TrueDoc run 97 on olmOCR-bench: 86.8, with all 1,403 page outputs and the scorer's log (part 3)

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+therefore, \(\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\prime})=\langle\boldsymbol{V}_{1}(\mathbf{B}),(\mathbf{B}-\omega_{0}\mathbf{C}^{\prime})(\boldsymbol{V}_{1}(\mathbf{B}))\rangle=2\mu_{1}-\omega_{0}\xi\) since \(\mathbf{C}^{\prime}\in C\). So the strict inequality is asserting that \(2\mu_{1}-\omega_{0}\xi<\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\star})\). However, the arguments in Lemmas 4 and 5, that apply to \(\mathbf{C}^{\star}\) (since \(\mathbf{C}^{\star}\in C)\), show that \(\lambda_{1}(\mathbf{B}-\omega\mathbf{C}^{\star})\) is a concave function of \(\omega\) and that the linear function \(2\mu_{1}-\omega\xi\) is an upper bound for it. This violates the strict inequality in: \(\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\prime})=2\mu_{1}-\omega_{0}\xi< \lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\star})\leq2\mu_{1}-\omega_{0}\xi\). Let \(S\) denote the two dimensional subspace spanned by \(\boldsymbol{V}_{1}(\mathbf{B})\text{and}\boldsymbol{V}\text{and}\mathbf{C}_{S}^{\prime}\) denote the restriction of \(\mathbf{C}^{\prime}\) to this subspace (i.e., \(\mathbf{C}_{S}^{\prime}\) agrees with \(\mathbf{C}^{\prime}\) for any vector in this subspace and assigns 0 to any element outside). Also \(\boldsymbol{V}_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\prime})\in S\) by the definition of \(\boldsymbol{V}\text{and}\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}_{S}^{\prime})=\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\prime})\). Since \(\mathbf{C}^{\prime}\in C\), its restriction to \(S\) satisfies \(\langle\boldsymbol{V}_{1}(\mathbf{B}),\mathbf{C}_{S}^{\prime}(\boldsymbol{V}_{1}(\mathbf{B}))\rangle=\xi\). Also, \(\mathbf{C}_{S}^{\prime}≼\mathbf{B}/2\) since \(\mathbf{C}^{\prime}≼\mathbf{B}/2\). To satisfy the kissing constraint in Properties 2, we can add a component along \(\boldsymbol{V}:\mathbf{C}_{S}^{\prime\prime}:=\mathbf{C}_{S}^{\prime}+\delta\boldsymbol{V}\otimes\boldsymbol{V}\in C\) for some \(\delta\geq0\). Since \(\mathbf{C}_{S}^{\prime\prime}≽\mathbf{C}_{S}^{\prime}\) we have \(\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}_{S}^{\prime\prime})\leq\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}_{S}^{\prime})\). Proposition 3 establishes \(\mathbf{C}^{\star}\uparrow\mathbf{C}_{S}^{\prime\prime}\), a relationship that applies to all \(\omega\in[0,2]\), that guarantees:
+
+\[\begin{aligned} \lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\star})&\leq\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}_{S}^{\prime\prime})\leq\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}_{S}^{\prime}). \\ \text{This is in contradiction with the strict inequality}\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\star})&>\lambda_{1}(\mathbf{B}-\omega_{0}\mathbf{C}^{\prime}). \end{aligned}\]
+
+Proof of Theorem 1. Theorem 2 implies \(\lambda_{1}(\boldsymbol{B}^{\star}-\omega\boldsymbol{C}^{\star})=\lambda_{1}(\mathbf{B}-\omega\mathbf{C}^{\star})\leq\lambda_{1}(\mathbf{B}-\omega\mathbf{C})\text{for}\omega\in[0,2]\) that establishes \(C(\omega)\geq\lambda_{\text{max}}(\mathbf{A}(\omega))\) based on (14). The covariance \(\boldsymbol{\Sigma}_{k}\) converges to 0 according to the rate given by \(\lambda_{\text{max}}(\mathbf{A}(\omega))\) and the expected error at each step is the trace of \(\boldsymbol{\Sigma}_{k}:\mathbb{E}\left[\Vert\varepsilon_{k}\Vert^{2}\right]=\text{tr}\boldsymbol{\Sigma}_{k}.\text{When}i\) is drawn independently and identically distributed at each step of (1), the geometric rate of convergence is bound by \(C(\omega)\text{for every}\omega\in[0,2]\).
