B)^{3}\) is of degree \(2^{m}3^{n}\). Notice that \((e(B)^{3})^{2^{p-m}3^{q-n}}=(e(B)^{2})^{2^{p-m-1}3^{q-n+1}}\) is common to both \(e(A)\) and \(e(\bar{A})\) (if \(\bar{A}\) exists), and is common to both \(e(A)\) and \(e(\hat{A})\) (if \(\hat{A}\) exists). We have \(\deg(e(A)-(e(B)^{3})^{2^{p-m}3^{q-n}})=\deg(\mathrm{maxt}(x^{\prime}))\), \(\deg(e(\bar{A})-(e(B)^{3})^{2^{p-m}3^{q-n}})\geq\deg(\mathrm{maxt}(U))\), and \(\deg(e(\hat{A})-(e(B)^{2})^{2^{p-m-1}3^{q-n+1}})\geq\deg(\mathrm{maxt}(V))\). Since \(U\neq x\), we have \(\deg(\mathrm{maxt}(U))>\deg(\mathrm{maxt}(x^{\prime}))\), so \(\deg(e(A)-(e(B)^{3})^{2^{p-m}3^{q-n}})<\deg(e(\bar{A})-(e(B)^{3})^{2^{p-m}3^{q-n}})\). It follows that \(e(A)<_{L}e(\bar{A})\) (in the case that \(\bar{A}\) exists) as desired. Since \(\mathrm{dgap}(x^{\prime},B)=2^{m}3^{n}-2>2^{m+1}3^{n-1}-3\geq 2^{m+1}3^{n-1}-3\,\mathrm{dege}(V)=\mathrm{dgap}(B,V)\), we have \(\deg(\mathrm{maxt}(x^{\prime}))=2^{p}3^{q}-\mathrm{dgap}(x^{\prime},B)<2^{p}3^{q}-\mathrm{dgap}(B,V)=\deg(\mathrm{maxt}(V))\), so \(\deg(e(A)-(e(B)^{3})^{2^{p-m}3^{q-n}})<\deg(e(\hat{A})-(e(B)^{2})^{2^{p-m-1}3^{q-n+1}})\). It follows that \(e(A)<_{L}e(\hat{A})\) (in the case that \(\hat{A}\) exists) as desired. ∎
+
+**Lemma 18****.**: _Let \(A=f(\ldots f(B,x^{\prime})\ldots)\) be an \(f\)-expression such that \(x^{\prime}:=x\), \(\mathrm{dege}(A)=2^{p}3^{q}\), and \(B\) contains a core of \(A\). Suppose \(\bar{A}=f(\ldots f(B,U)\ldots)\) is an \(f\)-expression such that \(U\neq x\), \(\mathrm{dege}(\bar{A})=2^{p}3^{q}\), and \(B\) contains a core of \(\bar{A}\). Suppose that for every \(f\)-expression \(C\) in the ellipses \((\ldots)\) of \(A\) we have \(\deg(\mathrm{maxt}(C))\leq\deg(\mathrm{maxt}(x^{\prime}))\). Then \(e(A)<_{L}e(\bar{A})\)._
+
+Proof.: By our assumption, the subexpressions in the ellipses of \(A\) give rise to terms with powers no higher than that of \(\mathrm{maxt}(x^{\prime})\). Suppose \(e(B)^{2}\) is of degree \(2^{m}3^{n}\). Notice that \((e(B)^{2})^{2^{p-m}3^{q-n}}\) is common to both \(e(A)\) and \(e(\bar{A})\). We have \(\deg(e(A)-(e(B)^{2})^{2^{p-m}3^{q-n}})=\deg(\mathrm{maxt}(x^{\prime}))\) and \(\deg(e(\bar{A})-(e(B)^{2})^{2^{p-m}3^{q-n}})\geq\deg(\mathrm{maxt}(U))\). Since \(U\neq x\), we have \(\deg(\mathrm{maxt}(U))>\deg(\mathrm{maxt}(x^{\prime}))\), so \(\deg(e(\bar{A})-(e(B)^{2})^{2^{p-m}3^{q-n}})>\deg(e(A)-(e(B)^{2})^{2^{p-m}3^{q-n}})\). It follows that \(e(A)<_{L}e(\bar{A})\) as desired. ∎
+
+**Lemma 19****.**: _Let \(U\in\mathrm{term}(f;x)\) where \(U\neq x\). Let \(B_{U}^{(1)}=f(x_{1},U)\) where \(x_{1}:=x\). For every positive integer \(n\), let \(B_{U}^{(n+1)}=f(x_{n+1},B_{U}^{(n)})\), where \(x_{n+1}:=x\). Let \(A=f(\ldots B_{U}^{(n)}\ldots)\) where \(n\in\mathbb{N}\). Then \(\deg(\mathrm{maxt}(x_{1}))>\deg(\mathrm{maxt}(x_{2}))>\ldots>\deg(\mathrm{maxt}(x_{n}))\)._
+
+Proof.: Since \(B_{U}^{(n)}\) contains a core of \(A\), \(U\) must contain a core of \(A\). Notice that \(\mathrm{dgap}(x_{1},U)<\mathrm{dgap}(x_{2},B_{U}^{(1)})<\ldots<\mathrm{dgap}(x_{n},B_{U}^{(n-1)})\). Since \(\deg(\mathrm{maxt}(x_{1}))=\mathrm{dege}(A)-\mathrm{dgap}(x_{1},U)\), \(\deg(\mathrm{maxt}(x_{2}))=\deg(A)-\mathrm{dgap}(x_{2},B_{U}^{(1)})\), …, and \(\deg(\mathrm{maxt}(x_{n}))=\mathrm{dege}(A)-\mathrm{dgap}(x_{n},B_{U}^{(n-1)})\), the conclusion immediately follows. ∎
+
+Analogously, we have the following
+
+**Lemma 20****.**: _Let \(V\in\mathrm{term}(f;x)\) where \(V\neq x\). Let \(C_{V}^{(1)}=f(V,x_{1})\) where \(x_{1}:=x\). For every positive integer \(n\), let \(C_{V}^{(n+1)}=f(C_{V}^{(n)},x_{n+1})\), where \(x_{n+1}:=x\). Let \(A=f(\ldots C_{V}^{(n)}\ldots)\) where \(n\in\mathbb{N}\). Then \(\deg(\mathrm{maxt}(x_{1}))>\deg(\mathrm{maxt}(x_{2}))>\ldots>\deg(\mathrm{maxt}(x_{n}))\)._
+
+**Lemma 21****.**: _Let \(A=f(\ldots f(f(x_{1},B),x_{2})\ldots)\) be an \(f\)-expression such that \(x_{1}:=x\), \(x_{2}:=x\) and \(B\neq x\). Then \(\deg(\mathrm{maxt}(x_{1}))>\deg(\mathrm{maxt}(x_{2}))\)._
+
+Proof.: Notice that \(B\) must contain a core of \(A\). We have \(\mathrm{dgap}(x_{1},B)=3\,\mathrm{dege}(B)-2<6\,\mathrm{dege}(B)-3=\mathrm{dgap}(f(x_{1},B),x_{2})\). Since \(\deg(\mathrm{maxt}(x_{1}))=\mathrm{dege}(A)-\mathrm{dgap}(x_{1},B)\) and \(\deg(\mathrm{maxt}(x_{2}))=\mathrm{dege}(A)-\mathrm{dgap}(f(x_{1},B),x_{2})\), the conclusion immediately follows. ∎
+
+**Proposition 22****.**: _Let \(A=f(\ldots f(x^{\prime},B)\ldots)\) be an \(f\)-expression such that \(x^{\prime}:=x\). Suppose that for every \(f\)-expression \(C\) in the ellipses \((\ldots)\) of \(A\) we have \(\deg(\mathrm{maxt}(C))\leq\deg(\mathrm{maxt}(x^{\prime}))\). Suppose that \(B\) is \(e\)-isolated with respect to \(A\). Then \(f(x^{\prime},B)\) is \(e\)-isolated with respect to \(A\)._
+
+Proof.: Suppose \(3\,\mathrm{dege}(B)=2^{m}3^{n}\). Let \(A^{\prime}\) be an \(f\)-expression \(e\)-equivalent to \(A\). Then \(A^{\prime[m+(n-1)]}=B\) by our assumption, so the \((m+n)\)th stage of every development of \(A^{\prime}\) is either \(f(U,B)\) or \(f(B,V)\) for some \(U,V\in\mathrm{term}(f;x)\). Suppose that either \(A^{\prime}=f(\ldots f(U,B)\ldots)\) where \(U\neq x\) or \(A^{\prime}=f(\ldots f(B,V)\ldots)\). By Lemma 17 we have \(e(A)<_{L}e(A^{\prime})\), which is a contradiction. It follows that the \((m+n)\)th stage of every development of \(A^{\prime}\) must be of the form \(f(U,B)\) where \(U=x\). Hence \(A^{\prime[m+n]}=f(x,B)\) as desired. ∎
+
+_Notation 23__._: Let \(B^{(1)}=f(x,x)\). For every positive integer \(n\), let \(B^{(n+1)}=f(x,B^{(n)})\). Also, we let \(B^{(0)}=x\).
+
+Notation 23 is to be used for the remainder of this paper, and is not to be confused with the _ad hoc_ notations set up in Lemma 19 and Lemma 20.
+
+**Corollary 24****.**: _Let \(A=f(\ldots f(x^{\prime},B^{(m)})\ldots)\) be an \(f\)-expression where \(x^{\prime}:=x\). Suppose that for every \(f\)-expression \(C\) in the ellipses \((\ldots)\) of \(A\) we have \(\deg(\mathrm{maxt}(C))\leq\deg(\mathrm{maxt}(x^{\prime}))\). Then \(f(x^{\prime},B^{(m)})\) is \(e\)-isolated with respect to \(A\)._
+
+Proof.: We know that \(B^{(1)}=f(x,x)\) is \(e\)-isolated with respect to \(A\). Suppose we know that \(B^{(k)}\) is \(e\)-isolated with respect to \(A\) for some \(1\leq k\leq m\). Writing \(A\) as \(f(\ldots f(x^{\prime\prime},B^{(k)})\ldots)\) where \(x^{\prime\prime}:=x\), we see by Lemma 19 that for every \(f\)-expression \(D\) in the ellipses of \(f(\ldots f(x^{\prime\prime},B^{(k)})\ldots)\) we have \(\deg(\mathrm{maxt}(D))\leq\deg(\mathrm{maxt}(x^{\prime\prime}))\). It then follows by Proposition 22 that \(f(x^{\prime\prime},B^{(k)})\) is \(e\)-isolated with respect to \(A\). This completes the induction. ∎
+
+**Lemma 25****.**: _Let \(A=f(\ldots f(x^{\prime},B^{(m)})\ldots)\) be an \(f\)-expression where \(x^{\prime}:=x\). Let \(\bar{A}=f(\ldots C\ldots)\) be an \(f\)-expression such that \(\mathrm{dege}(\bar{A})=\mathrm{dege}(A)\), \(C\neq f(x^{\prime},B^{(m)})\), and \(\mathrm{dege}(C)=\mathrm{dege}(f(x^{\prime},B^{(m)}))\). Suppose that for every \(f\)-expression \(D\) in the ellipses of \(A\) we have \(\deg(\mathrm{maxt}(D))\leq\deg(\mathrm{maxt}(x^{\prime}))\). Then \(e(A)<_{L}e(\bar{A})\)._
+
+Proof.: \(C\) has a unique core by Lemma 11. Since \(C\neq f(x^{\prime},B^{(m)})\) and \(\mathrm{dege}(C)=\mathrm{dege}(f(x^{\prime},B^{(m)}))\), we can find some \(k\leq m\) in \(\mathbb{N}\) such that \(C^{[k+1]}=f(U,B^{(k)})\) where \(U\neq x\). Then we can write \(\bar{A}\) as \(f(\ldots f(U,B^{(k)})\ldots)\). Since \(f(x^{\prime},B^{(m)})^{[k+1]}=f(x^{\prime\prime},B^{(k)})\) where \(x^{\prime\prime}:=x\), we can write \(A\) as \(f(\ldots f(x^{\prime\prime},B^{(k)})\ldots)\). We see by Lemma 19 that for every \(f\)-expression \(E\) in the ellipses of \(f(\ldots f(x^{\prime\prime},B^{(k)})\ldots)\) we have \(\deg(\mathrm{maxt}(E))\leq\deg(\mathrm{maxt}(x^{\prime\prime}))\). It follows by Lemma 17 that \(e(A)<_{L}e(\bar{A})\). ∎
+
+Lexicographic ordering on polynomials with nonnegative integer coefficients is a well-ordering. In particular, for all \(m\) and \(n\) the set of all polynomials of degree \(2^{m}3^{n}\) induced by \(f\)-expressions contains exactly one lexicographically minimal polynomial, and we will see that this polynomial corresponds to an \(e\)-isolated \(f\)-expression.
+
+As an illustration, we claim that the \(f\)-expression \(f(f(f(x,f(x,x)),x),x)\) leads to the lexicographically minimal polynomial with degree \(36=(2^{2})(3^{2})\). The following lemma gives the general rule.
+
+**Lemma 26****.**: _For any \(A\in\mathrm{term}(f;x)\), let \(u(A)=f(x,A)\) and \(v(A)=f(A,x)\). The \(f\)-expression that induces the lexicographically minimal polynomial of degree \(2^{m}3^{n}\) with \(n\geq 1\) is \(v(\ldots v(u(\ldots u(x)\ldots))\ldots)\) with \(m\)\(v\)’s followed by \(n\)\(u\)’s in left-to-right order. Moreover, this \(f\)-expression is \(e\)-isolated._
+
+Proof.: Let \(A:=v(\ldots v(u(\ldots u(x)\ldots))\ldots)\) with \(m\)\(v\)’s followed by \(n\)\(u\)’s in left-to-right order, and let \(A^{\prime}\in\mathrm{term}(f;x)\) be such that \(A^{\prime}\neq A\) and \(\mathrm{dege}(A^{\prime})=\mathrm{dege}(A)\). It is clear that \(A\) has exactly one core. Let \(k\) be the largest positive integer such that the development of \(A\) agrees with every development of \(A^{\prime}\) at the \(k\)th stage. Let \(C\) denote the \(f\)-expression at the \(k\)th stage of the development of \(A\). Suppose \(k<n\). Then we can write \(A\) as \(f(\ldots f(x_{1},C)\ldots)\) and we can write \(A^{\prime}\) as either \(f(\ldots f(U,C)\ldots)\) or \(f(\ldots f(C,V)\ldots)\) where \(x_{1}:=x\) and \(U,V\in\mathrm{term}(f;x)\) such that \(U\neq x\). By Lemma 19, Lemma 20, and Lemma 21, we have \(\deg(\mathrm{maxt}(x_{1}))>\deg(\mathrm{maxt}(D))\) for every \(f\)-expression \(D\) in the ellipses of \(f(\ldots f(x_{1},C)\ldots)\). It follows by Lemma 17 that \(e(A)<_{L}e(A^{\prime})\). Suppose \(k\geq n\). Then we can write \(A\) as \(f(\ldots f(C,x_{2})\ldots)\) and we can write \(A^{\prime}\) as \(f(\ldots f(C,W)\ldots)\), where \(x_{2}:=x\) and \(W\in\mathrm{term}(f;x)\) such that \(W\neq x\). By Lemma 20, we have \(\deg(\mathrm{maxt}(x_{2}))>\deg(\mathrm{maxt}(E))\) for every \(f\)-expression \(E\) in the ellipses of \(f(\ldots f(C,x_{2})\ldots)\). It follows by Lemma 18 that \(e(A)<_{L}e(A^{\prime})\). Thus, in all cases we have \(e(A)<_{L}e(A^{\prime})\). It immediately follows from this analysis that \(A\) is \(e\)-isolated, though we could well have proven this particular fact by repeatedly applying Proposition 6 and Proposition 7. ∎
+
+As we will see, this concept of lexicographic minimality can be applied to prove that many classes of \(f\)-expressions are \(e\)-isolated. This concept also illustrates one advantage of working with the single-variable case of Friedman’s problem, because it is less clear how one would lexicographically order multiple-variable polynomials.
+
+**Proposition 27****.**: _Let \(f(f(F,x^{\prime}),B)\) be an \(f\)-expression such that \(e(f(f(F,x^{\prime}),B))\) has degree \(2^{m}3^{n}\), where \(x^{\prime}:=x\). Suppose that_
+
+1. 1.\(3\,\mathrm{dege}(B)\leq\deg(\mathrm{maxt}(x^{\prime}))\)_, where we note that_ \(\deg(\mathrm{maxt}(x^{\prime}))=3+2\,\mathrm{dege}(F)\)__
+2. 2.\(e(F)\) _is lexicographically minimal among polynomials of degree_ \(2^{m-2}3^{n}\) _induced by_ \(f\)_-expressions as characterized in Lemma_ 26__
+3. 3.\(a\)_,_ \(b\) _are_ \(f\)_-expressions such that_ \(e(f(a,b))=e(f(f(F,x^{\prime}),B))\)_._
+
+_Then we must have \(a=f(F,x^{\prime})\) and \(e(b)=e(B)\)._
+
+Proof.: First notice that \(m\geq 2\) and that, by Lemma 26, \(e(f(F,x^{\prime}))\) is lexicographically minimal among polynomials of degree \(2^{m-1}3^{n}\) induced by \(f\)-expressions. Writing \(f(f(F,x^{\prime}),B)\) as \(f(\ldots f(x^{\prime\prime},B^{(n-1)})\ldots)\) where \(x^{\prime\prime}:=x\), we see by Lemma 20 and Lemma 21 that \(\deg(\mathrm{maxt}(C))\leq\deg(\mathrm{maxt}(x^{\prime\prime}))\) for every \(f\)-expression \(C\) in the ellipses of \(f(\ldots f(x^{\prime\prime},B^{(n-1)})\ldots)\). It follows by Corollary 24 that \(f(a,b)^{[n]}=B^{(n)}=f(f(F,x^{\prime}),B)^{[n]}\). Assume we know that \(f(a,b)^{[k]}=f(f(F,x^{\prime}),B)^{[k]}\) for some positive integer \(n\leq k<m-1+n\). Suppose \(f(a,b)=f(\ldots f(f(f(F,x^{\prime}),B)^{[k]},D)\ldots)\) where \(D\neq x\). Writing \(f(f(F,x^{\prime}),B)\) as \(f(\ldots f(f(f(F,x^{\prime}),B)^{[k]},x^{\prime\prime\prime})\ldots)\) where \(x^{\prime\prime\prime}:=x\), we see by Lemma 20 that \(\deg(\mathrm{maxt}(E))\leq\deg(\mathrm{maxt}(x^{\prime\prime\prime}))\) for every \(f\)-expression \(E\) in the ellipses of \(f(\ldots f(f(f(F,x^{\prime}),B)^{[k]},x^{\prime\prime\prime})\ldots)\). It follows by Lemma 18 that \(e(f(a,b))>_{L}e(f(f(F,x^{\prime}),B))\), which is a contradiction. Thus, we must have \(f(a,b)^{[k+1]}=f(f(f(F,x^{\prime}),B)^{[k]},x)=f(f(F,x^{\prime}),B)^{[k+1]}\). This completes the induction. Consequently, we have \(f(a,b)^{[(m-1)+n]}=f(F,x)\), so it follows that \(a=f(F,x)\). Finally, \(e(f(f(F,x),b))=e(f(f(F,x),B))\) implies that \(e(b)=e(B)\). ∎
+
+Next, we will apply the concept of lexicographic minimality to prove another general result. First, let \(A\), \(B\) be fixed \(f\)-expressions. We examine some of the restrictions \(e(f(A,B))=e(f(C,D))\) imposes on the \(f\)-expressions \(C\) and \(D\). To avoid triviality, assume \(e(B)\neq e(D)\). We have \(e(A)^{2}+e(B)^{3}=e(C)^{2}+e(D)^{3}\) iff
+
+\[e(B)^{3}-e(D)^{3}=e(C)^{2}-e(A)^{2}\] (5)
+
+iff \((e(B)-e(D))(e(B)^{2}+e(B)e(D)+e(D)^{2})=e(C)^{2}-e(A)^{2}\). Since \(e(B)-e(D)\neq 0\), we know that \(e(B)-e(D)\) is not a constant. So we have \(\deg(e(C)^{2}-e(A)^{2})>\max(2\,\mathrm{dege}(B),2\,\mathrm{dege}(D))\). In particular, if \(\mathrm{dege}(A)\leq\mathrm{dege}(B)\), then we must have
+
+\[2\,\mathrm{dege}(C)>\max(2\,\mathrm{dege}(B),2\,\mathrm{dege}(D)).\] (6)
+
+This implies that \(2\,\mathrm{dege}(C)>2\,\mathrm{dege}(B)\geq 2\,\mathrm{dege}(A)\), which implies that \(\mathrm{dege}(C)>\mathrm{dege}(A)\). It then follows by (5) that \(e(B)^{3}>_{L}e(D)^{3}\), so we have \(e(B)>_{L}e(D)\). Of course, from (6) we also obtain \(\mathrm{dege}(C)>\mathrm{dege}(D)\). Therefore, we have established the following
+
+**Lemma 28****.**: _Let \(A\), \(B\) be fixed \(f\)-expressions such that \(\mathrm{dege}(A)\leq\mathrm{dege}(B)\). If \(e(f(A,B))=e(f(C,D))\) and \(e(B)\neq e(D)\), then \(\mathrm{dege}(C)>\mathrm{dege}(A)\), \(e(B)>_{L}e(D)\), and \(\mathrm{dege}(C)>\mathrm{dege}(D)\)._
+
+_Notation 29__._: In what follows, we will denote the coefficient of the highest degree term of a polynomial \(p(x)\) by \(\mathrm{lead}(p(x))\). For example, we have \(\mathrm{lead}(e(f(f(x,f(x,x)),f(f(x,x),x))))=2\).
+
+For the next two propositions, fix \(f(A,f(E,B^{(m)}))\in\mathrm{term}(f;x)\) where \(\mathrm{dege}(E)\leq\mathrm{dege}(B^{(m)})\) and \(\mathrm{dege}(A)\leq\mathrm{dege}(f(E,B^{(m)}))\), so \(e(f(A,f(E,B^{(m)})))\) is monic, of degree \(3^{m+2}\). Writing \(f(A,f(E,B^{(m)}))\) as \(f(A,f(E,f(x^{\prime},B^{(m-1)})))\), we see that \(2\,\mathrm{dege}(A)<\deg(\mathrm{maxt}(E))<\deg(\mathrm{maxt}(x^{\prime}))\), because \(\mathrm{dgap}(A,f(E,B^{(m)}))=3^{m+2}-2\,\mathrm{dege}(A)\geq 3^{m+2}-2\,\mathrm{dege}(f(E,B^{(m)}))=3^{m+2}-2\cdot 3^{m+1}=3^{m+1}>3^{m+1}-2\,\mathrm{dege}(E)=\mathrm{dgap}(E,B^{(m)})\) and \(\mathrm{dgap}(E,B^{(m)})=3^{m+1}-2\,\mathrm{dege}(E)\geq 3^{m+1}-2\,\mathrm{dege}(B^{(m)})=3^{m+1}-2\cdot 3^{m}=3^{m}>3^{m}-2=\mathrm{dgap}(x^{\prime},B^{(m-1)})\). It follows that \(B^{(m)}\) is \(e\)-isolated with respect to \(f(A,f(E,B^{(m)}))\) by Corollary 24. Suppose that
+
+\[e(f(A,f(E,B^{(m)})))=e(f(C,D)).\]
+
+Then \(D=f(F,B^{(m)})\) for some \(F\in\mathrm{term}(f;x)\), so
+
+\[e(f(A,f(E,B^{(m)})))=e(f(C,f(F,B^{(m)}))).\] (7)
+
+In the following two propositions and their corollaries, we will use Lemma 28 along with the concept of lexicographic minimality to show that, for a large class of \(f\)-expressions that \(E\) may assume, the preceeding equality implies that
+
+\[F=E\;\;\;\&\;\;\;e(C)=e(A).\] (8)
+
+**Proposition 30****.**: _Let \(f(A,f(E,B^{(m)}))\in\mathrm{term}(f;x)\) have the property that \(\mathrm{dege}(E)\leq\mathrm{dege}(B^{(m)})\) and \(\mathrm{dege}(A)\leq\mathrm{dege}(f(E,B^{(m)}))\). Suppose that \(e(f(A,f(E,B^{(m)})))=e(f(C,D))\). Then \(D=f(F,B^{(m)})\) for some \(F\in\mathrm{term}(f;x)\). Furthermore, we must have \(e(E)\geq_{L}e(F)\) and \(\mathrm{dege}(E)=\mathrm{dege}(F)\)._
+
+Proof.: We have already proved the first part of the conclusion in the discussion leading up to (7), so it remains to show that \(e(E)\geq_{L}e(F)\) and \(\mathrm{dege}(E)=\mathrm{dege}(F)\). If \(e(F)=e(E)\), then we are done. Suppose \(e(F)\neq e(E)\). Then \(e(f(F,B^{(m)}))\neq e(f(E,B^{(m)}))\). By Lemma 28 we must have \(\mathrm{dege}(C)>\mathrm{dege}(A)\) and \(e(f(E,B^{(m)}))>_{L}e(f(F,B^{(m)}))\), and it follows that
+
+\[e(E)>_{L}e(F).\] (9)
+
+Therefore, if \(\mathrm{dege}(E)\neq\mathrm{dege}(F)\), we must have \(\mathrm{dege}(E)>\mathrm{dege}(F)\); so let us assume \(\mathrm{dege}(E)>\mathrm{dege}(F)\) and obtain a contradiction. We have
+
+\[e(A)^{2}+(e(E)^{2}+e(B^{(m)})^{3})^{3}=e(C)^{2}+(e(F)^{2}+e(B^{(m)})^{3})^{3}.\]
+
+After cancelling out the common \(e(B^{(m)})^{9}\) from both sides, we see that the degree of the left-hand side is \(\deg(e(E)^{2}e(B^{(m)})^{6})\) and the highest-degree term on the right-hand side must be \(e(C)^{2}\) because \(\deg(e(F)^{2}e(B^{(m)})^{6})<\deg(e(E)^{2}e(B^{(m)})^{6})\). This means that
+
+\[2\,\mathrm{dege}(C)=\deg(e(E)^{2}e(B^{(m)})^{6}).\] (10)
+
+Then the coefficient of the highest degree term of the left-hand side must be \(\mathrm{lead}(3e(E)^{2}e(B^{(m)})^{6})=3\,\mathrm{lead}(e(E))^{2}\mathrm{lead}(e(B^{(m)}))^{6}=3\,\mathrm{lead}(e(E))^{2}\), and the coefficient of the highest degree term of the right-hand side must be \(\mathrm{lead}(e(C)^{2})=\mathrm{lead}(e(C))^{2}\). We must have \(\mathrm{lead}(e(C))^{2}=3\,\mathrm{lead}(e(E))^{2}\), from which it follows that
+
+\[\mathrm{lead}(e(C))=\sqrt{3}\,\mathrm{lead}(e(E)),\] (11)
+
+which is not even rational. This is a contradiction, so we must have \(\mathrm{dege}(E)=\mathrm{dege}(F)\). ∎
+
+**Corollary 31****.**: _Under the hypotheses of Proposition 30, if \(E\) induces the lexicographically minimal polynomial with degree \(\mathrm{dege}(E)\), then \(F=E\) and \(e(C)=e(A)\)._
+
+Proof.: We must have \(e(E)=e(F)\) or \(e(E)<_{L}e(F)\). Since \(e(E)\geq_{L}e(F)\) by Proposition 30, this forces \(e(F)=e(E)\), from which the conclusion follows by Lemma 26. ∎
+
+The next proposition will be a generalization of Proposition 30. We will demonstrate in Corollary 34 that \((\ref{partial determination})\) follows even if \(f(E,B^{(m)})\) assumes values in a more general class of \(f\)-expressions than that in Corollary 31.
+
+_Notation 32__._: For every integer \(1\leq j\leq m\) define the function \(Y_{j}:\mathrm{term}(f;x)\)\(\longrightarrow\)\(\mathrm{term}(f;x)\) by \(Y_{j}(A)=f(A,B^{(j)})\) for every \(A\in\mathrm{term}(f;x)\); in other words, we can write \(Y_{j}=f(-,B^{(j)})\) as a one-argument function.
+
+**Proposition 33****.**: _Assume Notation 32 above. Let \(1\leq j\leq m\) be a positive integer and \(d_{1}\), …, \(d_{j-1}\), \(d_{j}\) be a sequence of positive integers such that \(m=d_{1}>d_{2}>\ldots>d_{j-1}>d_{j}\geq 1\). Let \(U\in\mathrm{term}(f;x)\) with \(\mathrm{dege}(U)\leq\mathrm{dege}(B^{(d_{j})})=3^{d_{j}}\), and let \(E=Y_{d_{2}}(Y_{d_{3}}(\ldots Y_{d_{j}}(U)\ldots))\), so \(f(E,B^{(m)})=Y_{d_{1}}(E)=Y_{d_{1}}(Y_{d_{2}}(\ldots Y_{d_{j}}(U)\ldots))=Y_{d_{1}}\circ Y_{d_{2}}\circ\ldots\circ Y_{d_{j}}(U)\). Let \(A\in\mathrm{term}(f;x)\) be such that \(\mathrm{dege}(A)\leq\mathrm{dege}(f(E,B^{(m)}))\). Suppose that \(e(f(C,D))=e(f(A,f(E,B^{(m)})))\). Then there exists \(V\in\mathrm{term}(f;x)\) such that \(D=Y_{d_{1}}\circ Y_{d_{2}}\circ\ldots\circ Y_{d_{j}}(V)\), \(e(U)\geq_{L}e(V)\), and \(\mathrm{dege}(V)=\mathrm{dege}(U)\)._
+
+Proof.: We will prove this by induction on \(j\). We have already proved the case \(j=1\) in Proposition 30. Now assume that the statement of this proposition holds for all \(1\leq k\leq j-1\). In what follows, we will prove this proposition for \(j\).
+
+Since \(e(f(C,D))=e(f(A,f(E,B^{(m)})))\) and \(f(E,B^{(m)})=Y_{d_{1}}\circ Y_{d_{2}}\circ\ldots\circ Y_{d_{j-1}}(f(U,B^{(d_{j})}))\), it follows from our inductive hypothesis (with \(f(U,B^{(d_{j})})\) in the role of \(U\)) that \(D=Y_{d_{1}}\circ Y_{d_{2}}\circ\ldots\circ Y_{d_{j-1}}(\tilde{V})\) for some \(\tilde{V}\in\mathrm{term}(f;x)\) such that \(e(f(U,B^{(d_{j})}))\geq_{L}e(\tilde{V})\) and \(\mathrm{dege}(\tilde{V})=\mathrm{dege}(f(U,B^{(d_{j})}))\). We claim that \(\tilde{V}=f(V,B^{(d_{j})})\) for some \(V\in\mathrm{term}(f;x)\) such that \(e(V)\leq_{L}e(U)\). Notice that we must have \(\tilde{V}=f(V,W)\) for some \(V,W\in\mathrm{term}(f;x)\), where \(\mathrm{dege}(W)=\mathrm{dege}(B^{(d_{j})})\). Notice also that \(f(U,B^{(d_{j})})=f(U,f(x^{\prime},B^{(d_{j}-1)}))\) where \(x^{\prime}:=x\), and \(\deg(e(U)^{2})=2\,\mathrm{dege}(U)\leq 2\,\mathrm{dege}(B^{(d_{j})})<2+2\,\mathrm{dege}(B^{(d_{j})})=\deg(x^{\prime 2})+2\,\deg(e(B^{(d_{j}-1)})^{3})=\deg(\mathrm{maxt}(x^{\prime}))\). Suppose \(W\neq B^{(d_{j})}\). Then by Lemma 25 we have \(e(f(U,B^{(d_{j})}))<_{L}e(f(V,W))=e(\tilde{V})\), which is a contradiction. Hence we must have \(W=B^{(d_{j})}\). Since \(e(f(U,B^{(d_{j})}))\geq_{L}e(\tilde{V})=e(f(V,B^{(d_{j})}))\), we have \(e(V)\leq_{L}e(U)\) as claimed.
+
+Now we want to show that \(\mathrm{dege}(V)=\mathrm{dege}(U)\). Suppose
+
+\[\mathrm{dege}(V)<\mathrm{dege}(U).\] (12)
+
+We will show that this leads to a contradiction. We have \(e(f(A,f(E,B^{(m)})))=e(f(A,Y_{d_{1}}\circ Y_{d_{2}}\circ\ldots\circ Y_{d_{j}}(U)))=e(A)^{2}+(((\ldots((e(U)^{2}+e(B^{(d_{j})})^{3})^{2}+e(B^{(d_{j-1})})^{3})^{2}\ldots)^{2}+e(B^{(d_{2})})^{3})^{2}+e(B^{(d_{1})})^{3})^{3}=e(C)^{2}+(((\ldots((e(V)^{2}+e(B^{(d_{j})})^{3})^{2}+e(B^{(d_{j-1})})^{3})^{2}\ldots)^{2}+e(B^{(d_{2})})^{3})^{2}+e(B^{(d_{1})})^{3})^{3}=e(f(C,Y_{d_{1}}\circ Y_{d_{2}}\circ\ldots\circ Y_{d_{j}}(V)))=e(f(C,D))\). Notice that, in the polynomial expansion of the preceeding five equalities, the terms (excluding the \(e(A)^{2}\) and \(e(C)^{2}\)) that do not contain \(e(U)^{2}\) as a factor or \(e(V)^{2}\) as a factor, \((e(B^{(d_{2})})^{3})^{2}(e(B^{(d_{1})})^{3})^{2}\) for example, are common to both sides of the third equality and can thus be subtracted off from these two sides. Subtracting off these common terms leaves
+
+\[\deg(e(U)^{2}e(B^{(d_{j})})^{3}e(B^{(d_{j-1})})^{3}\ldots e(B^{(d_{2})})^{3}e(B^{(d_{1})})^{6})\]
+
+as the degree of the left-hand side of the third equality. Since
+
+\[\deg(e(V)^{2}e(B^{(d_{j})})^{3}e(B^{(d_{j-1})})^{3}\ldots e(B^{(d_{2})})^{3}e(B^{(d_{1})})^{6})\]
+
+is less than the degree of the left-hand side of the third equality by (12), it follows that \(2\,\mathrm{dege}(C)\) must equal the degree of the left-hand side of the third equality. We see from the third equality that the coefficient of the highest degree term of the left-hand side is \(\mathrm{lead}(3\cdot 2^{j-1}e(U)^{2}e(B^{(d_{j})})^{3}e(B^{(d_{j-1})})^{3}\ldots e(B^{(d_{2})})^{3}e(B^{(d_{1})})^{6})=3\cdot 2^{j-1}\mathrm{lead}(e(U))^{2}\mathrm{lead}(e(B^{(d_{1})}))^{6}=3\cdot 2^{j-1}\mathrm{lead}(e(U))^{2}\) and that the coefficient of the highest degree term of the right-hand side is \(\mathrm{lead}(e(C)^{2})=\mathrm{lead}(e(C))^{2}\). It follows that we must have \(\mathrm{lead}(e(C))^{2}=3\cdot 2^{j-1}\mathrm{lead}(e(U))^{2}\), which implies that \(\mathrm{lead}(e(C))=\sqrt{3\cdot 2^{j-1}}\,\mathrm{lead}(e(U))\), which is not even rational. This is a contradiction, so we must have \(\mathrm{dege}(U)=\mathrm{dege}(V)\). This completes the induction. ∎
+
+**Corollary 34****.**: _Under the hypotheses of Proposition 33, if \(e(U)\) is lexicographically minimal among polynomials with degree \(\mathrm{dege}(U)\) induced by \(f\)-expressions, then \(V=U\) and \(e(A)=e(C)\)._
+
+Proof.: We must have \(e(V)=e(U)\) or \(e(V)>_{L}e(U)\). Since \(e(U)\geq_{L}e(V)\) by Proposition 33, we must have \(e(V)=e(U)\), from which the conclusion follows by Lemma 26. ∎
+
+In the proof of Proposition 30 (and analogously Proposition 33), we saw that, under the hypotheses of this proposition and given \(\mathrm{dege}(E)>\mathrm{dege}(F)\), the subexpression \(C\) is not able to “make up” for the difference between \(e(f(E,B^{(m)}))\) and \(e(f(F,B^{(m)}))\), and hence \(e(f(C,f(F,B^{(m)})))\neq e(f(A,f(E,B^{(m)})))\). We will analyze and make use of this phenomenon extensively in what follows, where we consider the developments of \(f\)-expressions more complicated than \(A\) of Proposition 22.
+
+Consider \(A,B_{0},E_{1}\in\mathrm{term}(f;x)\) where \(B_{0}\), \(E_{1}\) are subexpressions of \(A\), \(\mathrm{dege}(A)=2^{p}3^{q}\), \(\mathrm{dege}(B_{0})=2^{m}3^{n}\), and \(\mathrm{dege}(E_{1})=2^{i}3^{j}\). For the remainder of this paper, assume the following three
+
+_Assumption 35__._: \(B_{0}\) contains all the cores of \(A\) and is \(e\)-isolated with respect to \(A\).
+
+_Assumption 36__._: \(A^{[m+n+1]}\) is either \(f(E_{1},B_{0})\) or \(f(B_{0},E_{1})\). In other words, we can write \(A=f(\ldots f(E_{1},B_{0})\ldots)\) or \(A=f(\ldots f(B_{0},E_{1})\ldots)\) respectively.
+
+_Assumption 37__._: For every occurrence \(x^{\prime}\) of \(x\) in \(E_{1}\) and every occurrence \(x^{\prime\prime}\) of \(x\) in the ellipses of the expressions for \(A\) shown in Assumption 36 we have \(\deg(\mathrm{maxt}(x^{\prime}))>\deg(\mathrm{maxt}(x^{\prime\prime}))\).
+
+_Remark 38__._: Note that Assumption 37 is a more general version of the corresponding hypothesis in Proposition 22.
+
+In the restricted version of Friedman’s problem we will study below, we will show that whether or not \(A^{[m+n+1]}\) is \(e\)-isolated with respect to \(A\) is related to the solution sets of certain exponential Diophantine equations.
+
+In either of the cases \(A=f(\ldots f(E_{1},B_{0})\ldots)\) or \(A=f(\ldots f(B_{0},E_{1})\ldots)\) we have
+
+\[\mathrm{dgap}(E_{1},B_{0})=2^{m+\pi_{1}}3^{n+\pi_{2}}-2^{i+\pi_{2}}3^{j+\pi_{1}}\] (13)
+
+where \(\{\pi_{1},\pi_{2}\}=\{0,1\}\). Note that \(\pi_{1}=1\) corresponds to the case \(A=f(\ldots f(B_{0},E_{1})\ldots)\) and \(\pi_{2}=1\) corresponds to the case \(A=f(\ldots f(E_{1},B_{0})\ldots)\). We could attempt to prove, as in Proposition 22, that \(f(E_{1},B_{0})\) or \(f(B_{0},E_{1})\) is \(e\)-isolated with respect to \(A\), but this assertion, if true, may be very difficult to prove. Below, we will simply explore how this problem can be analyzed through the study of certain Diophantine equations. Here are some lemmas that will aid us in this effort.
+
+**Lemma 39****.**: _Let \(A=f(\ldots f(E_{1},B)\ldots)\) and \(A^{\prime}=f(\ldots f(E_{2},B)\ldots)\) be \(f\)-expressions such that \(\mathrm{dege}(A)=2^{p}3^{q}=\mathrm{dege}(A^{\prime})\), \(B\) contains all the cores of \(A\), and \(B\) contains at least one core of \(A^{\prime}\). Suppose that for every occurrence \(x_{1}\) of \(x\) in \(E_{1}\) and for every occurrence \(x_{2}\) of \(x\) in the ellipses of \(A\) we have \(\deg(\mathrm{maxt}(x_{1}))>\deg(\mathrm{maxt}(x_{2}))\). Suppose that \(e(E_{1})<_{L}e(E_{2})\). Then \(e(A)<_{L}e(A^{\prime})\)._
+
+Proof.: We can write \(e(E_{1})^{2}=p_{1}(x)+q(x)\) and \(e(E_{2})^{2}=p_{2}(x)+q(x)\), where \(p_{1}(x)\), \(p_{2}(x)\), \(q(x)\) are polynomials and \(\deg(p_{1}(x))<\deg(p_{2}(x))\). Examining \((e(E_{1})^{2}+e(B)^{3})^{\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}}\) from \(e(A)\) and \((e(E_{2})^{2}+e(B)^{3})^{\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}}\) from \(e(A^{\prime})\), we see that \((q(x)+e(B)^{3})^{\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}}\) is common to both \(e(A)\) and \(e(A^{\prime})\). We have \(\deg(e(A)-(q(x)+e(B)^{3})^{\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}})=\deg(p_{1}(x))+(\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}-1)(3\,\mathrm{dege}(B))<\deg(p_{2}(x))+(\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}-1)(3\,\mathrm{dege}(B))\leq\deg(e(A^{\prime})-(q(x)+e(B)^{3})^{\frac{2^{p}3^{q}}{3\,\mathrm{dege}(B)}})\), from which the conclusion follows. ∎
+
+Analogously we have the following
+
+**Lemma 40****.**: _Let \(A=f(\ldots f(B,E_{1})\ldots)\) and \(A^{\prime}=f(\ldots f(B,E_{2})\ldots)\) be \(f\)-expressions such that \(\mathrm{dege}(A)=2^{p}3^{q}=\mathrm{dege}(A^{\prime})\), \(B\) contains all the cores of \(A\), and \(B\) contains at least one core of \(A^{\prime}\). Suppose that for every occurrence \(x_{1}\) of \(x\) in \(E_{1}\) and for every occurrence \(x_{2}\) of \(x\) in the ellipses of \(A\) we have \(\deg(\mathrm{maxt}(x_{1}))>\deg(\mathrm{maxt}(x_{2}))\). Suppose that \(e(E_{1})<_{L}e(E_{2})\). Then \(e(A)<_{L}e(A^{\prime})\)._
+
+Suppose \(\bar{A}\in\mathrm{term}(f;x)\) such that \(e(\bar{A})=e(A)\). Then we must have \(\mathrm{dege}(\bar{A})=\mathrm{dege}(A)=2^{p}3^{q}\) and \(\bar{A}^{[m+n]}=B_{0}\). Since \(B_{0}\) contains all the cores of \(A\), \(B_{0}\) must contain all the cores of \(\bar{A}\); otherwise we would have \(\mathrm{lead}(e(\bar{A}))>\mathrm{lead}(e(A))\), which is a contradiction. \(\bar{A}^{[m+n+1]}\) must be either \(f(E_{2},B_{0})\) or \(f(B_{0},E_{2})\) for some \(E_{2}\in\mathrm{term}(f;x)\), so we have either \(\bar{A}=f(\ldots f(E_{2},B_{0})\ldots)\) or \(\bar{A}=f(\ldots f(B_{0},E_{2})\ldots)\). We will say that \(A\) **and \(\bar{A}\) have the same orientation at the \((m+n+1)\)st stage** if \(A=f(\ldots f(E_{1},B_{0})\ldots)\) and \(\bar{A}=f(\ldots f(E_{2},B_{0})\ldots)\), or if \(A=f(\ldots f(B_{0},E_{1})\ldots)\) and \(\bar{A}=f(\ldots f(B_{0},E_{2})\ldots)\). In each of the cases \(A=f(\ldots f(E_{1},B_{0})\ldots)\) or \(A=f(\ldots f(B_{0},E_{1})\ldots)\), our ultimate goal (which, by the way, will not be achieved in this paper) is to show that \(A\) and \(\bar{A}\) have the same orientation at the \((m+n+1)\)st stage and furthermore that \(\mathrm{dgap}(E_{2},B_{0})=\mathrm{dgap}(E_{1},B_{0})\). In other words, in both cases we want to show that \(\mathrm{dege}(E_{2})=\mathrm{dege}(E_{1})\), from which it would follow by Lemma 39 and Lemma 40 that \(E_{2}=E_{1}\) whenever \(e(E_{1})\) is lexicographically minimal for its degree. If \(\mathrm{dgap}(E_{2},B_{0})<\mathrm{dgap}(E_{1},B_{0})\), then \(\deg(\mathrm{maxt}(E_{2}))=\mathrm{dege}(A)-\mathrm{dgap}(E_{2},B_{0})>\mathrm{dege}(A)-\mathrm{dgap}(E_{1},B_{0})=\deg(\mathrm{maxt}(E_{1}))\), so under Assumption 37 we have \(e(\bar{A})>_{L}e(A)\), which is a contradiction. Therefore, we must have
+
+\[\mathrm{dgap}(E_{2},B_{0})\geq\mathrm{dgap}(E_{1},B_{0}).\] (14)
+
+Now, for the remainder of this paper, we make the following
+
+_Assumption 41__._: Either \(A\) and \(\bar{A}\) have the opposite orientation at the \((m+n+1)\)st stage or \(\mathrm{dgap}(E_{2},B_{0})>\mathrm{dgap}(E_{1},B_{0})\).