+
+### 4.4 Perron-Frobenius Theory For Positive Linear Maps
+
+The superoperator \(\mathbf{A}\) defined in (7) plays the role of the iteration matrix — whose spectrum provides convergence analysis in classical iterative methods [Saad, 2003] — for randomized iterations. In this section we discuss the theoretical foundations that provide necessary properties on the spectrum of \(\mathbf{A}\) in the covariance analysis we have seen. Recall the superoperator \(\mathbf{A}\), for a fixed \(\omega\), denotes a linear map over the space of \(n\times n\) matrices as:
+
+\[\mathbf{A}(\boldsymbol{X})= \frac{1}{m} \sum_{i=1}^{m}(\boldsymbol{I}-\omega\boldsymbol{P}_{i})\boldsymbol{X}(\boldsymbol{I}-\omega\boldsymbol{P}_{i}).\]
+
+Since orthogonal projection is a symmetric operator, for any symmetric positive semi-definite matrix \(\boldsymbol{X}\) the operation \((\boldsymbol{I}-\omega\boldsymbol{P}_{i})\boldsymbol{X}(\boldsymbol{I}-\omega\boldsymbol{P}_{i})\) preserves its positivity [Bhatia, 2009]. Hence the superoperator \(\mathbf{A}\) is a positive linear map, leaving the cone of symmetric positive semi-definite matrices invariant. The spectra of positive linear maps on general (noncommutative) matrix algebras was studied in [Evans and Høegh-Krohn, 1978] that generalized the Perron-Frobenius theorem to this context. The spectral radius of a positive linear map is attained by an eigenvalue for which there exists an eigenvector that is positive semi-definite (see Theorem 6.5 in [Wolf, 2012]). The notion of irreducibility for positive linear maps guarantees that the eigenvalue is simple and the corresponding eigenvector is well-defined (up to a sign). What is more is that the eigenvector can be chosen to be a positive definite matrix. This guarantees that the power iterations in (7) converge along this positive definite matrix with the corresponding simple eigenvalue giving the rate of convergence. For a system of equations in \(\boldsymbol{A}\mathbf{x}=\mathbf{b}\), we examine the irreducibility of its corresponding superoperator \(\mathbf{A}\) for any given relaxation value \(\omega\). The criteria for irreducibility of positive linear maps was developed in [Farenick, 1996] and involve invariant subspaces. A collection \(S\) of (closed) subspaces of the vector space of \(n\times n\) matrices is called trivial if it only contains \(\{\mathbf{0}\}\) and the space itself. Given a bounded linear operator \(\boldsymbol{M}\), let \(\text{Lat}(\boldsymbol{M})\) denote the invariant subspace lattice of \(\boldsymbol{M}\). The following theorem is a specialization of a more general result in [Farenick, 1996] (see Theorem 2) to our superoperator. Theorem 3 (Irreducibility of the superoperator \(\mathbf{A})\). The positive linear map \(\mathbf{A}\) is irreducible if and only if, \(\bigcap_{i=1}^{m}\text{Lat}(\boldsymbol{I}-\omega\boldsymbol{P}_{i})\) is trivial. Based on this theorem, we establish the equivalence of the irreducibility of \(\mathbf{A}\), in the sense of positive linear maps, to a geometric notion of irreducibility defined for alternating projections (2) that is inherently a geometric approach to solving a system of equations \(\boldsymbol{A}\mathbf{x}=\mathbf{b}\). We recall the Frobenius notion of irreducibility for symmetric matrices. Such a matrix \(\boldsymbol{M}\) is called irreducible if it can not be transformed to block diagonal form by a permutation matrix \(\boldsymbol{\Pi}\):
+
+\[\boldsymbol{M}=\boldsymbol{\Pi} \begin{bmatrix} \boldsymbol{M}^{\prime} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{M}^{\prime\prime} \end{bmatrix} \boldsymbol{\Pi}^{-1},\]

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+\(1)\Omega(t,t)=\) id, \(\Omega(t,t_{0})=\Omega(t,s)\Omega(s,t_{0})\text{forall}t,s,t_{0}\in[\tau,\infty)\);
+
+2. and for almost all \(t,s\in[\tau,\infty)\) it holds that
+
+\[\frac{\frac{\partial}{\partial\partial}}{\partial s} \overline{t} \Omega^{\Omega}(t^{(},^{t,}s^{s})^{)}=^{=}-^{A}A^{(t}(^{)}s^{\Omega})\Omega^{(t}(^{,}t^{s},^{)}s^{,}).\] (3)
+
+A non-empty, closed, convex subset \(K\subset\mathscr{H}\) is called a cone, if \(\mathbb{R}_{+}K\subset K\text{and}K\bigcap(-K)=\{0\}\) hold. A cone K is called
+
+generating if any \(x\in\mathscr{H}\) can be written as \(x=x^{+}-x^{-}\)
+
+for some \(x^{\pm}\in K\). In a Hilbert space any generating cone is nonflat, that is there exists a constant \(a_{K}>0\), independent
+
+on x, such that in the representation \(x=x^{+}-x^{-}\) the vectors
+
+\(x^{\pm}\) can be chosen so that
+
+\[\Vert x^{\pm}\Vert\leq a_{K}\Vert x\Vert.\] (4)
+
+The set \(K^{*}=\{f\in L(\mathscr{H};\mathbb{R}):f(K)\subset\mathbb{R}_{+}\}\subset\mathscr{H}^{*}=\mathscr{H}\) is
+
+called adjoint cone. If K is generating, then \(K^{*}\) is normal, i.e., there is a constant \(b_{K}>0\text{such that}x\in K,y-x\in K\) imply
+