+
+Notice that Assumption 41 is the negation of the statement “\(A\) and \(\bar{A}\) have the same orientation at the \((m+n+1)\)st stage and \(\mathrm{dgap}(E_{2},B_{0})=\mathrm{dgap}(E_{1},B_{0})\)” which we hope to eventually prove, so we will try to derive contradictions under Assumption 41. Also notice that Assumption 41 implies that \(A^{[m+n+1]}\) is _not_\(e\)-isolated with respect to \(A\). We will see in the following discussion that certain exponential Diophantine equations must hold, and we will study these exponential Diophantine equations to see how contradictions might be derived.
+
+There exists \(C\in\mathrm{term}(f;x)\) and positive integer \(l\geq m+n+1\) such that either \(\bar{A}^{[l]}=f(C,\bar{A}^{[l-1]})\) or \(\bar{A}^{[l]}=f(\bar{A}^{[l-1]},C)\), and such that \(\deg(e(\bar{A})-e(B_{0})^{\frac{2^{p}3^{q}}{\mathrm{dege}(B_{0})}})=\deg(\mathrm{maxt}(C))\). Since \(\deg(e(A)-e(B_{0})^{\frac{2^{p}3^{q}}{\mathrm{dege}(B_{0})}})=\deg(\mathrm{maxt}(E_{1}))\), we must also have \(\deg(e(\bar{A})-e(B_{0})^{\frac{2^{p}3^{q}}{\mathrm{dege}(B_{0})}})=\deg(\mathrm{maxt}(E_{1}))\), so \(\deg(\mathrm{maxt}(C))=\deg(\mathrm{maxt}(E_{1}))\). It follows that \(\mathrm{dgap}(C,\bar{A}^{[l-1]})=\mathrm{dgap}(E_{1},B_{0})\). Let us now introduce subscripts that will show more about the relation between \(C\) and \(B_{0}\). Thus, there will exist (possibly more than one choice of) \(k_{1},k_{2}\in\mathbb{N}\cup\{0\}\) and \(C_{k_{1},k_{2}}\in\mathrm{term}(f;x)\) (above called \(C\)), such that \(\bar{A}^{[m+n+k_{1}+k_{2}]}\) is either \(f(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})\) or \(f(\bar{A}^{[m+n+k_{1}+k_{2}-1]},C_{k_{1},k_{2}})\), \(k_{1}\geq 1\) or \(k_{2}\geq 1\),
+
+\[\mathrm{dege}(\bar{A}^{[m+n+k_{1}+k_{2}]})=2^{m+k_{1}}3^{n+k_{2}},\] (15)
+
+and
+
+\[\mathrm{dgap}(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})=\mathrm{dgap}(E_{1},B_{0});\] (16)
+
+note that, by our definition, \(k_{1}\), \(k_{2}\) denote the number of times \(e(B_{0})\) gets squared and the number of times \(e(B_{0})\) gets cubed, respectively, when we arrive at \(\bar{A}^{[m+n+k_{1}+k_{2}]}\). We will call such a \(C_{k_{1},k_{2}}\) a **supplementing subexpression for \(E_{1}\)**; this is a generalization of the “supplementing” \(C\) we mentioned in the paragraph following the proof of Corollary 34. Notice that the case \(k_{1}=\pi_{2}\), \(k_{2}=\pi_{1}\) corresponds to the case where \(A\), \(\bar{A}\) have the opposite orientation at the \((m+n+1)\)st stage and where \(\mathrm{dgap}(E_{2},B_{0})=\mathrm{dgap}(E_{1},B_{0})\); in this case we have \(C_{k_{1},k_{2}}=C_{\pi_{2},\pi_{1}}=E_{2}\), i.e. \(E_{2}\) is one such supplementing subexpression for \(E_{1}\). The case \(k_{1}=\pi_{1}\), \(k_{2}=\pi_{2}\) corresponds to the case where \(A\) and \(\bar{A}\) have the same orientation at the \((m+n+1)\)st stage and \(\mathrm{dgap}(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})=\mathrm{dgap}(E_{2},B_{0})\). Since \(\mathrm{dgap}(E_{2},B_{0})>\mathrm{dgap}(E_{1},B_{0})\) by Assumption 41, we have \(\mathrm{dgap}(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})>\mathrm{dgap}(E_{1},B_{0})\), which is a contradiction. Therefore, we can exclude the case \(k_{1}=\pi_{1}\), \(k_{2}=\pi_{2}\) (which we will call the **excluded case**) in our analysis below, because this case cannot happen under Assumption 41.
+
+Notice that either
+
+\[2^{m+k_{1}}3^{n+k_{2}}-\mathrm{dgap}(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})=3\mathrm{dege}(C_{k_{1},k_{2}})\]
+
+or
+
+\[2^{m+k_{1}}3^{n+k_{2}}-\mathrm{dgap}(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})=2\mathrm{dege}(C_{k_{1},k_{2}}),\]
+
+and both \(3\mathrm{dege}(C_{k_{1},k_{2}})\), \(2\mathrm{dege}(C_{k_{1},k_{2}})\) must be the product of a power of 2 and a power of 3. Since \(2^{m+k_{1}}3^{n+k_{2}}-\mathrm{dgap}(C_{k_{1},k_{2}},\bar{A}^{[m+n+k_{1}+k_{2}-1]})=2^{m+k_{1}}3^{n+k_{2}}-\mathrm{dgap}(E_{1},B_{0})=2^{m+k_{1}}3^{n+k_{2}}-(2^{m+\pi_{1}}3^{n+\pi_{2}}-2^{i+\pi_{2}}3^{j+\pi_{1}})\), we must have
+
+\[2^{m+k_{1}}3^{n+k_{2}}-(2^{m+\pi_{1}}3^{n+\pi_{2}}-2^{i+\pi_{2}}3^{j+\pi_{1}})=2^{l_{1}}3^{l_{2}},\] (17)
+
+where \(l_{1},l_{2}\in\mathbb{N}\cup\{0\}\). Solving (17) will make it easier for us to determine whether or not the supplementing subexpression \(C_{k_{1},k_{2}}\) exists and more generally whether or not Assumption 41 can be true. Notice that Equation (17) is essentially the equation
+
+\[2^{a}3^{b}+2^{c}3^{d}=2^{e}3^{f}+2^{g}3^{h}.\] (18)
+
+We will study (18) in [1], where some partial results are proven.
+
+## References
+
+* [1] Roger Tian, _On the Diophantine Equation \(2^{a}3^{b}+2^{c}3^{d}=2^{e}3^{f}+2^{g}3^{h}\)_. Preprint, 6 pp., 2009.
+

arxiv_ground_truth/0910.1576.md

@@ -0,0 +1,146 @@
+# On the Diophantine Equation \(2^{a}3^{b}+2^{c}3^{d}=2^{e}3^{f}+2^{g}3^{h}\)
+
+Roger Tian
+
+(July 5, 2024)
+
+###### Abstract
+
+This paper is a continuation of [1], in which I studied Harvey Friedman’s problem of whether the function \(f(x,y)=x^{2}+y^{3}\) satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the exponential Diophantine equation \(2^{a}3^{b}+2^{c}3^{d}=2^{e}3^{f}+2^{g}3^{h}\) into subcases that are easier to analyze. Then we will solve an equation obtained by imposing a restriction on one of these subcases, after which we will solve a generalization of this equation.
+
+**Acknowledgements**
+
+I would like to thank my thesis advisor, George Bergman, for giving advice on how to better organize this paper and make it more readable, and for pointing out areas of my paper that needed clarification.
+
+We will follow the convention that \(0\notin\mathbb{N}\).
+
+A resticted version of Friedman’s problem (mentioned in the Abstract) I studied in my paper [1] is related to the solution set of the equation
+
+\[2^{a}3^{b}+2^{c}3^{d}=2^{e}3^{f}+2^{g}3^{h}.\] (1)
+
+The results we will get on this equation, which are still very partial, will not be applied in this paper to Friedman’s problem.
+
+Suppose \(a_{0}\), \(b_{0}\), \(c_{0}\), \(d_{0}\), \(e_{0}\), \(f_{0}\), \(g_{0}\), \(h_{0}\) are nonnegative integers such that
+
+\[2^{a_{0}}3^{b_{0}}+2^{c_{0}}3^{d_{0}}=2^{e_{0}}3^{f_{0}}+2^{g_{0}}3^{h_{0}}.\] (2)
+
+Without loss of generality, we may assume that \(\min\{a_{0},c_{0},e_{0},g_{0}\}=0=\min\{b_{0},d_{0},f_{0},h_{0}\}\), or equivalently \(0\in\{a_{0},c_{0},e_{0},g_{0}\}\) and \(0\in\{b_{0},d_{0},f_{0},h_{0}\}\); we can always divide (2) by \(2^{\min\{a_{0},c_{0},e_{0},g_{0}\}}3^{\min\{b_{0},d_{0},f_{0},h_{0}\}}\). Suppose that there is exactly one zero in \(\{a_{0},c_{0},e_{0},g_{0}\}\). Then (2) reduces to an equation in which exactly three of its four terms contain a factor of 2, so one side of this resulting equation is divisible by 2 while the other side is not, which is a contradiction. Thus, there must be at least two zeros in \(\{a_{0},c_{0},e_{0},g_{0}\}\) and, by the same reasoning, with the factor 2 replaced by the factor 3, there must be at least two zeros in \(\{b_{0},d_{0},f_{0},h_{0}\}\). Then, depending on which terms of (2) the zeros occur in, we can reduce Equation (1) to 36 cases. However, merging the cases that are identical up to permutations of the summands, we get the following seven equations:
+
+\[1+1=2^{e}3^{f}+2^{g}3^{h}\] (3)
+
+\[1+3^{d}=2^{e}+2^{g}3^{h}\] (4)
+
+\[3^{b}+3^{d}=2^{e}+2^{g}\] (5)
+
+\[1+2^{c}=3^{f}+2^{g}3^{h}\] (6)
+
+\[1+2^{c}3^{d}=1+2^{g}3^{h}\] (7)
+
+\[3^{b}+2^{c}=1+2^{g}3^{h}\] (8)
+
+\[3^{b}+2^{c}=3^{f}+2^{g}\] (9)
+
+Note, for instance, that in the case \(a_{0}=b_{0}=c_{0}=f_{0}=0\), Equation (4) must have at least one solution. The solution to (3) is \((e,f,g,h)=(0,0,0,0)\). The solutions to (7) are \((c,d,g,h)=(s,t,s,t)\) for all nonnegative integers \(s\) and \(t\). We now solve (8) subject to the restiction \(b=h\), i.e. the equation \(2^{c}-1=3^{b}(2^{g}-1)\). We first prove a few lemmas.
+
+**Lemma 1****.**: _Let \(p,m,n\in\mathbb{N}\cup\{0\}\) where \(p>1\) and \(m>0\). If \(p^{m}-1\mid p^{n}-1\), then \(m\mid n\)._
+
+Proof.: We have \(n=qm+r\) where \(q,r\in\mathbb{N}\cup\{0\}\) and \(0\leq r<m\). We will prove this lemma by induction on \(q\). For \(q=0\), we have \(n<m\), so \(p^{n}-1<p^{m}-1\), from which it follows that \(p^{m}-1\mid p^{n}-1\)\(\Longrightarrow\)\(p^{n}-1=0\)\(\Longrightarrow\)\(n=0\)\(\Longrightarrow\)\(m\mid n\). Suppose the lemma is true for some \(q\), we will prove it for \(q+1\). Suppose \(p^{m}-1\mid p^{n}-1=p^{(q+1)m+r}-1\). Then \(p^{m}-1\) divides \(p^{(q+1)m+r}-1-(p^{m}-1)=p^{(q+1)m+r}-p^{m}=p^{m}(p^{qm+r}-1)\). Since \(p^{m}-1\) and \(p^{m}\) are relatively prime, we have \(p^{m}-1\mid p^{qm+r}-1\). It follows from our inductive hypothesis that \(m\mid qm+r\), so \(r=0\) and \(n=(q+1)m\). This completes the induction. ∎
+
+_Notation 2__._: Let \(n,m,k\in\mathbb{N}.\) By \(n^{k}\parallel m\) we will always mean that \(n^{k}\mid m\) and \(n^{k+1}\nmid m\).
+
+**Lemma 3****.**: _If \(k,n\in\mathbb{N}\) where \(k\) is odd, then \(2^{n+2}\parallel 3^{2^{n}k}-1\)._
+
+Proof.: We will first prove this claim for \(n=1\). We have \(3^{2k}-1=9^{k}-1=(8+1)^{k}-1=-1+\sum_{j=0}^{k}{{k\choose j}8^{j}}=\sum_{j=1}^{k}{{k\choose j}8^{j}}=8\sum_{j=1}^{k}{{k\choose j}8^{j-1}}\), and we see that \(\sum_{j=1}^{k}{{k\choose j}8^{j-1}}\) is odd because the \(j=1\) term of this sum is odd and all other terms of this sum are even. Now suppose the claim is true for some \(n\geq 1\). Then there exists an \(l\in\mathbb{N}\) such that \(l\) is odd and \(3^{2^{n+1}k}-1=(3^{2^{n}k})^{2}-1=(3^{2^{n}k}-1+1)^{2}-1=(2^{n+2}l+1)^{2}-1=2^{2(n+2)}l^{2}+2(2^{n+2}l)+1-1=2^{2(n+2)}l^{2}+2(2^{n+2}l)=2(2^{n+2}l)(2^{n+1}l+1)\), and we see that \(2^{n+1}l+1\) is odd and \(2^{n+3}\parallel 3^{2^{n+1}k}-1\). This completes the induction. ∎
+
+**Lemma 4****.**: _If \(m\) is odd, then \(2^{2}\parallel 3^{m}+1\)._
+
+Proof.: Notice that for \(m=1\) we have \(3^{m}+1=3+1=4\). Now suppose that the claim is true for some \(m\geq 1\), where \(m\) is odd. We have \(3^{m+2}+1=9\cdot 3^{m}+1=9(3^{m}+1-1)+1=9(4k-1)+1=36k-9+1=36k-8=4(9k-2)\) where \(k\) is odd, and we see that \(9k-2\) is also odd. This completes the induction. ∎
+
+**Lemma 5****.**: _If \(m_{1},l\in\mathbb{N}\) and \(m_{2}\in\mathbb{N}\cup\{0\}\) where \(2,3\nmid l\), then \(3^{m_{2}+1}\parallel 2^{2^{m_{1}}3^{m_{2}}l}-1\)._
+
+Proof.: We will first prove this claim for \(m_{2}=0\). Notice that \(2^{2^{m_{1}}l}-1=4^{2^{m_{1}-1}l}-1=(3+1)^{2^{m_{1}-1}l}-1=\sum_{i=1}^{2^{m_{1}-1}l}{{2^{m_{1}-1}l\choose i}3^{i}}+1-1=\sum_{i=1}^{2^{m_{1}-1}l}{{2^{m_{1}-1}l\choose i}3^{i}}=\sum_{i=2}^{2^{m_{1}-1}l}{{2^{m_{1}-1}l\choose i}3^{i}}+2^{m_{1}-1}l\cdot 3=3(\frac{1}{3}\sum_{i=2}^{2^{m_{1}-1}l}{{2^{m_{1}-1}l\choose i}3^{i}}+2^{m_{1}-1}l)\), and we see that \(3\nmid\frac{1}{3}\sum_{i=2}^{2^{m_{1}-1}l}{{2^{m_{1}-1}l\choose i}3^{i}}+2^{m_{1}-1}l\). Now suppose for some \(m_{2}\geq 0\) we have \(2^{2^{m_{1}}3^{m_{2}}l}-1=3^{m_{2}+1}l_{1}\) where \(3\nmid l_{1}\). Then \(2^{2^{m_{1}}3^{m_{2}}l}=3^{m_{2}+1}l_{1}+1\Longrightarrow 2^{2^{m_{1}}3^{m_{2}+1}l}=(3^{m_{2}+1}l_{1}+1)^{3}=3^{3(m_{2}+1)}l_{1}^{3}+3\cdot 3^{2(m_{2}+1)}l_{1}^{2}+3\cdot 3^{m_{2}+1}l_{1}+1\), so \(2^{2^{m_{1}}3^{m_{2}+1}l}-1=3\cdot 3^{m_{2}+1}l_{1}(3^{2(m_{2}+1)-1}l_{1}^{2}+3^{m_{2}+1}l_{1}+1)\). Since \(3\nmid l_{1}(3^{2(m_{2}+1)-1}l_{1}^{2}+3^{m_{2}+1}l_{1}+1)\), we have \(3^{m_{2}+2}\parallel 2^{2^{m_{1}}3^{m_{2}+1}l}-1\). This completes the induction. ∎
+
+**Lemma 6****.**: _If \(m_{1},l\in\mathbb{N}\cup\{0\}\) where \(2,3\nmid l\), then \(3^{m_{1}+1}\parallel 2^{3^{m_{1}}l}+1\)._
+
+Proof.: We will first prove this claim for \(m_{1}=0\). Notice that \(2^{l}+1=(3-1)^{l}+1=\sum_{k=1}^{l}{{l\choose k}3^{k}(-1)^{l-k}}+(-1)^{l}+1=\sum_{k=1}^{l}{{l\choose k}3^{k}(-1)^{l-k}}-1+1=\sum_{k=1}^{l}{{l\choose k}3^{k}(-1)^{l-k}}=3\sum_{k=1}^{l}{{l\choose k}3^{k-1}(-1)^{l-k}}=3(\sum_{k=2}^{l}{{l\choose k}3^{k-1}(-1)^{l-k}}+l)\), and we see that \(3\nmid\sum_{k=2}^{l}{{l\choose k}3^{k-1}(-1)^{l-k}}+l\). Suppose for some \(m_{1}\geq 0\) we have \(2^{3^{m_{1}}l}+1=3^{m_{1}+1}l_{1}\) where \(3\nmid l_{1}\). Then we have \(2^{3^{m_{1}}l}=3^{m_{1}+1}l_{1}-1\Longrightarrow 2^{3^{m_{1}+1}l}=(3^{m_{1}+1}l_{1}-1)^{3}=3^{3(m_{1}+1)}l_{1}^{3}-3\cdot 3^{2(m_{1}+1)}l_{1}^{2}+3\cdot 3^{m_{1}+1}l_{1}-1\), so \(2^{3^{m_{1}+1}l}+1=3\cdot 3^{m_{1}+1}l_{1}(3^{2(m_{1}+1)-1}l_{1}^{2}-3^{m_{1}+1}l_{1}+1)\). Since \(3\nmid l_{1}(3^{2(m_{1}+1)-1}l_{1}^{2}-3^{m_{1}+1}l_{1}+1)\), we have \(3^{m_{1}+2}\parallel 2^{3^{m_{1}+1}l}+1\). This completes the induction. ∎
+
+**Proposition 7****.**: _The only solutions \((k,m,n)\) in the positive integers of the exponential Diophantine equation \(3^{k}(2^{m}-1)=2^{n}-1\) are \((1,1,2)\) and \((2,3,6)\)._
+
+Proof.: We know from Lemma 1 that \(n=lm\) for some \(l\in\mathbb{N}\). Since \(k>0\), we must have \(l\geq 2\). Notice that \(2^{n}-1=2^{lm}-1=(2^{m}-1)(2^{(l-1)m}+2^{(l-2)m}+\ldots+2^{m}+1)=3^{k}(2^{m}-1)\), so
+
+\[3^{k}=2^{(l-1)m}+2^{(l-2)m}+\ldots+2^{m}+1>2^{m}-1.\] (10)
+
+It follows that \(3^{2k}>3^{k}(2^{m}-1)\), so
+
+\[3^{2k}>2^{n}-1.\] (11)
+
+Now, by Lemma 5 we have \(3^{k}\mid 2^{n}-1\)\(\Longrightarrow\)\(n=2^{m_{1}}3^{m_{2}}l_{1}\) where \(m_{1},l_{1}\in\mathbb{N}\) and \(m_{2}\in\mathbb{N}\cup\{0\}\) such that \(2,3\nmid l_{1}\) and \(m_{2}\geq k-1\). Note that \(m_{2}=k-1\) if \(3^{k}\parallel 2^{n}-1\).
+
+We shall now prove by induction that \(3^{2k}<2^{2^{m_{1}}3^{k-1}l_{1}}-1\) for every \(k\geq 3\) and all choices of \(m_{1}\) and \(l_{1}\). It is easy to check that, for all choices of \(m_{1}\) and \(l_{1}\), we have \(3^{2k}<2^{2^{m_{1}}3^{k-1}l_{1}}-1\) for \(k=3\). Suppose we know, for some value of \(k\geq 3\), that \(3^{2k}<2^{2^{m_{1}}3^{k-1}l_{1}}-1\), or equivalently \(3^{2k}+1<2^{2^{m_{1}}3^{k-1}l_{1}}\), for all choices of \(m_{1}\) and \(l_{1}\). Then we have \(2^{2^{m_{1}}3^{k}l_{1}}=(2^{2^{m_{1}}3^{k-1}l_{1}})^{3}>(3^{2k}+1)^{3}>3^{2(k+1)}+1\) for all choices of \(m_{1}\) and \(l_{1}\). This completes the induction. If \(k\geq 3\) and \(3^{k}\mid 2^{n}-1\), then \(2^{n}-1=2^{2^{m_{1}}3^{m_{2}}l_{1}}-1\geq 2^{2^{m_{1}}3^{k-1}l_{1}}-1>3^{2k}\), contradicting (11). Thus, we see that there are no solutions for \(k\geq 3\).
+
+It remains to determine the possible solutions when \(k=1,2\). Suppose \(k=1\). Then \(3^{2k}=3^{2}>2^{n}-1\) implies that \(n=1\), 2, or 3, so we must have \(n=2\) because \(n\) is even, hence \(m=1\). Suppose \(k=2\). Then by (10) we have \(3^{2}=9>2^{m}-1\), so \(m=1\), 2, or 3. It is easy to check that the cases \(m=1\) and \(m=2\) do not yield solutions. For \(m=3\), we have \(n=6\). ∎
+
+_Remark 8__._: Central to our proof is the fact that, for \(k\) sufficiently large, \(2^{n}-1=3^{k}\cdot q\) implies \(q\) must be much larger than \(3^{k}\).
+
+We will now solve a more general exponential Diophantine equation using the same ideas in the proof of the preceding proposition. First, we generalize Lemma 5.
+
+**Lemma 9****.**: _Let \(m\geq 3\) be an odd positive integer. Then the following statements hold._
+
+1. 1._If_ \(n\in\mathbb{N}\) _is odd, then_ \(m\nmid(m-1)^{n}-1\)_._
+2. 2._If_ \(m_{1},l\in\mathbb{N}\)_,_ \(m_{2}\in\mathbb{N}\cup\{0\}\)_, and_ \(2,m\nmid l\)_, then_ \(m^{m_{2}+1}\parallel(m-1)^{2^{m_{1}}m^{m_{2}}l}-1\)_._
+
+Proof.: We consider the two cases separately.
+
+1. 1.We have \((m-1)^{n}-1\equiv(-1)^{n}-1\equiv-1-1\equiv-2\pmod{m}\), from which the conclusion follows.
+2. 2.Our proof is by induction on \(m_{2}\).
+
+For \(m_{2}=0\), we have \((m-1)^{2^{m_{1}}l}-1=\sum_{i=0}^{2^{m_{1}}l}{{2^{m_{1}}l\choose i}m^{i}(-1)^{2^{m_{1}}l-i}}-1=-1+1+\sum_{i=1}^{2^{m_{1}}l}{{2^{m_{1}}l\choose i}m^{i}(-1)^{2^{m_{1}}l-i}}=\sum_{i=1}^{2^{m_{1}}l}{{2^{m_{1}}l\choose i}m^{i}(-1)^{2^{m_{1}}l-i}}=\)
+
+\[m\left(\left(\sum_{i=2}^{2^{m_{1}}l}{{2^{m_{1}}l\choose i}m^{i-1}(-1)^{2^{m_{1}}l-i}}\right)-2^{m_{1}}l\right).\] (12)
+
+Notice that \(m\mid\sum_{i=2}^{2^{m_{1}}l}{{2^{m_{1}}l\choose i}m^{i-1}(-1)^{2^{m_{1}}l-i}}\), but \(m\nmid 2^{m_{1}}l\) because \(m\nmid l\) and \(\gcd(m,2^{m_{1}})=1\). Thus, \(m\nmid\sum_{i=2}^{2^{m_{1}}l}{{2^{m_{1}}l\choose i}m^{i-1}(-1)^{2^{m_{1}}l-i}}-2^{m_{1}}l\) and so \(m\parallel(m-1)^{2^{m_{1}}l}-1\) as desired.
+
+Suppose for some \(m_{2}\geq 0\) we have \((m-1)^{2^{m_{1}}m^{m_{2}}l}-1=m^{m_{2}+1}l_{1}\) where \(m\nmid l_{1}\). Then \((m-1)^{2^{m_{1}}m^{m_{2}}l}=m^{m_{2}+1}l_{1}+1\)\(\Longrightarrow\)\((m-1)^{2^{m_{1}}m^{m_{2}+1}l}=(m^{m_{2}+1}l_{1}+1)^{m}=\sum_{i=0}^{m}{{m\choose i}(m^{m_{2}+1}l_{1})^{i}}=1+\sum_{i=1}^{m}{{m\choose i}(m^{m_{2}+1}l_{1})^{i}}\), so \((m-1)^{2^{m_{1}}m^{m_{2}+1}l}-1=\sum_{i=1}^{m}{{m\choose i}(m^{m_{2}+1}l_{1})^{i}}=m\cdot m^{m_{2}+1}l_{1}+\sum_{i=2}^{m}{{m\choose i}(m^{m_{2}+1}l_{1})^{i}}=m^{m_{2}+2}l_{1}+\sum_{i=2}^{m}{{m\choose i}m^{(m_{2}+1)i}l_{1}^{i}}=m^{m_{2}+2}(l_{1}+\sum_{i=2}^{m}{{m\choose i}m^{(m_{2}+1)(i-1)-1}l_{1}^{i}})\). By assumption \(m\nmid l_{1}\). However, \(m\mid\sum_{i=2}^{m}{{m\choose i}m^{(m_{2}+1)(i-1)-1}l_{1}^{i}}\) because \(m\) clearly divides \({m\choose i}m^{(m_{2}+1)(i-1)-1}l_{1}^{i}\) for all \(i\geq 3\) and \({m\choose 2}m^{m_{2}}l_{1}^{2}=\frac{m(m-1)}{2}m^{m_{2}}l_{1}^{2}=\frac{m-1}{2}m^{m_{2}+1}l_{1}^{2}\) is divisible by \(m\) since \(m-1\) is even. It follows that \(m\nmid l_{1}+\sum_{i=2}^{m}{{m\choose i}m^{(m_{2}+1)(i-1)-1}l_{1}^{i}}\). This completes the induction.
+
+∎
+
+_Remark 10__._: Part 1 of the lemma holds for all integers \(m\geq 3\). If \(m\) is even, then we can write \(m=2^{k}p\) where \(k,p\in\mathbb{N}\) and \(2\nmid p\), and we can observe (taking \(m_{1}=k\) and \(l=p\)) from (12) that \(m^{2}\mid(m-1)^{2^{k}p}-1\). Thus, Part 2 of the lemma is false for even \(m\).
+
+**Proposition 11****.**: _The only solutions \((k,p,q,n)\) in the positive integers of the exponential Diophantine equation \((2n+1)^{k}((2n)^{p}-1)=(2n)^{q}-1\) are \((2,3,6,1)\) and \((1,1,2,n)\) for all positive \(n\)._
+
+Proof.: We know from Lemma 1 that \(q=lp\) for some \(l\in\mathbb{N}\). Since \(k>0\), we must have \(l\geq 2\). Notice that \((2n)^{q}-1=(2n)^{lp}-1=((2n)^{p}-1)((2n)^{(l-1)p}+(2n)^{(l-2)p}+\ldots+(2n)^{p}+1)=(2n+1)^{k}((2n)^{p}-1)\), so we have \((2n+1)^{k}=(2n)^{(l-1)p}+(2n)^{(l-2)p}+\ldots+(2n)^{p}+1>(2n)^{p}-1\). It follows from
+
+\[(2n+1)^{k}>(2n)^{p}-1\] (13)
+
+that \((2n+1)^{2k}>(2n+1)^{k}((2n)^{p}-1)\), so
+
+\[(2n+1)^{2k}>(2n)^{q}-1.\] (14)
+
+Since \((2n+1)^{k}\mid(2n)^{q}-1\), by Lemma 9 we have \(q=2^{m_{1}}(2n+1)^{m_{2}}l_{1}\) where \(m_{1},l_{1}\in\mathbb{N}\) such that \(2,2n+1\nmid l_{1}\) and \(m_{2}\in\mathbb{N}\cup\{0\}\) such that \(m_{2}\geq k-1\).
+
+We will prove by induction that for every \(k\geq 3\) we have \((2n+1)^{2k}<(2n)^{2^{m_{1}}(2n+1)^{k-1}l_{1}}-1\) for all choices of \(n\), \(m_{1}\), and \(l_{1}\). For \(k=3\), observe that \((2n+1)^{2k}<(2n)^{2^{m_{1}}(2n+1)^{k-1}l_{1}}-1\) for all choices of \(n\), \(m_{1}\), and \(l_{1}\); take \(x:=2n+1\), \(m_{1}=1=l_{1}\) and note that \(x^{6}<x^{x^{2}}-1<((x-1)^{2})^{x^{2}}-1=(x-1)^{2x^{2}}-1\) for all \(x\geq 3\). Suppose we know, for some value of \(k\geq 3\), that \((2n+1)^{2k}<(2n)^{2^{m_{1}}(2n+1)^{k-1}l_{1}}-1\)\(\Longleftrightarrow\)\((2n+1)^{2k}+1<(2n)^{2^{m_{1}}(2n+1)^{k-1}l_{1}}\) for all choices of \(n\), \(m_{1}\), and \(l_{1}\). Then \((2n)^{2^{m_{1}}(2n+1)^{k}l_{1}}>((2n+1)^{2k}+1)^{2n+1}\geq(2n+1)^{2k(2n+1)}+1\geq(2n+1)^{2(k+1)}+1\) for all choices of \(n\), \(m_{1}\), and \(l_{1}\). This completes the induction. If \(k\geq 3\) and \((2n+1)^{k}\mid(2n)^{q}-1\), then we have \((2n)^{q}-1=(2n)^{2^{m_{1}}(2n+1)^{m_{2}}l_{1}}-1\geq(2n)^{2^{m_{1}}(2n+1)^{k-1}l_{1}}-1>(2n+1)^{2k}\), contradicting (14). Thus, there are no solutions for \(k\geq 3\).
+
+It remains to determine the possible solutions when \(k=1,2\). Consider \(k=1\). Then by (13) we have
+
+\[2n+1>(2n)^{p}-1,\] (15)
+
+which for \(n=1\) becomes \(4>2^{p}\), which is false for every \(p\geq 2\). Notice that as \(n\) increases, \((2n)^{p}-1\) where \(p\geq 2\) increases faster than \(2n+1\). It follows that (15) is false for every \(p\geq 2\), so we must have \(p=1\). For \(p=1\), we have \((2n+1)^{k}((2n)^{p}-1)=(2n+1)(2n-1)=(2n)^{2}-1=(2n)^{q}-1\)\(\Longrightarrow\)\(q=2\), and \(n\) can be any positive integer. Consider \(k=2\). Then by (13) we have
+
+\[(2n+1)^{2}>(2n)^{p}-1,\] (16)
+
+which for \(n=1\) becomes \(9>2^{p}-1\), which is false for every \(p\geq 4\). Notice that as \(n\) increases, \((2n)^{p}-1\) where \(p\geq 4\) increases faster than \((2n+1)^{2}\). It follows that (16) is false for every \(p\geq 4\), so we must have \(p=1\), \(2\), or \(3\). If \(p=3\), then by (16) we must have \(n=1\), so \(3^{2}(2^{3}-1)=2^{q}-1\)\(\Longrightarrow\)\(q=6\). Take \(x:=2n\) for simplicity of notation as we check the remaining two cases. Suppose \(p=2\). Then we have \((x+1)^{2}(x^{2}-1)=x^{q}-1\)\(\Longrightarrow\)\(x^{4}+2x^{3}-2x-1=x^{q}-1\)\(\Longrightarrow\)\(x^{4}+2x^{3}-2x=x^{q}\), so \(q\geq 5\). However, it is easy to see that \(x^{4}+2x^{3}-2x<x^{q}\) for \(q\geq 5\), so there are no solutions for \(p=2\). Suppose \(p=1\). Then \((x+1)^{2}(x-1)=x^{q}-1\)\(\Longrightarrow\)\(x^{3}+x^{2}-x-1=x^{q}-1\)\(\Longrightarrow\)\(x^{3}+x^{2}-x=x^{q}\), so \(q\geq 4\). However, it is easy to see that \(x^{3}+x^{2}-x<x^{q}\) for \(q\geq 4\), so there are no solutions for \(p=1\). ∎
+
+_Notation 12__._: For all \(n,m\in\mathbb{N}\) where \(n\geq m\geq 2\) there exists a unique \(k\in\mathbb{N}\cup\{0\}\) such that \(m^{k}\parallel n\), and we will denote this \(k\) by \(v_{m}(n)\).
+
+Since Lemma 9 allowed us to solve the Diophantine equation of Proposition 11, a natural question is whether this lemma can be extended to _all_ positive integers \(m\geq 3\). If this lemma can be extended thus, then we may be able to solve the more general Diophantine equation
+
+\[(m+1)^{k}(m^{p}-1)=m^{q}-1\] (17)
+
+using arguments similar to those in the proof of Proposition 11. There may be many different such extensions, some of them stronger than others. For the time being, we state one such possible (rather weak) extension for even integers \(m>3\) as the following
+
+**Conjecture 13****.**: _There exists a positive integer \(N\) such that for all integers \(n\geq N\) and all even integers \(m>3\) we have \((m^{v_{m}((m-1)^{n}-1)})^{2}\leq(m-1)^{n}-1\)._
+
+## References
+
+* [1] Roger Tian, _Identities of the Function \(f(x,y)=x^{2}+y^{3}\)_. Preprint, 18 pp., 2009.
+

arxiv_ground_truth/0910.2938.md

@@ -0,0 +1,366 @@
+**How to understand the cell by breaking it: network analysis of gene perturbation screens**
+
+Florian Markowetz∗
+
+Cancer Research UK Cambridge Research Institute, Cambridge CB2 0RE, UK
+
+\(\ast\) florian.markowetz@cancer.org.uk
+
+## Abstract
+
+Modern high-throughput gene perturbation screens are key technologies at the forefront of genetic research. Combined with rich phenotypic descriptors they enable researchers to observe detailed cellular reactions to experimental perturbations on a genome-wide scale. This review surveys the current state-of-the-art in analyzing single gene perturbation screens from a network point of view. We describe approaches to make the step from the parts list to the wiring diagram by using phenotypes for network inference and integrating them with complementary data sources. The first part of the review describes methods to analyze one- or low-dimensional phenotypes like viability or reporter activity; the second part concentrates on high-dimensional phenotypes showing global changes in cell morphology, transcriptome or proteome.
+
+## Introduction
+
+Functional genomics has demonstrated considerable success in inferring the inner working of a cell through analysis of its response to various perturbations. In recent years several technological advances have pushed gene perturbation screens to the forefront of functional genomics. Most importantly, modern technologies make it possible to probe gene function on a genome-wide scale in many model organisms and human. For example, large collections of knock-out mutants play a prominent role in the study of _S. cerevisiae_[1] and RNA interference (RNAi) has become a widely used high-throughput method to knock-down target genes in a wide range of organisms, including _Drosophila melanogaster_, _C. elegans_, and human [2, 3, 4].
+
+Another major advance is the development of rich phenotypic descriptions by imaging or measuring molecular features globally. Observed phenotypes can reveal which genes are essential for an organism, or work in a particular pathway, or have a specific cellular function. Combining high-throughput screening techniques with rich phenotypes enables researchers to observe detailed reactions to experimental perturbations on a genome-wide scale. This makes gene perturbation screens one of the most promising tools in functional genomics.
+
+Advances in the design and analysis of gene perturbation screens may have an immediate impact on many areas of biological and medical research. New screening and phenotyping techniques often directly translate into new insights in gene and protein functions. Results of perturbation screens can also reveal unexploited areas of potential therapeutic intervention. For example, a recent RNAi screen showed that some of the most critical protein kinases for the proliferation and survival of cancer cell lines are also the least studied [5].
+
+A goal becoming more and more prominent in both experimental as well as computational research is to leverage gene perturbation screens to the identification of molecular interactions, cellular pathways and regulatory mechanisms. Research focus is shifting from understanding the phenotypes of single proteins to understanding _how_ proteins fulfill their function, _what_ other proteins they interact with and _where_ they act in a pathway. Novel ideas on how to use perturbation screens to uncover cellular wiring diagrams can lead to a better understanding of how cellular networks are de-regulated in diseases like cancer. This knowledge is indispensable for finding new drug targets to attack the drivers of a disease and not only the symptoms.
+
+PhenotypesA phenotype can be any observable characteristic of an organism. Analysis strategies strongly depend on how rich and informative phenotype descriptors are. We will call phenotypes resulting from a single reporter (or a small number of reporters) _low-dimensional_ phenotypes and the genes showing significant results _hits_[6, 7]. Examples of such low-dimensional phenotypes are cell viability versus cell death [1], growth rates [8] or the activity of reporter constructs, e.g. a luciferase, downstream of a pathway of interest [9]. Low-dimensional phenotyping screens can identify candidate genes on a genome-wide scale and are often used as a first step for follow-up analysis. We will discuss methods to functionally interpret hits from low-dimensional phenotyping screens and to place them in the context of cellular networks in the first part of this review.
+
+The second part will be devoted to _high-dimensional_ phenotyping screens, which evaluate a large number of cellular features at the same time. Observing system-wide changes promises key insights into cellular mechanisms and pathways that can not be supplied by low-dimensional screens. For example, high-dimensional phenotypes can include changes in cell morphology [10, 11, 12, 13], or growth rates under a wide range of conditions [14], or transcriptional changes measured on microarrays [15, 16, 17, 18], or changes in the metabolome and proteome [19] measured by mass spectrometry [20] or flow cytometry [21, 22]. Morphological and growth phenotypes can be obtained on a genome-wide scale [13, 14], while transcriptional and proteomic phenotypes are often restricted to individual pathways or processes [16, 21, 17].
+
+The distinction between low- and high-dimensional phenotypes may sound technical, but it is crucial for choosing potential analysis methods. The central difference is that high-dimensional phenotypes allow to compute correlations and other similarity measures, which are not applicable for low-dimensional phenotypes. Another important distinction is between _static_ phenotypes, providing a ‘snapshot’ of a cell’s reaction to a gene perturbation, and _dynamic_ phenotypes showing a cell’s reaction over time. We expect more and more studies in the future to produce dynamic output and in the following note explicitly which methods can be applied to dynamic phenotypes. For the biological interpretation of screening results it is very important to keep in mind which level of ‘cellular granularity’ a phenotype describes: growth rates or cell morphologies are much more ‘high-level’ features of the cell than gene or protein expressions. As soon as more studies produce dynamic phenotypes on many different cellular levels, integrative analysis of inter-connected phenotypes [23] will become more important. In the following, however, we concentrate on the current state-of-the art, which almost always uses a single type of readout in a perturbation screen.
+
+Pre-processing pipelineIn this review we focus on single gene perturbations by knockouts [1] or RNA interference [4] that allow targeting individual genes or combinations of genes. Before network analysis, the raw data needs to pass an initial analysis and quality control pipeline specific to the perturbation and phenotyping technologies used. Low-dimensional screens are mostly performed in multiple-well-plates and a typical analysis pipeline [4] includes data pre-processing, removal of spatial biases per plates, normalization between plates, and finally detection of significant hits [6, 7, 24]. In vertebrates, genes need to be targeted with multiple siRNAs to ensure effective down-regulation [4] and the multiple phenotypes per gene can afterwards be integrated into a statistical score [25]. High-dimensional morphological screens depend on computational analysis like image segmentation [26, 27] and phenotype discovery [28, 29, 30] for rapid and consistent phenotyping. Molecular high-dimensional phenotypes need pre-processing depending on their platform and different approaches exist e.g. for flow-cytometry data [31] or microarrays [32].
+
+From phenotypes to cellular networksThe phenotypes we have discussed above allow only an indirect view on how different genes in the same process interact to achieve a particular phenotype. Cell morphology or sensitivity to stresses, for example, are global features of the cell and hard to relate directly to how individual genes contribute to them (see Fig. 1a). Gene expression phenotypes show transcriptional changes in the genes downstream of a perturbed pathway but offer only an indirect view of pathway structure due to the high number of non-transcriptional regulatory events like protein modifications [33]. For example, different protein activation states by phosphorylation may not be visible by changes in mRNA concentrations (see Fig. 1b).