+\(\Vert x\Vert\leq b_{K}\Vert y\Vert\). A cone K is called selfadjoint if \(K^{*}=K\). In a Hilbert space any selfadjoint cone is normal [13]. Definition 1: Let K be a generating and selfadjoint cone in \(\mathscr{H}\) . System (1) is called Wazewski [14] with respect to K (also called monotone [15]) if its evolution operator \(\Omega\) satisfies \(\Omega(t,s)K\subset K\text{for all}t\geq s\geq\tau\). Definition 2: We say that (1) is stable in K if for any \(\varepsilon>0\) and any \(t_{0}\geq\tau\) there is a \(\delta=\delta(\varepsilon,t_{0})>0\) such that \(x_{0}\in B_{\delta}\cap K\Longrightarrow \Vert x(t;t_{0},x_{0})\Vert<\varepsilon,t\geq t_{0}\); if, in addition, the \(\delta\) can be chosen independent on t0, then (1) is called uniformly stable in K; asymptotically stable in K if it is stable in K and there is \(\eta=\eta(t_{0})>0\text{such that}x_{0}\in B_{\eta}\cap K\Longrightarrow \lim_{t\rightarrow\infty}\Vert x(t;t_{0},x_{0})\Vert=0\); if, in addition, \(\eta\) can be chosen independent on t0, then (1) is called uniformly asymptotically stable in K. If any of the above properties holds with \(\mathscr{H}\) on the place of K, then we drop ”in K” in their definitions. Let system (1) be decomposed in two subsystems as follows
+
+\[\dot{x}_{i}=A_{ii}(t)x_{i}+A_{ij}(t)x_{j},i\neq j,i,j=1,2,(5)\]
+
+with \(x_{i}\in\mathscr{H}_{i},A_{ij}\in L(H_{j},H_{i})\text{for}i,j=1,2\). Let Ki be a solid selfadjoint cone in \(\mathscr{H}_{i},\text{then}K=K_{1}\bigoplus K_{2}\) is a solid and selfadjoint cone in \(\mathscr{H}\) . Let \(\Omega_{i}\in C([\tau,\infty)\times[\tau,\infty);L(\mathscr{H}_{i}))\) denote the evolution operator of the decoupled subsystem
+
+\[\dot{y}_{i}=A_{ii}(t)y_{i},y_{i}\in\mathscr{H}_{i},i=1,2\] (6)
+
+It can be verified by definition that (1), written as (5), is a Wazewski system for the cone K, if and only if
+
+\[\Omega_{i}(t,s)K_{i}\subset K_{i}\text{and}A_{ij}(t)K_{j}\subset K_{i},t\geq s\geq\tau.(7)\]
+
+Definition 3: An operator \(O\in L(\mathscr{H}_{j};\mathscr{H}_{i})\) is called positive, if it satisfies \(OK_{j}\subset K_{i}\) and it is denoted by \(O\geq0\). For the evolution operator \(\Omega_{i}\) of (6) let \(\alpha_{i},\beta_{i},\gamma_{i},\delta_{i}\in C([\tau,\infty);\mathbb{R}_{>0})\) be such that
+
+\[\Vert\Omega_{i}(^{\Vert}t^{\Omega},s^{i})^{(}\Vert^{t,}\leq^{s)\Vert}(\gamma^{\leq_{i}}(t^{\alpha}))^{(_{i_{-}}t_{1})}(^{\beta}\delta_{i}^{(_{i}}(^{s}s^{)})^{,})^{t_{-1}},^{\geq s}s^{\geq}\geq^{\tau}t^{,}\geq\tau,^{(8)}\]
+
+for example we can fix any \(p\in(s,t)\) and take
+
+\(\alpha_{i}(t)=\Vert\Omega_{i}(t,p)\Vert,\beta_{i}(s)=\Vert\Omega_{i}(p,s)\Vert\) due to \(\Omega_{i}(t,s)= \Omega_{i}(t,p)\Omega_{i}(p,s)\). Let \(q_{i}\in C([\tau,\infty);\mathbb{R}_{>0})\) be suitable weight functions guaranteeing convergence of the next integrals for \(t\in[\tau,\infty)\)
+
+\[\begin{aligned} \phi_{i}(t):&=\gamma_{i}^{2}(t)\int_{t}^{\infty}q_{i}(s)\delta_{i}^{2}(s)ds, \\ g_{i}(t):&=\beta_{i}^{2}(t)\int_{t}^{\infty}q_{i}(s)\alpha_{i}^{2}(s)ds. \end{aligned}\]