+
+This gap between observed phenotypes and underlying cellular networks is the main problem in the analysis of perturbation screens and applies to both low- and high-dimensional screens. The goal of computational analysis is to bridge this gap by inferring gene function and recovering pathways and mechanism from observed phenotypes. The following methods address the challenge in different ways, mostly by integrating the perturbation effects and phenotypes with additional sources of information like collections of functionally related gene sets or protein-interaction networks.
+
+## Network analysis of low-dimensional phenotypes
+
+Global overview by enrichment analysisA simple way to link phenotypes to gene function is to test whether pathways or functional groups of genes (e.g. defined by Gene Ontology terms [34] or MSigDB [35]) are enriched in the list of hits. Most methods use a hypergeometric test statistic (see Fig. 2a) and many can be used online [36, 37, 38] or as Bioconductor packages [39]. An alternative global functional annotation method tests whether functional groups show a trend towards especially strong or weak phenotypes without using a cutoff to define hits [35] (see Fig. 2b). Enrichment analysis can also be very useful to analyze high-dimensional phenotypes, for example when functionally annotating the results of a clustering method.
+
+Enrichment analysis results in a list of \(p\)-values describing how significantly each gene set was represented in the hits. Enrichment analysis reduces complexity and improves interpretability of results by moving from single genes to functionally related gene sets. This type of analysis is often called ’un-biased’ and ’hypothesis-free’ and is ideal for a comprehensive first overview. However, enrichment analysis loses its value for complexity reduction if the number of gene sets becomes too big. Also, overlap and dependencies between gene lists that could potentially bias the results have so far only been addressed for the GO graph [39, 38] but not for more general collections of gene lists like MSigDB [35].
+
+Good data analysis asks specific questions. A hypothesis-free method can only be the very first starting point for a deeper exploration of the data. For example, all enrichment methods rely on known gene sets and cannot uncover new pathways or components. Enrichment methods treat pathways as bags of unconnected genes without considering connections within and between pathways. Thus, enrichment methods can only deliver a very crude picture of the cell. In the following we will discuss approaches to overcome some of the limitations of enrichment analysis by integrating the observed phenotypes with complementary sources of information.
+
+Mapping phenotypes to networksAnother valuable source of information to interpret RNAi hits are gene and protein networks obtained either experimentally [40, 41] or computationally by literature mining [42] or integrating heterogeneous genomic data [43, 44, 45]. All computational networks are available online on supplementary webpages and the experimental networks can be obtained from databases like STRING [46] or BioGRID [47].
+
+Using these complementary data sources can improve hit identification [48, 49, 50] and even provide a more refined view of the pathways the hits contribute to. One strategy is to search for sub-networks containing a surprisingly large number of hits (see Fig. 3a). While this strategy is already useful when evaluating interesting sub-networks by eye [51, 52] its true power comes from the availability of efficient search algorithms to find sub-networks enriched for RNAi hits and assess their significance [53, 54, 55, 56, 57]. An additional application of mapping hits to a network is that known phenotypes can be used to predict phenotypes of genes not included in the screen, e.g. by assuming that a gene connected to many hits should also show a strong phenotype [51]. The success of all network-mapping strategies strongly depends on the quality and coverage of both the screen and the linkage in the network.
+
+Gene prioritizationOther approaches complement genomic data with biological prior knowledge showing how ‘interesting’ hits look like. Gene prioritization [49, 58] ranks genes according to how promising they would be for follow-up studies. Because it uses prior knowledge to fine-tune the algorithm, gene prioritization can be more focussed than a global un-informed search for enriched subnetworks.
+
+## Network analysis of high-dimensional phenotypes
+
+Global overview by clustering and ranking.Most state-of-the-art analysis techniques rely on a guilt-by-association paradigm: genes with similar phenotypes will most probably have a similar biological function. This explains the prevalence of clustering techniques in analyzing high-dimensional phenotyping screens [10, 14, 17, 13]. Clustering is a convenient first analysis and visualization step that can can highlight strong trends and patterns in the data and can thus yield a global first impression of functional units. Another analysis strategy relying on guilt-by-association is to rank genes by their phenotypic similarity compared to a gene of interest [11]. Clustering and ranking can be combined with enrichment analysis (as discussed above) for functional interpretation.
+
+Graph methods linking causes to effectsAnother useful data visualization especially for transcriptional phenotypes is to build a directed (not necessarily acyclic) graph by drawing an arrow between two genes if perturbing one results in a significant expression change at the other [59]. This graph representation can be then used as a starting point for further analysis, for example by using graph-theoretic methods of transitive reduction [60] to distinguish between direct and indirect effects of a perturbation [61, 62].
+
+Probabilistic graphical models.Most approaches to infer pathway structure from experimental data rely on probabilistic graphical models. For low-dimensional phenotypes they often suffer from non-uniqueness and un-identifiability issues [63], but can be applied very successfully in high-dimensional settings. A prominent approach are (static or dynamic) Bayesian networks, which describe probabilistically how a gene is controlled by its regulators [64, 65]. To model experimental perturbations most approaches rely on the concept of ‘ideal interventions’ [66] which deterministically fix a target gene to a particular state (e.g ‘0’ for a gene knockout). Ideal interventions were applied in Bayesian networks [67, 21, 68], factor graphs [69] and dependency networks [70]. In simulations [71, 72] and on real data [21, 71] it was found that interventions are critical for effective inference.
+
+The model of ideal interventions contains a number of idealizations (hence the name), most importantly that manipulations only affect single genes and that perturbation strength can be controlled deterministically. The first assumption may not be true if there are off-target or compensatory effects involving other genes. The second assumption may also not hold true in realistic biological scenarios; in particular for RNAi screens where experimentalists often lack knowledge about the exact knock-down efficiency. Probabilistic generalizations of ideal interventions can be used to cope with this uncertainty [73].
+
+Probabilistic data integrationHigh-dimensional phenotypic profiles can be mapped to given graphs and networks by finding subgraphs that are connected in the background network and at the same time show high similarity of phenotypic profiles. These approaches already exist for mapping gene expression data onto protein interaction networks [74] and the same algorithms could easily be applied to any other kind of high-dimensional phenotypic profiles (see Fig. 3b). Other approaches use data integration to construct potential pathways from protein interactions and transcription factor binding data to relate perturbed genes to the observed downstream effects [75, 76, 77].
+
+Multiple Input - Multiple Output (MIMO) modelsMany of the approaches discusses so far–like clustering or graphical models–can be applied to both static ‘snapshots’ as well as dynamic time-course measurements. Another approach to model specifically the dynamics of networks comes from a branch of control theory called ‘systems identification’ [78] and uses so called Multiple Input - Multiple Output (MIMO) models. MIMO models represent the evolution of a perturbed cell over time by linear differential equations [79, 80, 81, 82, 83] and can represent non-linear effects by transfer functions [84]. The models can be inferred by regression techniques in the linear case [80] or Monte Carlo stochastic search in the non-linear case [84]. The framework is very flexible and can incorporated single as well as combinatorial perturbations.
+
+Nested Effects Models (NEMs)One of the key problems in analyzing perturbation screens is that the observed phenotypes are downstream of the perturbed pathway and may not show the direct influence of one pathway component on another. A class of models explicitly addressing this problem are Nested Effects Models [33, 85]. They reconstruct pathway structure from _subset relations_ based on the following rationale: Perturbing some genes may have an influence on a global process, while perturbing others affects sub-processes of it. Imagine, for example, a signaling pathway activating several transcription factors. Blocking the entire pathway will most probably affect all targets of all transcription factors, while perturbing a single transcription factor will only affect its direct targets, which are a subset of the phenotype obtained by blocking the complete pathway. Given high-dimensional phenotypes showing a subset structure, NEMs find the most likely pathway topology explaining the data. They differ from other statistical approaches like Bayesian networks by encoding subset relations instead of correlations or other similarity measures. The theory of NEMs has been applied and extended in several studies [86, 87, 88, 89]. An implementation is available as an R/Bioconductor package [90]. Other extensions to the NEM framework distinguish between activating and inhibiting regulation [91] or include dynamic information from time-series measurements [92].
+
+## Discussion and Outlook
+
+In this review we have discussed two main approaches to describe the reaction of a cell to an experimental gene perturbation: low-dimensional phenotypes measure individual reporters for cell viability or pathway activation, while high-dimensional phenotypes show global effects on cell morphology, transcriptome or proteome. Table 1 lists examples of freely available software implementing some of these approaches. All of them can be directly applied to gene perturbation screens, even though some of them have been introduced in different contexts. While this review has focused on single gene knock-outs and knock-downs, similar approaches can be applied to gene over-expression screens [93, 94, 22, 83], drug treatment [84], environmental stresses changing many genes [95, 96] or even natural genetic variation [97].
+
+Predicting phenotypes from metabolic networksThe focus of this review is on functionally annotating hits in a network context and reconstructing networks from high-dimensional phenotypes. In a complementary direction of research, genome-wide reconstructions of metabolic networks [98, 99] are used to predict effects of gene perturbations. Instead of predicting networks from phenotypes, these approaches predict phenotypes from networks. For example, in _S. cerevisiae_ and _E. coli_ computational models very accurately predict fitness effects of gene knock-outs [100, 101] as well as compensatory rescue effects [102]. However, recent developments in metabolic network modeling have led to linear programming algorithms to extract relevant context-specific sub-networks of activity from a genome-wide network [103, 104]. In the same way as the probabilistic data integration methods discussed above, e.g. [74], these algorithms could be used in the future to find metabolic sub-networks active under certain gene perturbations.
+
+From single to combinatorial perturbationsWhile single gene perturbation screens have been immensely successful in extending our knowledge of pathway components and interactions, an important limitation can be caused by compensatory effects, genetic buffering and redundancy of cellular mechanisms and pathways [105, 106]. This can only be overcome by perturbing several genes at the same time. The number of possible combinations grows rapidly and thus current approaches are mainly limited to perturbing pairs of genes and observing low-dimensional phenotypes like fitness estimates [107]. The analysis of combinatorial perturbations is the topic of another review [108].
+
+The end of the screen is the beginning of the experimentGlobal phenotyping and pathway screening can be combined in the same study. For example, a first genome-wide screen identifies key genes representative for pathways and cellular mechanisms involved in the phenotype. In a second step the hits of the first screen could be assayed for high-dimensional molecular phenotypes to infer a pathway diagram using Nested Effects Models or other statistical approaches.
+
+In a further step this preliminary pathway models could be used to plan an additional round of experimentation. Different modeling frameworks propose future experiments to most effectively refine a pathway hypothesis, e.g. Bayesian networks [109, 110], physical network models [76], logical models [111], Boolean networks [112], and dynamical modeling [79].
+
+Iteratively integrating experimentation and computation may lead to a virtuous circle and is one of the most promising approaches to refine our understanding of the inner working of the cell.
+
+## Acknowledgments
+
+I thank the organizers of the ISMB 2009 tutorial sessions for the opportunity to present this review. Yinyin Yuan, Roland Schwarz, Gregoire Pau provided helpful comments on drafts of the manuscript. My research is funded by Cancer Research UK.
+
+## References
+
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+
+## Figures and legends
+
+Figure 1: **Cellular networks underlying observable phenotypes.****(a)** Phenotypes are the response of the cell to external signals mediated by cellular networks and pathways. The goal of computation is to reconstruct these networks from the observed phenotypes. **(b)** Global molecular phenotypes like gene expression allow a view inside the cell but also have limitations. This is exemplified here in a cartoon pathway adapted from [61] showing a cascade of five genes/proteins (A-E). Proteins A-C form a kinase cascade, D is a transcription factor acting on E. Up-regulation of A starts information flow in the cascade and results in E being turned on. In gene expression data this is visible as a correlation between A and E (represented as an undirected edge in the model). Experimentally perturbing a genes, say C, removes the corresponding protein from the cascade, breaks the information flow and results in an expression change at E (represented as an arrow in the model). However, the different phosphorylation and activation states of proteins B-D will most probably not be visible as changes in gene expression. Thus, due to the pathway mostly acting on the protein level most parts of the cascade (dashed arrows in the model) can not be inferred from gene expression data directly.
+
+Figure 2: **Functional annotation of hits by enrichment analysis.****(a)** In the first approach [38] a cutoff is applied to select the hits with strongest phenotypes. A hyper-geometric test then evaluates if the overlap between the hits and a given gene set is surprisingly large (or small) compared to the overlap with a random set. **(b)** A second approach [35] does not need a cutoff. It maps the gene set (black bars) onto the observed phenotypes and quantifies if there is a significant trend or if the genes are spread out uniformly over the whole range.
+
+Figure 3: **Extracting rich sub-networks.** Different patterns in the graph point to a common cellular mechanism causing a phenotype: **(a)** hits in a low-dimensional screen (red nodes) clustering in highly connected sub-networks, and **(b)** high correlation between high-dimensional phenotypes of target genes connected in the background network. The black graph represents any type of background network.
+
+## Tables
+
+\begin{table}
+\begin{tabular}{|c|c|l|}
+\hline
+\multicolumn{3}{c}{**General data analysis and network visualization**} \\
+\hline
+Bioconductor & Software environment for the analysis of genomic data featuring hundreds of contributed packages [113] & www.bioconductor.org \\
+Cytoscape & Software platform for visualizing molecular interaction networks and integrating them with other data types [114] & www.cytoscape.org \\
+\hline \hline
+\multicolumn{3}{c}{**Setting up data for network analysis**} \\
+\hline
+cellHTS2 & End-to-end analysis of cell-based screens: from raw intensity readings to the annotated hit list [6] & www.bioconductor.org \\
+RNAither & Analysis of cell-based RNAi screens, includes quality assessment and customizable normalization [7] & www.bioconductor.org \\
+EBImage & Cell image analysis and feature extraction [27] & www.bioconductor.org \\
+CellProfiler & Cell image analysis and feature extraction [26] & www.cellprofiler.org \\
+\hline \hline
+\multicolumn{3}{c}{**Enrichment analysis**} \\
+\hline
+DAVID & Tools for data annotation, visualization and integration [36] & david.abcc.ncifcrf.gov \\
+GOLEM & Enrichment analysis and visualization of GO graph (Fig 2a) [37] & function.princeton.edu/GOLEM \\
+Ontologizer & Enrichment analysis with dependencies between GO nodes (Fig 2a) [38] & compbio.charite.de/ontologizer \\
+GSEA & Gene set enrichment analysis (Fig 2b) [35] & www.broadinstitute.org/gsea/ \\
+\hline \hline
+\multicolumn{3}{c}{**Clustering and ranking**} \\
+\hline
+Cell Profiler Analyst & Interactive exploration and analysis of multidimensional data from image-based experiments [28] & www.cellprofiler.org \\
+PhenoBlast & Ranking of phenotype profiles according to similarity with given profile [11] & www.rnai.org \\
+Endeavour & Prioritizes hits for further analysis [58] & www.esat.kuleuven.be/endeavour/ \\
+\hline \hline
+\multicolumn{3}{c}{**Finding rich sub-networks**} \\
+\hline
+heinz & Finds optimal subnetworks rich in hits (Fig 3a)[55] & www.planet-lisa.net \\
+jActiveModules & Finds heuristic subnetworks rich in hits (Fig 3a) [53] & www.cytoscape.org \\
+Matisse & Finds subnetworks with high phenotypic similarity (Fig 3b) [74] & acgt.cs.tau.ac.il/matisse/ \\
+\hline \hline
+\multicolumn{3}{c}{**Network reconstruction**} \\
+\hline
+nem & Nested Effects Models reconstruct pathway features from subset relations in high-dim phenotypes [90] & www.bioconductor.org \\
+copia & Copia uses MIMO models to reconstruct networks from perturbations [84] & cbio.mskcc.org/copia/ \\ \hline
+\end{tabular}
+The table contains the name of the method, a short description with reference, and a webpage where it can be obtained. This list is far from comprehensive, but hopefully provides a starting point even for non-coding experimentalists.
+
+\end{table}
+Table 1: **Examples of software for network analysis of gene perturbation screens.**
+

arxiv_ground_truth/0911.4272.md

@@ -0,0 +1,62 @@
+Angle measures of some cones associated with finite reflection groups
+
+P. V. Bibikov¹, V. S. Zhgoon²
+
+Footnote 1: The first author was partially supported by the Moebius Contest Foundation for Young Scientists
+
+Footnote 2: The second author was partially supported by the following grants RFBR 09-01-00287, 09-01-12169.
+
+Let \(V\) be a Euclidean vector space of dimension \(n\) with the inner product \((\cdot,\cdot)\). For a convex polyhedral cone \(C\) we denote by \(C^{\circ}\) its open kernel, by \(\langle C\rangle\) its linear span, and by \(C^{*}\) its _dual cone_, i.e. \(C^{*}=\{v\in V:\,(v,u)\geqslant 0\;\forall\,u\in C\}\). Denote by \(\sigma(C)\)_the relative angle measure of \(C\)_, i.e. \(\sigma(C)=\mathop{\mathrm{vol}}(C\cap B)/\mathop{\mathrm{vol}}(B)\), where \(B\subset\langle C\rangle\) is the unit ball centered at the origin. Let \(F\) be a \(k\)-dimensional face of some solid cone \(C\). By \(C/F\) we denote the orthogonal projection of \(C\) to the subspace \(\langle F\rangle^{\bot}\). We note that \((C/F)^{*}\) is an \((n-k)\)-dimensional face of \(C^{*}\) that is orthogonal to \(F\).
+
+Let now \(W\subset O(V)\) be a finite reflection group. Set \(W^{k}=\{w\in W:\dim\ker(1-w)=k\}\) and denote \(W^{\mathrm{reg}}:=W^{0}\). For a subspace \(U\subset V\) we denote by \(W_{U}\) the subgroup of \(W\) that fixes \(U\) pointwise.
+
+The aim of this paper is to prove the following theorem conjectured by the first author in [1].
+
+**Theorem 1****.**: _For a fundamental cone \(C\) of a finite reflection group \(W\) and for each \(k=0,\ldots,n\) we have_
+
+\[\sum\limits_{{F\subset C,\ \dim F=k}}\sigma(F)\cdot\sigma((C/F)^{*})=|W^{k}|/|W|,\]
+
+_where \(F\) runs over the \(k\)-dimensional faces of \(C\)._
+
+The following two results are crucial for the proof of Theorem 1. The first one is the fundamental result of Waldspurger [7], for the simplest proof of which we refer the reader to [2, 3]. The second one is the so-called “Curious Identity” of De Concini and Procesi [5] (see also [1, 6]).
+
+**Theorem 2******(Walsdpurger)**.**: \(C^{*}=\bigsqcup\limits_{w\in W}(1-w)C^{\circ}\)_._
+
+**Theorem 3******(Curious Identity)**.**: \(\sigma(C^{*})=|W^{\mathrm{reg}}|/|W|.\)__
+
+Proof of Theorem 3.: Theorem 2 implies that \(wC^{*}=\bigcup_{w^{\prime}\in W^{\mathrm{reg}}}(1-w^{\prime})wC.\) Thus we obtain
+
+\[|W|\sigma(C^{*})=\sum_{w\in W}\sigma(wC^{*})=\sum_{w^{\prime}\in W^{\mathrm{reg}}}\sum_{w\in W}\sigma((1-w^{\prime})wC)=\sum_{w^{\prime}\in W^{\mathrm{reg}}}\sigma((1-w^{\prime})V)=|W^{\mathrm{reg}}|.\]
+
+∎
+
+**Remark****.**: It is easy to see from the previous proof that the number of cones \(wC^{*}\) covering a generic point of \(V\) is equal to \(|W^{\mathrm{reg}}|\).
+
+Proof of Theorem 1.: Consider the sum \(S:=\sum_{(C,F)}\sigma(F)\cdot\sigma((C/F)^{*})\) over all pairs \((C,F)\), where \(C\) is a fundamental cone of \(W\) and \(F\) is a \(k\)-dimensional face of \(C\). This sum is equal to the left hand side of the required formula multiplied by \(|W|\). We shall calculate \(S\) in a different way.
+
+Let us recall that any two fundamental cones of \(W\) with a common face \(F\) are conjugate by a unique element of the reflection subgroup \(W_{F}\) that fixes \(F\) pointwise. We also note that \(C/F\) is a fundamental cone for the action of \(W_{F}\) on \(\langle F\rangle^{\bot}\). By Theorem 3 we have \(\sigma((C/F)^{*})=|W^{\mathrm{reg}}_{F}|/|W_{F}|\). If we take the sum of \(\sigma((C/F)^{*})\) over all \(C\) that contain a fixed face \(F\), we get \(|W^{\mathrm{reg}}_{F}|\). The group \(W_{F}\) and the measure \(\sigma((C/F)^{*})\) depend only on the subspace \(U:=\langle F\rangle\) that is an intersection of reflection hyperplanes. Let us take the sum of \(\sigma(F)\cdot\sigma((C/F)^{*})\) over all pairs \((C,F)\) such that \(F\subset U\) for a fixed \(k\)-dimensional space \(U\). We get \(|W^{\mathrm{reg}}_{U}|\) multiplied by the total measure of faces \(F\subset U\), which is equal to one, since the faces \(F\) decompose \(U\). Taking the sum over all subspaces \(U\) we obtain \(S=|W^{k}|\). ∎
+
+**Remarks****.**: 1. Given a cone \(C\) and its face \(F\), we define the cone \(F\oplus(C/F)^{*}\). It follows from the previous proof that for any \(k\)-dimensional subspace \(U\), which is an intersection of reflection hyperplanes, a generic point of \(V\) is covered by \(|W^{\mathrm{reg}}_{U}|\) cones \(F\oplus(C/F)^{*}\) with \(F\subset U\).
+
+2. The sums from Theorem 1 can be expressed in terms of the exponents \(m_{1},\ldots,m_{n}\) of \(W\) with a help of the Solomon formula [4]: \(\sum\limits_{k=0}^{n}|W^{n-k}|t^{k}=\prod\limits_{k=0}^{n}(1+m_{k}t)\).
+
+**Definition****.**: Let \(C\) be a fundamental cone of \(W\). We say that two \(k\)-dimensional faces \(F\) and \(F^{\prime}\) of \(C\) are equivalent if there exists an element \(w\in W\) such that \(w\langle F\rangle=\langle F^{\prime}\rangle\).
+
+Denote by \(N_{F}\) the subgroup of \(W\) normalizing \(\langle F\rangle\). Consider the sum \((|W|/|W_{F}|)\cdot\sum_{F^{\prime}\sim F}\sigma(F^{\prime})\) of relative angle measures for the \(W\)-translates of the \(k\)-dimensional faces \(F^{\prime}\subset C\) that are equivalent to \(F\). It is the same as the total measure of the \(W\)-translates of \(F^{\prime}\subset\langle F\rangle\). The latter is equal to \(|W|/|N_{F}|\) multiplied by the total measure of the faces \(F^{\prime}\subset\langle F\rangle\) which is equal to \(1\). Thus we get \(\sum_{F^{\prime}\sim F}\sigma(F^{\prime})=|W_{F}|/|N_{F}|\). Using Theorems 1, 3 and applying the previous equality we get:
+
+\[\sum_{F}|W^{\mathrm{reg}}_{F}|/|N_{F}|=\sum_{F}\Big{(}\sum\limits_{F^{\prime}\sim F}\sigma(F^{\prime})\Big{)}\cdot\sigma((C/F)^{*})=|W^{k}|/|W|,\]
+
+where the first sum is taken over representatives of all equivalence classes of \(k\)-dimensional faces \(F\subset C\).
+
+The authors are grateful to E. B. Vinberg for valuable discussions.
+
+## References
+
+* [1]Bibikov P. V. _Relative Angle Measure of the Dual Cone of the Fundamental Cone of a Finite Reflection Group_, Uspehi Mat. Nauk, (2009).
+* [2]Bibikov P. V., Zhgoon V. S. _On the Waldspurger theorem_, Uspehi Mat. Nauk, 64:5 (389) (2009), 177–178.
+* [3]Bibikov P. V., Zhgoon V. S. _On Tilings Defined by Discrete Reflection groups_, Izvestiya RAN. Ser. Mat., (2009).
+* [4]Solomon L. _Invariants of finite reflection groups_, Nagoya Math. J. 22 (1963) 57–64.
+* [5]De Concini C., Procesi C. _A curious identity and the volume of the root spherical simplex_, with appendix by J.Stembridge, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), no. 2, 155–165.
+* [6]Denham G. _A note of De Concini and Procesi’s curious identity_, Mat. Appl. v.19 (2008), n.1, 55-63.
+* [7]Waldspurger J.-L. _Une remarque sur les systèmes de racines_, Journal of Lie Theory, Volume 17 (2007), Number 3. P. 597–603.
+

arxiv_ground_truth/0911.5485.md

@@ -0,0 +1,209 @@
+# Steady-state molecular dynamics simulation of vapour to liquid nucleation with McDonald’s dæmon
+
+Martin Horsch
+
+Svetlana Miroshnichenko
+
+Jadran Vrabec¹
+
+Footnote 1: Author to whom correspondence should be addressed: Prof. Dr.-Ing. habil. J. Vrabec. E-mail: jadran.vrabec@upb.de. 
+
+###### Abstract
+
+The most interesting step of condensation is the cluster formation up to the critical size. In a closed system, this is an instationary process, as the vapour is depleted by the emerging liquid phase. This imposes a limitation on direct molecular dynamics (MD) simulation of nucleation by affecting the properties of the vapour to a significant extent so that the nucleation rate varies over simulation time. Grand canonical MD with McDonald’s dæmon is discussed in the present contribution and applied for sampling both nucleation kinetics and steady-state properties of a supersaturated vapour.
+
+The idea behind that approach is to simulate the production of clusters up to a given size for a specified supersaturation. In that way, nucleation is studied by a steady-state simulation. A series of simulations is conducted for the truncated and shifted Lennard-Jones fluid which accurately describes the fluid phase coexistence of noble gases and methane. The classical nucleation theory is found to overestimate the free energy of cluster formation and to deviate by two orders of magnitude from the nucleation rate below the triple point at high supersaturations.
+
+Keywords: Non-equilibrium statistical mechanics, nucleation, molecular dynamics : 05.70.Ln, 64.70.F-, 36.40.Sx
+
+## 1 Introduction
+
+The key properties of nucleation processes in a supersaturated vapours are the height \({\Delta\mathnormal{\Omega}}^{\star}\) of the free energy barrier that must be overcome to form stable clusters and the nucleation rate \(\mathcal{J}\) that indicates how many macroscopic droplets emerge in a given volume per time. The most widespread approach for calculating these quantities is the classical nucleation theory (CNT) [1], which has significant shortcomings, e.g., it overestimates the free energy of cluster formation [2; 3]. An important problem of CNT in case of vapour to liquid nucleation is that the underlying basic assumptions for the liquid do not apply to nanoscopic clusters [4; 5; 6].
+
+Molecular simulation permits the investigation of nanoscopic surface effects and the stability of supersaturated states from first principles, using effective pair potentials. For instance, the spinodal line can be detected with Monte Carlo (MC) [7] simulation methods; in experiments, it can only be approximated as it is impossible to discriminate an unstable state from a metastable state where \({\Delta\mathnormal{\Omega}}^{\star}\) is low. Equilibria [8] and vapourization processes [9; 10] of single clusters can also be simulated to obtain the surface tension as well as heat and mass transfer properties of strongly curved interfaces. Moreover, molecular dynamics (MD) [11; 12; 13] and MC [14] simulation of supersaturated systems with a large number of particles are useful for the study of very fast nucleation processes, whereas lower nucleation rates can be calculated by transition path sampling based methods [15; 16].
+
+Equilibrium simulations fail to reproduce kinetic properties of nucleation processes such as the overheating of growing clusters due to latent heat. On the other hand, direct MD simulation of nucleation, where cluster formation is observed directly in a near-spinodal supersaturated vapour, has its limits: if nucleation occurs too fast, it affects the properties of the vapour to a significant extent so that the nucleation rate obtained according to the method of Yasuoka and Matsumoto [11] and other properties of the system vary over simulation time [17]. In the present work, nucleation is studied as a steady-state process by combining grand canonical MD (GCMD) and McDonald’s dæmon [18; 19], an ‘intelligent being’ that eliminates large droplets from the system.
+
+## 2 Simulation method
+
+Supersaturated states can be characterized in terms of the difference between the chemical potential \(\mathnormal{\mu}\) of the vapour and the saturated chemical potential \({\mathnormal{\mu}}_{\mathnormal{\sigma}}(\mathnormal{T})\). The chemical potential of the vapour can be regulated by simulating the grand canonical ensemble with GCMD: alternating with canonical ensemble MD steps, particles are inserted into and deleted from the system probabilistically, with the usual grand canonical acceptance criterion [20]. For a test insertion, random coordinates are chosen for an additional particle, and for a test deletion, a random particle is removed from the system. The potential energy difference \(\Delta\mathcal{V}\) due to the test action is determined and compared with the chemical potential. The acceptance probability for insertions is
+
+\[\mathcal{P}=\min\left(1,\exp\left[\frac{\mathnormal{\mu}-\Delta\mathcal{V}}{\mathnormal{k}_{\mathrm{B}}\mathnormal{T}}\right]\frac{\mathnormal{V}}{\mathnormal{\Lambda}^{3}(\mathnormal{N}+1)}\right),\] (1)
+
+while for deletions it is
+
+\[\mathcal{P}=\min\left(1,\exp\left[\frac{-\mathnormal{\mu}-\Delta\mathcal{V}}{\mathnormal{k}_{\mathrm{B}}\mathnormal{T}}\right]\frac{\mathnormal{V}}{\mathnormal{\Lambda}^{3}\mathnormal{N}}\right),\] (2)
+
+wherein \(\mathnormal{\Lambda}\) is the thermal wavelength. Of course, care must be taken that the momentum of the inserted particles is consistent with the simulated ensemble and does not introduce any artifical velocity gradients. The MD integration time step was \(\Delta\mathnormal{t}\) = 0.00404 in reduced time units, i.e., \(\mathnormal{\sigma}(\mathnormal{m}/\mathnormal{\varepsilon})^{1/2}\), wherein \(\mathnormal{\varepsilon}\) is the energy parameter of the fluid model and \(\mathnormal{m}\) is the mass of a particle. The number of test actions per simulation time step was chosen between \(10^{-6}\) and \(10^{-3}\)\(\mathnormal{N}\), a value which was occasionally decreased after equilibration if very low nucleation rates were observed.
+
+Molecular simulation of nucleation has to rely on a cluster criterion to distinguish the emerging liquid from the surrounding supersaturated vapour [21]. In the present case, the Stillinger criterion [22] was used to define the liquid phase and clusters were determined as biconnected components. Whenever a cluster exceeded the specified threshold size \(\mathnormal{\Theta}\), an intervention of McDonald’s dæmon removed it from the system, leaving a vacuum behind [18; 19].
+
+## 3 Nucleation theory
+
+The free energy of cluster formation is the same for the grand canonical and the isothermal-isobaric ensemble [23]. At specified values of the chemical potential \(\mathnormal{\mu}\) of the supersaturated vapour, the total system volume \(\mathnormal{V}\) and the temperature \(\mathnormal{T}\), it is related to the surface energy \(\mathnormal{\eta}\) by [24]
+
+\[\Delta{\mathnormal{\Omega}}_{\mathnormal{\nu}}=\int_{\mathnormal{V}_{\mathrm{\ell}}(1)}^{\mathnormal{V}_{\mathrm{\ell}}(\mathnormal{\nu})}(\mathnormal{p}-\mathnormal{p}_{\mathrm{\ell}})\mathnormal{d}\mathnormal{V}_{\mathrm{\ell}}+\int_{{\mathcal{F}}({1})}^{{\mathcal{F}}({\mathnormal{\nu}})}\left(\frac{\partial\mathnormal{\eta}}{\partial\mathcal{F}}\right)\mathnormal{d}\mathcal{F}+\int_{1}^{\mathnormal{\nu}}({\mathnormal{\mu}}_{\mathrm{\ell}}-\mathnormal{\mu})\mathnormal{d}\mathnormal{\nu},\] (3)
+
+where \(\mathnormal{\nu}\) is the number of particles in the cluster, \(\mathnormal{p}\) is the supersaturated vapour pressure, \(\mathnormal{V}_{\mathrm{\ell}}(\mathnormal{\nu})\) is the volume and \({\mathcal{F}}({\mathnormal{\nu}})\) the surface area of a cluster containing \(\mathnormal{\nu}\) particles. Note that \({\mathnormal{\mu}}_{\mathrm{\ell}}\) as well as \(\mathnormal{p}_{\mathrm{\ell}}\) are the chemical potential and the pressure of the liquid phase at the conditions prevailing inside the cluster. In CNT, it is assumed that the bulk liquid density at saturation \({\mathnormal{\rho}}^{\prime}\) and the density of a nanoscopic cluster are the same and all clusters are treated as spheres, i.e., \(\mathnormal{\rho}_{\mathrm{\ell}}={\mathnormal{\rho}}^{\prime}\) and \({\mathcal{F}}({\mathnormal{\nu}})={\mathcal{F}_{\bullet}}({\mathnormal{\nu}})=\left(6\sqrt{\pi}\mathnormal{\nu}/{\mathnormal{\rho}}^{\prime}\right)^{2/{}3}\). Accordingly, the chemical potential of the liquid inside the nucleus is approximated by
+
+\[{\mathnormal{\mu}}_{\mathrm{\ell}}={\mathnormal{\mu}}_{\mathnormal{\sigma}}(\mathnormal{T})+\int_{{\mathnormal{p}}_{\mathnormal{\sigma}}}^{\mathnormal{p}_{\mathrm{\ell}}}\frac{\mathnormal{d}\mathnormal{p}}{\mathnormal{\rho}_{\mathrm{\ell}}}\approx{\mathnormal{\mu}}_{\mathnormal{\sigma}}(\mathnormal{T})+\frac{\mathnormal{p}_{\mathrm{\ell}}-{\mathnormal{p}}_{\mathnormal{\sigma}}(\mathnormal{T})}{{\mathnormal{\rho}}^{\prime}},\] (4)
+
+and the cluster surface tension \(\tilde{\mathnormal{\gamma}}=(\partial\mathnormal{\eta}/\partial\mathcal{F})\) by the surface tension \(\mathnormal{\gamma}\) of the planar vapour-liquid interface, leading to [25; 26]
+
+\[\mathnormal{d}\mathnormal{\Omega}=\left[\mathnormal{\gamma}\,\sqrt[3]{\frac{2\pi}{3\mathnormal{\nu}}\left(\frac{4}{{\mathnormal{\rho}}^{\prime}}\right)^{2}}+{\mathnormal{\mu}}_{\mathnormal{\sigma}}(\mathnormal{T})-\mathnormal{\mu}+\frac{\mathnormal{p}-{\mathnormal{p}}_{\mathnormal{\sigma}}(\mathnormal{T})}{{\mathnormal{\rho}}^{\prime}}\right]\mathnormal{d}\mathnormal{\nu}.\] (5)
+
+The free energy of formation has a maximum \({\Delta\mathnormal{\Omega}}^{\star}\) which lies at the size \({\mathnormal{\nu}}^{\star}\) of the critical nucleus. Including the Zel’dovič factor \(\mathnormal{f}_{\mathrm{Z}}\) and the thermal non-accomodation factor \(\mathnormal{f_{\Delta\mathnormal{T}}}\) of Feder _et al. [1]_, the nucleation rate is
+
+\[\mathcal{J}=\mathnormal{f_{\Delta\mathnormal{T}}}\mathnormal{f}_{\mathrm{Z}}\frac{{\mathnormal{N}}_{1}}{\mathnormal{V}}\exp(-\mathnormal{\beta}{\Delta\mathnormal{\Omega}}^{\star})\frac{\mathnormal{p}\mathnormal{\Lambda}}{\mathnormal{h}}{\mathcal{F}}({{\mathnormal{\nu}}^{\star}}),\] (6)
+
+where \({\mathnormal{N}}_{1}\) is the number of vapour molecules in the system and \(\mathnormal{h}\) is the Planck constant.
+
+Instead of using the surface tension of the planar interface, Laaksonen, Ford, and Kulmala (LFK) [27] proposed an expression equivalent to
+
+\[\int_{0}^{{\mathcal{F}}({\mathnormal{\nu}})}\tilde{\mathnormal{\gamma}}\mathnormal{d}\mathcal{F}=\mathnormal{\gamma}{\mathcal{F}}({\mathnormal{\nu}})\left(1+\alpha_{1}\mathnormal{\nu}^{-1/{}3}+\alpha_{2}\mathnormal{\nu}^{-2/{}3}\right).\] (7)
+
+The two parameters \(\alpha_{1}\) and \(\alpha_{2}\) are determined from the assumption that almost all particles are arranged either as monomers or as dimers and that the Fisher [28] equation of state correctly relates \(\mathnormal{p}/\mathnormal{T}\) to the number of monomers and clusters present per volume. Effectively, LFK theory modifies CNT only by the introduction of the parameter \(\alpha_{1}\), since \(\alpha_{2}\) cancels out for all free energy differences if the usual assumption \(\mathcal{F}\sim\mathnormal{\nu}^{2/{}3}\) is applied.
+
+The Hale scaling law (HSL) is based on a different approach [29]. In agreement with experimental data on nucleation of water and toluene [29], it predicts
+
+\[\mathcal{J}\sim\mathnormal{\rho}^{-2/{}3}\left(\frac{\mathnormal{\gamma}}{\mathnormal{T}}\right)^{1/{}2}\mathnormal{p}^{2}\exp\left[\frac{4\mathnormal{\gamma}^{3}}{27(\ln\mathnormal{S})^{2}}\right],\] (8)
+
+with a proportionality constant depending only on properties of the critical point.
+
+In the present work, these theories are evaluated using Gibbs-Duhem integration over the metastable part of the vapour pressure isotherm collected by canonical ensemble MD simulation of small systems. The fluid model under consideration is the truncated and shifted Lennard-Jones (t. s. LJ) potential with a cutoff radius of \(2.5\mathnormal{\sigma}\)[30]. Note that the chemical potential supersaturation, i.e., \(\mathnormal{S}=\exp\left(\mathnormal{\beta}[\mathnormal{\mu}-{\mathnormal{\mu}}_{\mathnormal{\sigma}}(\mathnormal{T})]\right)\), deviates considerably from the pressure supersaturation \(\mathnormal{p}/{\mathnormal{p}}_{\mathnormal{\sigma}}\) and the density supersaturation \(\mathnormal{\rho}/{\mathnormal{\rho}}_{\mathnormal{\sigma}}\), with respect to the saturated vapour pressure \({\mathnormal{p}}_{\mathnormal{\sigma}}(\mathnormal{T})\) and density \({\mathnormal{\rho}}^{\prime\prime}(\mathnormal{T})\) of the bulk, cf. Fig. 1. For the saturated chemical potential of the t. s. LJ fluid, a correlation based on previously published data [8] gives
+
+\[\frac{{\mathnormal{\mu}}_{\mathnormal{\sigma}}(\mathnormal{T})-\mathnormal{\mu}_{\mathrm{id}}(\mathnormal{T})}{\mathnormal{k}_{\mathrm{B}}\mathnormal{T}}=-0.2367-\frac{1.7106\mathnormal{\varepsilon}}{\mathnormal{k}_{\mathrm{B}}\mathnormal{T}}-\frac{1.1514\mathnormal{\varepsilon}^{2}}{(\mathnormal{k}_{\mathrm{B}}\mathnormal{T})^{2}}.\] (9)
+
+In Fig. 2, the chemical potential supersaturation is shown as a function of the vapour density determined by GCMD simulation with McDonald’s dæmon. These values agree well with the metastable vapour pressure isotherm of the t. s. LJ fluid obtained by canonical ensemble simulation.
+
+Figure 1: Chemical potential supersaturation \(\mathnormal{S}\) (—), pressure supersaturation \(\mathnormal{p}/{\mathnormal{p}}_{\mathnormal{\sigma}}\)(– –), and density supersaturation \(\mathnormal{\rho}/{\mathnormal{\rho}}_{\mathnormal{\sigma}}\)(\(\cdot\)\(\cdot\)\(\cdot\)) in dependence of the excess pressure \(\Delta\mathnormal{p}=\mathnormal{p}-{\mathnormal{p}}_{\mathnormal{\sigma}}\) at \(\mathnormal{T}\) = 0.7 and 0.8 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\).
+
+Figure 2: Density dependence of the chemical potential supersaturation for the vapour of the t. s. LJ fluid, obtained from GCMD simulation with McDonald’s dæmon (\(\square\)) and by integration of the Gibbs-Duhem equation using data from canonical ensemble MD simulation with \(\mathnormal{T}\) = 0.7 (– –) and 0.85 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\) (—).