+
+## III. MAIN RESULTS
+
+For simplicity and clearness in this work we consider the case of r = 2 interconnected subsystems in (5). An extension for \(r\in\mathbb{N}\) will developed elsewhere. We introduce the following notation for \(1\leq i\neq j\leq2\): linear weighted integral gains for the interconnection (5) are defined as
+
+\[\begin{aligned} \pi_{ii}(t_{0})&=2\sup_{tt_{0}}\omega_{i}(t)\beta_{i}^{2}(t) \\ \times\int_{t}^{\infty}\alpha_{i}(s)\Vert A_{ji}(s)\Vert\beta_{j}(s)\int_{s}^{\infty\geq} \frac{\alpha_{i}(p)\alpha_{j}(p)\Vert A_{ij}(p)\Vert}{\omega_{i}(p)} dpds,^{(10)} \end{aligned}\]
+
+\[\begin{aligned} \pi_{ji}(t_{0})&=2\sup_{tt_{0}}\omega_{i}(t)\beta_{i}^{2}(t) \\ \times\int_{t}^{\infty}\alpha_{i}(s)\Vert A_{ji}(s)\Vert\beta_{j}(s)\int_{s}^{\infty\geq} \frac{\alpha_{i}(p)\alpha_{j}(p)\Vert A_{ji}(p)\Vert}{\omega_{j}(p)} dpds,^{(11)} \end{aligned}\]
+
+where \(\omega_{i}\) are suitable weight functions, which can help to enable convergence of the integrals. Also for \(f\in C([t_{0},T];L(\mathscr{H}_{i}))\) we introduce the weighted norm
+
+\[\Vert f\Vert_{\omega_{i},T}:=\max_{t\in[t_{0},T]}\omega_{i}(t)\Vert f(t)\Vert.\] (12)
+
+Theorem 1: Let (1) be a Wazewski system with respect to a selfadjoint solid cone K and written as (5) with r = 2. Let \(\alpha_{i},\beta_{i},\gamma_{i},\delta_{i}\) be as in (8) and \(\phi_{i},g_{i},q_{i},\omega_{i},\pi_{ij}\) as in (9),(10),(11).
+
+If the spectral radius of the matrix \(\Pi(t_{0})\in\mathbb{R}^{2\times2}\) defined by the weighted integral gains \(\pi_{ij}(t_{0})\) satisfies
+
+\[r_{\sigma}(\Pi(t_{0}))<1\] (13)
+
+then solutions to (1) satisfy the estimate
+
+\[\Vert x(t;t_{0},x_{0})\Vert\leq2a_{K} \sqrt{\frac{h(t_{0},t_{0})}{\phi(t)}} \exp\left(-\int_{t_{0}}^{t} \frac{q(s)}{2h(s,t_{0})} ds\right)\Vert x_{(^{0}1}\Vert,_{4)}\]
+
+where
+
+aK is from \((4),q(t):=\min\{q_{1}(t),q_{2}(t)\}\), \(\phi(t):=\min\{\phi_{1}(t),\phi_{2}(t)\}\text{and}h(t,t_{0}):=\max\{h_{11}(t,t_{0})+\)

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+# so we can get
+
+\[\begin{cases} \tfrac{dx_{1}}{dt} =x_{2}+ \tfrac{b_{11}}{b_{10}} x_{1}x_{2}+ \tfrac{b_{20}}{b_{10}^{2}} x_{2}^{2}+ \tfrac{b_{12}}{b_{10}} x_{1}^{2}x_{2}+ \tfrac{b_{21}}{b_{10}^{2}} x_{1}x_{2}^{2}+ \tfrac{b_{30}}{b_{10}^{3}} x_{1}^{3}+O(\Vert x\Vert^{4}) \\ \tfrac{dx_{2}}{dt} =a_{11}x_{1}x_{2}+ \tfrac{a_{20}}{b_{10}} x_{2}^{2}+ \tfrac{a_{21}}{b_{10}} x_{1}x_{2}^{2}+a_{12}x_{1}^{2}x_{2}+ \tfrac{a_{30}}{b_{10}^{2}} x_{2}^{3}+O(\Vert x\Vert^{4}) \end{cases}\]
+