+
+## 4 Intervention rate and nucleation rate
+
+The size evolution of any given cluster can be considered as a random walk over the order parameter \(\mathnormal{\nu}\), changing only by relatively small amounts \(\Delta\mathnormal{\nu}\), usually by the absorption or emission of monomers. As discussed by Smoluchowski [31; 32] during his scientifically most productive period in L’viv and Kraków, the probabilities for the growth and decay transitions are proportional to the respective values of the partition function \(\mathnormal{W}\), resulting in
+
+\[\mathnormal{\mathcal{P}^{+}({\mathnormal{\nu}})}=\frac{1}{2}+\frac{(\mathnormal{d}\mathnormal{W}/\mathnormal{d}\mathnormal{\nu})\Delta\mathnormal{\nu}}{2\mathnormal{W}+\mathcal{O}(\mathnormal{\nu}^{2})}+\mathcal{O}(\mathnormal{\nu}^{2}),\] (10)
+
+and
+
+\[\mathnormal{\mathcal{P}^{-}({\mathnormal{\nu}})}=\frac{1}{2}-\frac{(\mathnormal{d}\mathnormal{W}/\mathnormal{d}\mathnormal{\nu})\Delta\mathnormal{\nu}}{2\mathnormal{W}+\mathcal{O}(\mathnormal{\nu}^{2})}+\mathcal{O}(\mathnormal{\nu}^{2}).\] (11)
+
+The probability \(\mathnormal{\mathcal{P}^{\mathsf{F}}({\mathnormal{\nu}})}\) that a certain size is _eventually_ reached (at any time during the random walk process), given that the current size is \(\mathnormal{\nu}\), has the property
+
+\[\mathnormal{\mathcal{P}^{\mathsf{F}}({\mathnormal{\nu}})}=\mathnormal{\mathcal{P}^{+}({\mathnormal{\nu}})}\mathnormal{\mathcal{P}^{\mathsf{F}}({\mathnormal{\nu}+\Delta\mathnormal{\nu}})}+\mathnormal{\mathcal{P}^{-}({\mathnormal{\nu}})}\mathnormal{\mathcal{P}^{\mathsf{F}}({\mathnormal{\nu}-\Delta\mathnormal{\nu}})}.\] (12)
+
+By substituting
+
+\[\mathnormal{\mathcal{P}^{\mathsf{F}}({\mathnormal{\nu}\pm\Delta\mathnormal{\nu}})}=\mathnormal{\mathcal{P}^{\mathsf{F}}({\mathnormal{\nu}})}\pm\frac{\mathnormal{d}\mathcal{P}^{\mathsf{F}}}{\mathnormal{d}\mathnormal{\nu}}\Delta\mathnormal{\nu}+\frac{\mathnormal{d}^{2}\mathcal{P}^{\mathsf{F}}}{2\mathnormal{d}\mathnormal{\nu}^{2}}\Delta\mathnormal{\nu}^{2}+\mathcal{O}\left(\Delta\mathnormal{\nu}^{3}\right),\] (13)
+
+it follows for small \(\Delta\mathnormal{\nu}\) neglecting terms of third order and beyond, that
+
+\[\frac{\mathnormal{d}\mathnormal{W}}{\mathnormal{W}\mathnormal{d}\mathnormal{\nu}}=\frac{-\mathnormal{d}\left(\mathnormal{d}\mathcal{P}^{\mathsf{F}}/\mathnormal{d}\mathnormal{\nu}\right)}{2\left(\mathnormal{d}\mathcal{P}^{\mathsf{F}}/\mathnormal{d}\mathnormal{\nu}\right)\mathnormal{d}\mathnormal{\nu}}.\] (14)
+
+Using the partition function for the grand canonical ensemble, the derivative of the probability is given by
+
+\[\frac{\mathnormal{d}\mathcal{P}^{\mathsf{F}}}{\mathnormal{d}\mathnormal{\nu}}=\mathnormal{\digamma}\exp\left(2\mathnormal{\beta}\Delta{\mathnormal{\Omega}}_{\mathnormal{\nu}}\right),\] (15)
+
+where \(\mathnormal{\digamma}\) is an integration constant. Obtaining the two remaining parameters from the boundary conditions
+
+\[q_{1} = 0,\] (16)
+\[\lim_{\mathnormal{\Theta}\to\infty}q_{\mathnormal{\Theta}} = 1,\] (17)
+
+the probability \(q_{\mathnormal{\Theta}}\) for a cluster containing \(\mathnormal{\Theta}\) molecules of eventually reaching macroscopic size, i.e., \(\mathcal{J}\to\infty\), is
+
+\[q_{\mathnormal{\Theta}}=\frac{\int_{1}^{\mathnormal{\Theta}}\exp\left(2\mathnormal{\beta}\Delta{\mathnormal{\Omega}}_{\mathnormal{\nu}}\right)\mathnormal{d}\mathnormal{\nu}}{\int_{1}^{\infty}\exp\left(2\mathnormal{\beta}\Delta{\mathnormal{\Omega}}_{\mathnormal{\nu}}\right)\mathnormal{d}\mathnormal{\nu}}.\] (18)
+
+The intervention rate \({\mathcal{J}}_{\mathnormal{\Theta}}\) of McDonald’s dæmon is related to the nucleation rate \(\mathcal{J}\) by
+
+\[\mathcal{J}={\mathcal{J}}_{\mathnormal{\Theta}}q_{\mathnormal{\Theta}}.\] (19)
+
+Thus, with an intervention threshold far below the critical size, the intervention rate is many orders of magnitude higher the steady-state nucleation rate. However, as confirmed by the present simulation results shown in Tab. 1, it reaches a plateau for \(\mathnormal{\Theta}>{\mathnormal{\nu}}^{\star}\), where \({\mathnormal{\nu}}^{\star}=41\) according to CNT and 39 according to SPC.
+
+\begin{table}
+\begin{tabular}{c c c|c c c|c}
+\(\mathnormal{V}\) & \(\mathnormal{N}\) & \(\mathnormal{p}/{\mathnormal{p}}_{\mathnormal{\sigma}}\) & \(\mathnormal{\Theta}\) & \(\ln q_{\mathnormal{\Theta}}(\mathrm{CNT})\) & \(\ln q_{\mathnormal{\Theta}}(\mathrm{LFK})\) & \(\ln{\mathcal{J}}_{\mathnormal{\Theta}}\) \\
+\hline
+5.38 \(\times\) 10⁶ & 0124000 & 2.70 & 10 & -16.700 & -12.700 & -13.6 \\
+4.32 \(\times\) 10⁷ & 1020000 & 2.75 & 20 & 0-8.140 & 0-6.330 & -17.0 \\
+5.38 \(\times\) 10⁶ & 0129000 & 2.78 & 25 & 0-5.550 & 0-4.340 & -17.6 \\
+5.38 \(\times\) 10⁶ & 0129000 & 2.78 & 35 & 0-2.320 & 0-1.820 & -19.9 \\
+4.32 \(\times\) 10⁷ & 1040000 & 2.78 & 48 & 0-0.508 & 0-0.400 & -21.7 \\
+4.32 \(\times\) 10⁷ & 1040000 & 2.78 & 65 & 0-0.022 & 0-0.019 & -21.9 \\
+2.15 \(\times\) 10⁷ & 0518000 & 2.77 & 74 & 0-0.002 & 0-0.002 & -22.1 \\
+\end{tabular}
+\end{table}
+Table 1: Dependence of the intervention rate \({\mathcal{J}}_{\mathnormal{\Theta}}\) as well as the probability \(q_{\mathnormal{\Theta}}\) according to CNT and LFK on the intervention threshold size \(\mathnormal{\Theta}\) for McDonald’s dæmon during GCMD simulation at \(\mathnormal{T}=0.7\)\(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\) and \(\mathnormal{S}=2.4958\), where the rates are given in units of \((\mathnormal{\varepsilon}/\mathnormal{m})^{1/{}2}\mathnormal{\sigma}^{-4}\). The number of particles in the system and the values for the pressure supersaturation \(\mathnormal{p}/{\mathnormal{p}}_{\mathnormal{\sigma}}\) refer to the steady state and the constant volume of the system is given in units of \(\mathnormal{\sigma}^{3}\).
+
+## 5 Results and discussion
+
+Homogeneous nucleation of the t. s. LJ fluid was studied by a series of GCMD simulations with McDonald’s dæmon for systems containing up to 17 million particles.
+
+After a temporal delay, depending on the threshold size, the pressure and the intervention rate reached a constant value, cf. Fig. 3. In a canonical ensemble MD simulation under similar conditions as the GCMD simulation that is also shown in Fig. 3, the pressure supersaturation decreased from about 3 to 1.5 and the rate of formation was significantly lower for larger nuclei, due to the free energy effect accounted for by Eqs. (18) and (19) as well as the depletion of the vapour [19].
+
+The constant supersaturation of the GCMD simulation agreed approximately with the time-dependent supersaturation in the canonical ensemble about \(\mathnormal{t}\) = 400 after simulation onset, cf. Fig. 3. At this stage, the number of small clusters present per volume was similar in both cases, and the rate of formation for clusters with \(\mathnormal{\nu}>150\) at \(\mathnormal{t}\) = 400 in the canonical ensemble simulation was of the same order of magnitude as the intervention rate of the dæmon.
+
+Figure 3: Top: Number per unit volume \(\mathnormal{\rho}_{\mathrm{n}}\) of clusters containing more than 25 (\(\cdot\) – \(\cdot\)), 50 (—), and 150 (– –) particles in a canonical ensemble MD simulation at \(\mathnormal{T}\) = 0.7 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\) and \(\mathnormal{\rho}\) = 0.004044 \(\mathnormal{\sigma}^{-3}\) using a hybrid geometric-energetic cluster criterion, number per unit volume \(\mathnormal{\rho}_{\mathrm{n}}\) of clusters with \(\mathnormal{\nu}\geq 25\) (\(\square\)) in a GCMD simulation with \(\mathnormal{T}\) = 0.7 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\), \(\mathnormal{S}\) = 2.8658, and \(\mathnormal{\Theta}\) = 50, using the Stillinger [22] cluster criterion with clusters determined as biconnected components, as well as the aggregated number of McDonald’s dæmon interventions per unit volume in the GCMD simulation, over simulation time. Bottom: Pressure over simulation time for the canonical ensemble MD simulation (– –) and the GCMD simulation with McDonald’s dæmon (—) [19].
+
+Van Meel _et al._[16] determined by MC simulation with forward flux sampling that supersaturated vapours of the t. s. LJ fluid at a temperature of \(\mathnormal{T}\) = 0.45 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\), i.e., significantly below the triple point \(\mathnormal{T}_{3}\) = 0.65 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\), initially undergo vapour to liquid nucleation, and CNT is known to underestimate the vapour to liquid nucleation rate of unpolar fluids [13]. The present dæmon intervention rates confirm this conclusion. LFK and HSL are significantly more accurate than CNT. Note that in Tab. 2, the nucleation rate according to Eq. (19) based on the CNT value of \(q_{\mathnormal{\Theta}}\) is given.
+
+From Tab. 2 it is also confirmed that the ‘direct observation method’ (DOM) [17], which in the present case corresponds to assuming
+
+\[\ln{\mathcal{J}}_{\mathnormal{\Theta}}=\ln\mathcal{J}-\ln q_{\mathnormal{\Theta}}=-\ln\tau\mathnormal{V},\] (20)
+
+where \(\tau\) is the temporal delay of formation for the first sufficiently large cluster, is inadequate for nucleation near the spinodal line.
+
+\begin{table}
+\begin{tabular}{c c|c c c c c c c c}
+\(\mathnormal{p}/{\mathnormal{p}}_{\mathnormal{\sigma}}\) & \(10^{-6}\mathnormal{N}\) & \(\mathnormal{\Theta}\) & \(-\ln\tau\mathnormal{V}\) & \(\ln q_{\mathnormal{\Theta}}(\mathrm{CNT})\) & \(\ln\mathcal{J}\) & \multicolumn{1}{c|}{} & \(\ln\mathcal{J}_{\mathrm{CNT}}\) & \(\ln\mathcal{J}_{\mathrm{LFK}}\) & \(\ln\mathcal{J}_{\mathrm{HSL}}\) \\
+\hline
+30.20 & 00.397 & 09 & -23.1 & -4.570 & 0-26.4 & \multicolumn{1}{c|}{} & 0-31.5 & 0-26.2 & 0-24.7 \\
+32.40 & 00.429 & 09 & -23.0 & -3.800 & 0-25.0 & \multicolumn{1}{c|}{} & 0-30.5 & 0-25.4 & 0-24.0 \\
+55.90 & 01.070 & 12 & -22.5 & -0.062 & 0-18.0 & \multicolumn{1}{c|}{} & 0-24.2 & 0-20.2 & 0-19.5 \\
+74.70 & 17.100 & 24 & -17.1 & \(\approx\) 0 & 0-18.8 & \multicolumn{1}{c|}{} & 0-21.8 & 0-18.6 & 0-17.7 \\
+\end{tabular}
+\end{table}
+Table 2: Vapour to liquid nucleation rate at \(\mathnormal{T}\) = 0.45 \(\mathnormal{\varepsilon}/\mathnormal{k}_{\mathrm{B}}\) from GCMD simulation with McDonald’s dæmon. The theories were evaluated with respect to the metastable vapour-liquid equilibrium at \({\mathnormal{p}}_{\mathnormal{\sigma}}=4.28\times 10^{-5}\)\(\mathnormal{\varepsilon}/\mathnormal{\sigma}^{3}\)[16], and the vapour-liquid surface tension \(\mathnormal{\gamma}=1.07\)\(\mathnormal{\varepsilon}/\mathnormal{\sigma}^{2}\)[16] was used.
+
+## 6 Conclusion
+
+GCMD with McDonald’s dæmon was established as a method for steady-state simulation of nucleating vapours at high supersaturations. A series of simulations was conducted for the t. s. LJ fluid. CNT was found to underpredict the nucleation rate below the triple point, whereas LFK and HSL more accurately describe vapour to liquid nucleation of the t. s. LJ fluid.
+
+ The authors would like to thank G. Chkonia, H. Hasse, S. Sastry, C. Valeriani, and J. Wedekind for fruitful discussions and Deutsche Forschungsgemeinschaft for funding SFB 716. The presented research was conducted under the auspices of the Boltzmann-Zuse Society of Computational Molecular Engineering (BZS), and the simulations were performed on the HP XC4000 supercomputer at the Steinbuch Centre for Computing, Karlsruhe, under the grant LAMO, as well as the _phoenix_ supercomputer at Höchstleistungsrechenzentrum Stuttgart (HLRS) under the grant MMHBF.
+
+## References
+
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+* (9) T. Ikeshoji, B. Hafskjold, Y. Hashi, and Y. Kawazoe, _Phys. Rev. Lett._**76**, 1792–1795 (1996).
+* (10) S. Sumardiono and J. Fischer, “Molecular dynamics simulations of mixture droplet evaporation,” in _Proc. Eurotherm Seminar 77, Parma – Heat and Mass Transfer in Food Processing_ (2005).
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+* (16) J. A. van Meel, A. J. Page, R. P. Sear, and D. Frenkel, _J. Chem. Phys._**129**, 204505 (2008).
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+* (18) J. E. McDonald, _Am. J. Phys._**31**, 31–41 (1962).
+* (19) M. Horsch and J. Vrabec, “Grand canonical steady-state simulation of nucleation,” submitted to _J. Chem. Phys._ (2009).
+* (20) M. M. Cielinski, M. Sc. thesis, University of Maine (1985).
+* (21) R. Badahur and R. B. McClurg, _J. Phys. Chem. B_**105**, 11893–11900 (2001).
+* (22) F. H. Stillinger, _J. Chem. Phys._**38**, 1486–1494 (1963).
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+* (24) P. G. Debenedetti, _Metastable liquids: concepts and principles_, Princeton University Press (1996).
+* (25) D. Reguera, R. K. Bowles, Y. Djikaev, and H. Reiss, _J. Chem. Phys._**118**, 340–353 (2003).
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+

arxiv_ground_truth/1001.1811.md

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+# The Physical Properties of the Cosmic Acceleration
+
+Spyros Basilakos
+
+Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece
+
+###### Abstract
+
+The observed late-time acceleration of the cosmic expansion constitutes a fundamental problem in modern theoretical physics and cosmology. In an attempt to weight the validity of a large number of dark energy models, I use the recent measurements of the expansion rate of the Universe, the clustering of galaxies the CMB fluctuations as well as the large scale structure formation, to put tight constraints on the different models.
+
+Cosmology; dark matter; dark energy pacs: 98.80.-k, 95.35.+d, 95.36.+x
+
+## I Introduction
+
+Recent studies in observational cosmology lead to the conclusion that the available high quality cosmological data (Type Ia supernovae, CMB, etc.) are well fitted by an emerging “standard model”. This standard model, assuming flatness, is described by the Friedman equation:
+
+\[H^{2}(a)=\left(\frac{{\dot{a}}}{a}\right)^{2}=\frac{8\pi G}{3}\left[\rho_{m}(a)+\rho_{X}(a)\right]\;,\] (1)
+
+where \(a(t)\) is the scale factor of the universe, \(\rho_{m}(a)\) is the density corresponding to the sum of baryonic and cold dark matter, with the latter needed to explain clustering, and an extra component \(\rho_{X}(a)\) with negative pressure, called dark energy, needed to explain the observed accelerated cosmic expansion (eg., Davis et al. 2007; Kowalski et al. 2008; Komatsu et al. 2009 and references therein).
+
+The nature of the dark energy is one of the most fundamental and difficult problems in physics and cosmology. Indeed, during the last decade there has been a theoretical debate among the cosmologists regarding the nature of the exotic “dark energy”. Various candidates have been proposed in the literature, such as a cosmological constant \(\Lambda\) (vacuum), time-varying \(\Lambda(t)\) cosmologies, quintessence, \(k-\)essence, vector fields, phantom, tachyons, Chaplygin gas and the list goes on (for recent reviews see Peebles & Ratra 2003; Copeland, Sami & Tsujikawa 2006; Caldwell & Kamionkowski 2009). Within this framework, high energy field theories generically indicate that the equation of state of such a dark energy is a function of cosmic time. To identify this type of evolution of the equation of state, a detailed form of the observed \(H(a)\) is required which may be obtained by a combination of multiple dark energy observational probes.
+
+In this cosmological framework, a serious issue is how (and when) the large scale structures form. Galaxies and large-scale structure grew gravitationally from tiny, nearly scale-invariant adiabatic Gaussian fluctuations. In this paper we focus also on galaxy clusters that occupy an eminent position in the structure hierarchy, being the most massive virialized systems known and therefore they appear to be ideal tools for testing theories of structure formation and extracting cosmological information. The cluster distribution basically traces scales that have not yet undergone the non-linear phase of gravitationally clustering, thus simplifying their connections to the initial conditions of cosmic structure formation.
+
+The structure of the paper is as follows. In section 2 we briefly discuss the dark energy issue. In sections 3 and 4 we present the various dark energy models and we use a joint statistical analysis, in order to place constraints on the main cosmological parameters. In section 5 we present the corresponding theoretical predictions regarding the formation of the galaxy clusters. Finally, we draw our conclusions in section 6. Throughout the paper we will use \(H_{0}\simeq 71\)km/sec/Mpc.
+
+## II The dark energy equation of state
+
+In the context of general relativity it is well known that for homogeneous and isotropic flat cosmologies, driven by non-relativistic matter and dark energy with equation of state \(p_{X}=w(a)\rho_{X}\), the expansion rate of the Universe can be written as (see eq.1)
+
+\[E^{2}(a)=\frac{H^{2}(a)}{H_{0}^{2}}=\Omega_{m}a^{-3}+\Omega_{X}{\rm e}^{3\int^{1}_{a}d{\rm lny}[1+w(y)]}\;\;.\] (2)
+
+Note, that \(E(a)\) is the normalized Hubble flow, \(\Omega_{m}\) is the dimensionless matter density at the present epoch, \(\Omega_{X}=1-\Omega_{m}\) is the corresponding dark energy density parameter and \(w(a)\) is the dark energy equation of state. Inverting the above equation we simply derive
+
+\[w(a)=\frac{-1-\frac{2}{3}a\frac{{d\rm lnE}}{da}}{1-\Omega_{m}a^{-3}E^{-2}(a)}\;\;.\] (3)
+
+The exact nature of the dark energy has yet to be found and thus the dark energy equation of state parameter includes our ignorance regarding the physical mechanism which leads to a late cosmic acceleration.
+
+On the other hand, it is possible to extent the previous methodology in the framework of modified gravity (see Linder & Jenkins 2003; Linder 2004). In this scenario, it is assumed that the dark energy may be an illusion, indicating the need to revise the general relativity and thus also the Friedmann equation. From the mathematical point of view, it can be shown that instead of using the exact Hubble flow through a modification of the Friedmann equation we can utilize a Hubble flow that looks like the nominal one (see eq.1). The key point here is to consider that the accelerated expansion of the universe can be attributed to a “geometrical” dark energy component. Due to the fact that the matter density in the universe (baryonic+dark) can not accelerate the cosmic expansion, we perform the following parametrization (Linder & Jenkins 2003; Linder 2004):
+
+\[E^{2}(a)=\frac{H^{2}(a)}{H_{0}^{2}}=\Omega_{m}a^{-3}+\delta H^{2}\;\;.\] (4)
+
+Obviously, with the aid of the latter approach we include any modification to the Friedmann equation of general relativity in the last term of eq.(4). Now from eqs.(3, 4) we can derive, after some algebra, the “geometrical” dark energy equation of state
+
+\[w(a)=-1-\frac{1}{3}\;\frac{d{\rm ln}\delta H^{2}}{d{\rm ln}a}\;\;.\] (5)
+
+From now on, for the modified cosmological models we will use the above formulation.
+
+## III Likelihood Analysis
+
+We will use various cosmological observations in order to constrain the dark energy models described in section 4. First of all, we use the Baryonic Acoustic Oscillations (BAOs). BAOs are produced by pressure (acoustic) waves in the photon-baryon plasma in the early universe, generated by dark matter overdensities. Evidence of this excess was recently found in the clustering properties of the luminous SDSS red-galaxies (Eisenstein et al. 2005; Padmanabhan et al. 2007) and it can provide a “standard ruler” with which we can constraint the dark energy models. In particular, we use the following estimator: \(A({\bf p})=\frac{\sqrt{\Omega_{m}}}{[z^{2}_{s}E(a_{s})]^{1/3}}\left[\int_{a_{s}}^{1}\frac{da}{a^{2}E(a)}\right]^{2/3}\), measured from the SDSS data to be \(A=0.469\pm 0.017\), where \(z_{s}=0.35\) [or \(a_{s}=(1+z_{s})^{-1}\simeq 0.75\)]. Therefore, the corresponding \(\chi^{2}_{\rm BAO}\) function is simply written
+
+\[\chi^{2}_{\rm BAO}({\bf p})=\frac{[A({\bf p})-0.469]^{2}}{0.017^{2}}\] (6)
+
+where \({\bf p}\) is a vector containing the cosmological parameters that we want to fit.
+
+On the other hand, a very accurate and deep geometrical probe of dark energy is the angular scale of the sound horizon at the last scattering surface as encoded in the location \(l_{1}^{TT}\) of the first peak of the Cosmic Microwave Background (CMB) temperature perturbation spectrum. This probe is described by the CMB shift parameter (Bond, Efstathiou & Tegmark, 1997; Nesseris & Perivolaropoulos 2007) and is defined as \(R=\sqrt{\Omega_{m}}\int_{a_{ls}}^{1}\frac{da}{a^{2}E(a)}\). The shift parameter measured from the WMAP 5-years data (Komatsu et al. 2009) is \(R=1.71\pm 0.019\) at \(z_{ls}=1090\) [or \(a_{ls}=(1+z_{ls})^{-1}\simeq 9.17\times 10^{-4}\)]. In this case, the \(\chi^{2}_{\rm cmb}\) function is given
+
+\[\chi^{2}_{\rm cmb}({\bf p})=\frac{[R({\bf p})-1.71]^{2}}{0.019^{2}}\] (7)
+
+Finally, we use the Union08 sample of 307 supernovae of Kowalski et al. (2008). The corresponding \(\chi^{2}_{\rm SNIa}\) function becomes:
+
+\[\chi^{2}_{\rm SNIa}({\bf p})=\sum_{i=1}^{307}\left[\frac{{\cal\mu}^{\rm th}(a_{i},{\bf p})-{\cal\mu}^{\rm obs}(a_{i})}{\sigma_{i}}\right]^{2}\;\;.\] (8)
+
+where \(a_{i}=(1+z_{i})^{-1}\) is the observed scale factor of the Universe, \(z_{i}\) is the observed redshift, \({\cal\mu}\) is the distance modulus \({\cal\mu}=m-M=5{\rm log}d_{L}+25\) and \(d_{L}(a,{\bf p})\) is the luminosity distance \(d_{L}(a,{\bf p})=\frac{c}{a}\int_{a}^{1}\frac{{\rm d}y}{y^{2}H(y)}\) where \(c\) is the speed of light. We can combine the above probes by using a joint likelihood analysis: \({\cal L}_{tot}({\bf p})={\cal L}_{\rm BAO}\times{\cal L}_{\rm cmb}\times{\cal L}_{\rm SNIa}\) or \(\chi^{2}_{tot}({\bf p})=\chi^{2}_{\rm BAO}+\chi^{2}_{\rm cmb}+\chi^{2}_{\rm SNIa}\) in order to put even further constraints on the parameter space used. Note, that we define the likelihood estimator ¹ as: \({\cal L}_{j}\propto{\rm exp}[-\chi^{2}_{j}/2]\).
+
+Footnote 1: Likelihoods are normalized to their maximum values. Note, that the step of sampling is 0.01 and the errors of the fitted parameters represent \(2\sigma\) uncertainties. Note that the overall number of data points used is \(N_{tot}=309\) and the degrees of freedom: \(dof=N_{tot}-n_{\rm fit}\), with \(n_{\rm fit}\) the number of fitted parameters, which vary for the different models.
+
+## IV Constraints on the flat dark energy models
+
+In this section, we consider a large family of flat dark energy models and with the aid of the above cosmologically relevant observational data, we attempt to put tight constraints on their free parameters. In the following subsections, we briefly present these cosmological models which trace differently the nature of the dark energy.
+
+### Constant equation of state - QP model
+
+In this case the equation of state is constant (see for a review, Peebles & Ratra 2003; hereafter QP-models). Such dark energy models do not have much physical motivation. In particular, a constant equation of state parameter requires a fine tuning of the potential and kinetic energies of the real scalar field. Despite the latter problem, these dark energy models have been used in the literature due to their simplicity. Notice that dark energy models with a canonical kinetic term have \(-1\leq w<-1/3\). On the other hand, models of phantom dark energy (\(w<-1\)) require exotic nature, such as a scalar field with a negative kinetic energy. Now using eq.(2) the normalized Hubble parameter becomes
+
+\[E^{2}(a)=\Omega_{m}a^{-3}+(1-\Omega_{m})a^{-3(1+w)}\;\;.\] (9)
+
+Comparing the QP-models with the observational data (we sample \(\Omega_{m}\in[0.1,1]\) and \(w\in[-2,-0.4]\)) we find that the best fit values are \(\Omega_{m}=0.28\pm 0.02\) and \(w=-1.02\pm 0.06\) with \(\chi_{tot}^{2}(\Omega_{m},w)/dof\simeq 309.2/307\) in very good agreement with the 5 years WMAP data Komatsu et al. (2009). Also Davis et al. (2007) utilizing the Essence-SNIa+BAO+CMB and a Bayesian statistics found \(\Omega_{m}=0.27\pm 0.04\), while Kowalski et al. (2008) using the Union08-SNIa+BAO+CMB obtained \(\Omega_{m}\simeq 0.285^{+0.03}_{-0.03}\). Obviously, our results coincide within \(1\sigma\) errors. It is worth noting that the concordance \(\Lambda\)-cosmology can be described by QP models with \(w\) strictly equal to -1. In this case we find: \(\Omega_{m}=0.28\pm 0.02\) with \(\chi_{tot}^{2}(\Omega_{m})/dof\simeq 309.3/308\).
+
+### The Braneworld Gravity - BRG model
+
+In the context of a braneworld cosmology (hereafter BRG) the accelerated expansion of the universe can be explained by a modification of gravity in which gravity itself becomes weak at very large distances (close to the Hubble scale) due to the fact that our four dimensional brane survives into an extra dimensional manifold (Deffayet, Dvali, & Cabadadze 2002). The interesting point in this scenario is that the corresponding functional form of the normalized Hubble flow, eq. (4), is affected only by one free parameter (\(\Omega_{m}\)). Notice, that the quantity \(\delta H^{2}\) is given by
+
+\[\delta H^{2}=2\Omega_{bw}+2\sqrt{\Omega_{bw}}\sqrt{\Omega_{m}a^{-3}+\Omega_{bw}}\;,\] (10)
+
+where \(\Omega_{bw}=(1-\Omega_{m})^{2}/4\). The geometrical dark energy equation of state parameter (see eq.5) reduces to
+
+\[w(a)=-\frac{1}{1+\Omega_{m}(a)}\;,\] (11)
+
+where \(\Omega_{m}(a)\equiv\Omega_{m}a^{-3}/E^{2}(a)\). The joint likelihood analysis provides a best fit value of \(\Omega_{m}=0.24\pm 0.02\), but the fit is rather poor \(\chi_{tot}^{2}(\Omega_{m})/dof\simeq 369/308\).
+
+### The parametric Dark Energy model - CPL model
+
+In this model we use the Chevalier-Polarski-Linder (Chevallier & Polarski 2001; Linder 2003, hereafter CPL) parametrization. The dark energy equation of state parameter is defined as a first order Taylor expansion around the present epoch:
+
+\[w(a)=w_{0}+w_{1}(1-a)\;.\] (12)
+
+The normalized Hubble parameter is given by (see eq.2):
+
+\[E^{2}(a)=\Omega_{m}a^{-3}+(1-\Omega_{m})a^{-3(1+w_{0}+w_{1})}e^{3w_{1}(a-1)}\;,\] (13)
+
+where \(w_{0}\) and \(w_{1}\) are constants. We sample the unknown parameters as follows: \(w_{0}\in[-2,-0.4]\) and \(w_{1}\in[-2.6,2.6]\). We find that for the prior of \(\Omega_{m}=0.28\) the overall likelihood function peaks at \(w_{0}=-1.1^{+0.22}_{-0.16}\) and \(w_{1}=0.60^{+0.62}_{-1.54}\) while the corresponding \(\chi_{tot}^{2}(w_{0},w_{1})/dof\) is 307.6/307.
+
+### The low Ricci dark energy - LRDE model
+
+In this modified cosmological model, we use a simple parametrization for the Ricci scalar which is based on a Taylor expansion around the present time: \({\cal R}=r_{0}+r_{1}(1-a)\) [for more details see Linder 2004]. It is interesting to point that at the early epochs the cosmic evolution tends asymptotically to be matter dominated. In this framework, we have
+
+\[E^{2}(a)=\left\{\begin{array}[]{cc}a^{4(r_{0}+r_{1}-1)}{\rm e}^{4r_{1}(1-a)}&\;\;\;\;a\geq a_{t}\\ \Omega_{m}a^{-3}&\;\;\;\;a<a_{t}\end{array}\right.\] (14)
+
+where \(a_{t}=1-(1-4r_{0})/4r_{1}\). The matter density parameter at the present epoch, is related with the above constants via \(\Omega_{m}=\left(\frac{4r_{0}-4r_{1}-1}{4r_{1}}\right)^{4r_{0}+4r_{1}-1}{\rm e}^{1-4r_{0}}\). The equation of state parameter that corresponds to the current geometrical dark energy model is given by
+
+\[w(a)=\frac{1-4{\cal R}}{3}\left[1-\Omega_{m}{\rm e}^{-\int_{a}^{1}(1-4{\cal R})(dy/y)}\right]^{-1}\;\;.\] (15)
+
+Note, that we sample the unknown parameters as follows: \(r_{0}\in[0.5,1.5]\) and \(r_{1}\in[-2.4,-0.1]\) and here are the results: \(r_{0}=0.82\pm 0.04\) and \(r_{1}=-0.74^{+0.10}_{-0.08}\) (\(\Omega_{m}\simeq 0.28\)) with \(\chi_{tot}^{2}(r_{0},r_{1})/dof\simeq 309.8/307\).
+
+### The high Ricci dark energy - HRDE model
+
+A different than the previously described geometrical method was defined by Linder & Cahn (2007), in which the Ricci scalar at high redshifts evolves as
+
+\[{\cal R}\simeq\frac{1}{4}\left[1-3w_{1}\frac{\delta H^{2}}{H^{2}}\right]\;,\] (16)
+
+where \(\delta H^{2}=E^{2}(a)-\Omega_{m}a^{-3}\). In this geometrical pattern the normalized Hubble flow becomes:
+
+\[E^{2}(a)=\Omega_{m}a^{-3}\left(1+\beta a^{-3w_{1}}\right)^{-{\rm ln}\Omega_{m}/{\rm ln}(1+\beta)}\;,\] (17)
+
+where \(\beta=\Omega_{m}^{-1}-1\). Using the same sampling as in the QP-models, the joint likelihood peaks at \(\Omega_{m}=0.28\pm 0.03\) and \(w_{1}=-1.02\pm 0.1\) with \(\chi_{tot}^{2}(\Omega_{m},w_{1})/dof\simeq 309.2/307\). To this end, the effective equation of state parameter is related to \(E(a)\) according to eq.(3).
+
+### The tension of cosmological magnetic fields - TCM model
+
+Recently, Contopoulos & Basilakos (2007) proposed a novel idea which is based on the following consideration (hereafter TCM): if the cosmic magnetic field is generated in sources (such as galaxy clusters) whose overall dimensions remain unchanged during the expansion of the Universe, the stretching of this field by the cosmic expansion generates a tension (negative pressure) that could possibly explain a small fraction of the dark energy (\(\sim 5-10\%\)). In this flat cosmological scenario the normalized Hubble flow becomes:
+
+\[E^{2}(a)=\Omega_{m}a^{-3}+\Omega_{\Lambda}+\Omega_{B}a^{-3+n}\;,\] (18)
+
+where \(\Omega_{B}\) is the density parameter for the cosmic magnetic fields and \(\Omega_{\Lambda}=1-\Omega_{m}-\Omega_{B}\). The equation of state parameter which is related to magnetic tension is (see eq.3)
+
+\[w(a)=-\frac{3\Omega_{\Lambda}+n\Omega_{B}a^{-3+n}}{3(\Omega_{\Lambda}+\Omega_{B}a^{-3+n})}\;\;.\] (19)
+
+To this end, we sample \(\Omega_{B}\in[0,0.3]\) and \(n\in[0,10]\) and we find that for the prior of \(\Omega_{m}=0.28\) the best fit values are: \(\Omega_{B}=0.10\pm 0.10\) and \(n=3.60^{+4.5}_{-2.6}\) with \(\chi_{tot}^{2}(\Omega_{B},n)/dof\simeq 308.9/307\).
+
+### The Pseudo-Nambu Goldstone boson - PNGB model
+
+In this cosmological model the dark energy equation of state parameter is expressed with the aid the potential \(V(\phi)\propto[1+cos(\phi/F)]\) (Abrahamse et al 2008 and references therein):
+
+\[w(a)=-1+(1+w_{0})a^{F}\;,\] (20)
+
+where \(w_{0}\in[-2,-0.4]\) and \(F\in[0,8]\). In case of \(a\ll 1\) we get \(w(a)\longrightarrow-1\). Based on this parametrization the normalized Hubble function is given by
+
+\[E^{2}(a)=\Omega_{m}a^{-3}+(1-\Omega_{m})\rho_{X}(a)\;.\] (21)
+
+In this context, the corresponding dark energy density is
+
+\[\rho_{X}(a)={\rm exp}\left[\frac{3(1+w_{0})}{F}(1-a^{F})\right]\;\;.\] (22)
+
+Notice, that the likelihood function peaks at \(w_{0}=-1.04\pm 0.17\) and \(F=5.9\pm 3.2\) with \(\chi_{tot}^{2}(w_{0},F)/dof\simeq 317/307\).
+
+### The early dark energy - EDE model
+
+Another cosmological scenario that we include in our paper is the early dark energy model (hereafter EDE). The basic assumption here is that at early epochs the amount of dark energy is not negligible (Doran, Stern & Thommes 2006 and references therein). In this model, the total dark energy component is given by:
+
+\[\Omega_{X}(a)=\frac{1-\Omega_{m}-\Omega_{e}(1-a^{-3w_{0}})}{1-\Omega_{m}-\Omega_{m}a^{3w_{0}}}+\Omega_{e}(1-a^{-3w_{0}})\;,\] (23)
+
+where \(\Omega_{e}\) is the early dark energy density and \(w_{0}\) is the equation of state parameter at the present epoch. Notice, that the EDE model was designed to simultaneously (a) mimic the effects of the late dark energy and (b) provide a decelerated expansion of the universe at high redshifts. The normalized Hubble parameter is written as:
+
+\[E^{2}(a)=\frac{\Omega_{m}a^{-3}}{1-\Omega_{X}(a)}\;,\] (24)
+
+while using eq.(3), we can obtain the equation of state parameter as a function of the scale factor. From the joint likelihood analysis we find that \(\Omega_{e}=0.05\pm 0.04\) and \(w_{0}=-1.14^{+0.18}_{-0.10}\) (for the prior of \(\Omega_{m}=0.28\)) with \(\chi_{tot}^{2}(\Omega_{e},w_{0})/dof\simeq 308.7/307\).
+
+### The Variable Chaplygin Gas - VCG model
+
+Let us consider now a completely different model namely the variable Chaplygin gas (hereafter VCG) which corresponds to a Born-Infeld tachyon action (Bento, Bertolami & Sen 2004). Recently, an interesting family of Chaplygin gas models was found to be consistent with the current observational data (Dev, Alcaniz & Jain 2003). In the framework of a spatially flat geometry, it can be shown that the normalized Hubble function takes the following formula:
+
+\[E^{2}(a)=\Omega_{b}a^{-3}+(1-\Omega_{b})\sqrt{B_{s}a^{-6}+(1-B_{s})a^{-n}}\;,\] (25)
+
+where \(\Omega_{b}\simeq 0.021h^{-2}\) is the density parameter for the baryonic matter (see Kirkman et al. 2003) and \(B_{s}\in[0.01,0.51]\) and \(n\in[-4,4]\). The effective matter density parameter is: \(\Omega^{eff}_{m}=\Omega_{b}+(1-\Omega_{b})\sqrt{B_{s}}\). We find that the best fit parameters are \(B_{s}=0.05\pm 0.02\) and \(n=1.58^{+0.35}_{-0.43}\) (\(\Omega^{eff}_{m}\simeq 0.26\)) with \(\chi_{tot}^{2}(B_{s},n)/dof\simeq 314.7/307\).
+
+## V Evolution of matter perturbations
+
+The evolution equation of the growth factor for models where the dark energy fluid has a vanishing anisotropic stress and the matter fluid is not coupled to other matter species is given by (Linder & Jenkins 2003)
+
+\[D^{{}^{\prime\prime}}+\frac{3}{2}\left[1-\frac{w(a)}{1+X(a)}\right]\frac{D^{{}^{\prime}}}{a}-\frac{3}{2}\frac{X(a)}{1+X(a)}\frac{D}{a^{2}}=0\;,\] (26)
+
+where
+
+\[X(a)=\frac{\Omega_{m}}{1-\Omega_{m}}{\rm e}^{-3\int_{a}^{1}w(y)d{\rm ln}y}=\frac{\Omega_{m}a^{-3}}{\delta H^{2}}\;.\] (27)
+
+Note, that the prime denotes derivatives with respect to the scale factor. Useful expressions of the growth factor can be found for the \(\Lambda\)CDM cosmology in Peebles (1993), for dark energy models with \(w=const\) in Silveira, & Waga (1994) for dark energy models with a time varying equation of state in Linder & Cahn (2007) and for the scalar tensor models in Gannouji & Polarski (2008). In this work the growth factor evolution for the current cosmological model is derived by solving numerically eq. (26). Note, that the growth factors are normalized to unity at the present time.
+
+### The formation of galaxy clusters
+
+In this section we briefly investigate the cluster formation processes by generalizing the basic equations which govern the behavior of the matter perturbations within the framework of the current dark energy models. Also we compare our predictions with those found for the traditional \(\Lambda\) cosmology. This can help us understand better the theoretical expectations of the dark energy models as well as the variants from the usual \(\Lambda\) cosmology.
+
+Figure 1: The predicted fractional rate of cluster formation as a function of redshift for the current cosmological models (\(\sigma_{8}=0.80\)). The points represent the following cosmological models: (a) BRG (open stars), (b) LRDE (open squares), (c) TCM (open triangles), (d) EDE (open circles) and (e) PNGB (solid triangles). The lines represent: (a) CPL model (long dashed), (b) HRDE model (dot line) and VCG model (dashed line).
+
+The concept of estimating the fractional rate of cluster formation has been proposed by different authors (eg., Weinberg 1987; Richstone, Loeb & Turner 1992). In particular, these authors introduced a methodology which computes the rate at which mass joins virialized structures, which grow from small initial perturbations in the universe. The basic tool is the Press & Schechter (1974) formalism which considers the fraction of mass in the universe contained in gravitationally bound structures (such as galaxy clusters) with matter fluctuations greater than a critical value \(\delta_{c}\), which is the linearly extrapolated density threshold above which structures collapse (Eke, Cole & Frenk 1996). Assuming that the density contrast is normally distributed with zero mean and variance \(\sigma^{2}(M,z)\) we have:
+
+\[{\cal P}(\delta,z)=\frac{1}{\sqrt{2\pi}\sigma(M,z)}{\rm exp}\left[-\frac{\delta^{2}}{2\sigma^{2}(M,z)}\right]\;\;.\] (28)
+
+In this kind of studies it is common to parametrize the rms mass fluctuation amplitude at 8 \(h^{-1}\)Mpc which can be expressed as a function of redshift as \(\sigma(M,z)=\sigma_{8}(z)=D(z)\sigma_{8}\). The current cosmological models are normalized by the analysis of the WMAP 5 years data \(\sigma_{8}=0.80\) (Komatsu et al. 2009). The integration of eq.(28) provides the fraction of the universe, on some specific mass scale, that has already collapsed producing cosmic structures (galaxy clusters) at redshift \(z\) and is given by Richstone et al. (1992):
+
+\[P(z)=\int_{\delta_{c}}^{\infty}{\cal P}(\delta,z)d\delta=\frac{1}{2}\left[1-{\rm erf}\left(\frac{\delta_{c}}{\sqrt{2}\sigma_{8}(z)}\right)\right]\;.\] (29)
+
+Notice, that for the model of modified gravity (BRG) we use \(\delta_{c}\simeq 1.47\) (see Schafer & Koyama 2008), for the EDE model we use \(\delta_{c}\simeq 1.4\) (see Bartelmann, Doran & Wetterich 2006). For the rest of the dark energy models, due to the fact that \(w\simeq-1\) close to the present epoch, we utilize a \(\delta_{c}\) approximation which is given by Weinberg & Kamionkowski (2003 see their eq.18).
+
+Obviously the above generic form of eq.(29) depends on the choice of the background cosmology. The next step is to normalize the probability to give the number of clusters which have already collapsed by the epoch \(z\) (cumulative distribution), divided by the number of clusters which have collapsed at the present epoch (\(z=0\)), \(F(z)=P(z)/P(0)\). In figure 1 we present, in a logarithmic scale, the behavior of normalized cluster formation rate as a function of redshift for the various dark energy models. In general, prior to \(z\sim 0.2\) the cluster formation has terminated due to the fact that the matter fluctuation field, \(D(z)\), effectively freezes. For the traditional \(\Lambda\) cosmology we find the known behavior in which galaxy clusters appear to be formed at high redshifts \(z\sim 2\) (Basilakos 2003; Basilakos, Sanchez & Perivolaropoulos 2009 and references therein). From figure 1 it becomes also clear, that the vast majority of the dark energy models seem to have a cluster formation rate which is close to that predicted by the usual \(\Lambda\) cosmology (see solid line in figure 1). However, for the BRG and EDE cosmological scenarios we find that galaxy clusters appear to form earlier (\(z\sim 3\)) than in the CPL, LRDE, HRDE, TCM, PNGB and VCG dark energy models.
+
+## VI Conclusions
+
+In this work we have studied analytically and numerically the overall dynamics of the universe for a large number of dark energy models beyond the _concrdance_\(\Lambda\) cosmology, by using several parameterizations for the dark energy equation of state. We performed a joint likelihood analysis, using the current high-quality observational data (SNIa, CMB shift parameter and BAOs), and we put tight constraints on the main cosmological parameters. We also find that for the vast majority of the dark energy models, the large scale structures (such as galaxy clusters) start to form prior to \(z\sim 1-2\).
+
+**Acknowledgments.** For this paper, I have benefited from discussions with L.Perivolaropoulos and M. Plionis. Therefore, I would like to thank both of them.
+
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