+\[\begin{cases} x_{3}=x_{1} \\ x_{4}=x_{2}+ \tfrac{b_{11}}{b_{10}} x_{1}x_{2}+ \tfrac{b_{20}}{b_{10}^{2}} x_{2}^{2}+ \tfrac{b_{12}}{b_{10}} x_{1}^{2}x_{2}+ \tfrac{b_{21}}{b_{10}^{2}} x_{1}x_{2}^{2}+ \tfrac{b_{30}}{b_{10}^{3}} x_{1}^{3}+O(\Vert x\Vert^{4}) \end{cases}\]
+
+\[\begin{cases} x_{1}=x_{3} \\ x_{2}=x_{4}+v_{11}x_{3}x_{4}+v_{02}x_{4}^{2}+v_{21}x_{3}^{2}x_{4}+v_{12}x_{3}x_{4}^{2}+v_{03}x_{4}^{3}+O(\Vert x\Vert^{4}) \end{cases}\]
+
+# wherein:
+
+\[\begin{aligned} v_{11}&=- \frac{b_{11}}{b_{10}} ,v_{02}=- \frac{b_{20}}{b_{10}^{2}} , \\ v_{21}&= \frac{b_{11}^{2}-b_{12}b_{10}}{b_{10}^{2}} . \\ v_{12}&= \frac{3b_{11}b_{20}-b_{21}b_{10}}{b_{10}^{3}} ,v_{03}= \frac{2b_{20}^{2}}{b_{10}^{4}} . \end{aligned}\]
+
+# Substitute the new variables \((x_{3},x_{4})\) into the original system, and we can get:
+
+\[\begin{cases} \tfrac{dx_{3}}{dt} =x_{4} \\ \tfrac{dx_{4}}{dt} =e_{11}x_{3}x_{4}+e_{02}x_{4}^{2}+e_{21}x_{3}^{2}x_{4}+e_{12}x_{3}x_{4}^{2}+e_{03}x_{4}^{3}+O(\Vert x\Vert^{4}) \end{cases}\] (14)
+
+# wherein:
+
+\[\begin{aligned} e_{11}&=a_{11},e_{02}= \frac{a_{20}+b_{21}}{b_{10}} , \\ e_{21}&=a_{12}+ \frac{a_{11}b_{11}}{b_{10}} +a_{11}v_{11}, \\ e_{12}&=a_{12}v_{02}+ \frac{(a_{20}+b_{11})b_{10}+a_{21}b_{10}+2a_{11}b_{20}+a_{20}b_{11}+b_{11}^{2}+2b_{10}b_{12}}{b_{10}^{2}} , \\ e_{03}&= \frac{2(a_{20}+b_{11})b_{10}^{2}v_{02}+a_{30}+b_{21}+2a_{20}b_{20}+b_{11}b_{20}}{b_{10}^{3}} . \end{aligned}\]
+
+# Then make the following transformation for the system (14):
+
+\[\begin{cases} x_{5}=x_{3} \\ x_{6}=x_{4}-e_{02}x_{3}x_{4} \end{cases}\]

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+[19] F. A. Potra and V. Pták, Nondiscrete Induction and Iterative Processes, Pitman, Boston, Massachusetts, 1984.
+
+[20] L. B. Rall, Convergence of the Newton process to multiple solutions, Numer. Math., 9 (1966), pp. 23–37.
+
+[21] S. J. Reddi, S. Kale, and S. Kumar, On the convergence of adam and beyond, in International Conference on Learning Representations, 2018.
+
+[22] G. W. Reddien, On Newton’s method for singular problems, SIAM J. Numer. Anal., 15 (1978), pp. 993–996.
+
+[23] H. Royden and P. Fitzpatrick, Real Analysis, Pearson, 4th ed., 2010.
+
+[24] H. Schwetlick, Numerische Lösung Nichtlinearer Gleichungen, VEB, Berlin, Germany, 1979.
+
+[25] W. Sun and Y.-X. Yuan, Optimization Theory and Methods: Nonlinear Programming, vol. 1, Springer, 2006.
+
+[26] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964. Reprinted by Chelsea Publishing Company, 1982; Reprinted by AMS Chelsea Publishing, 2016.
+
+## Appendix A. Q-Superlinear Implies UP-Superlinear.
+
+Proof of Theorem 3.2. Define
+
+\[\xi_{k}=\Vert\boldsymbol{x}_{k}-\boldsymbol{x}_{*}\Vert,f(k)=-\ln\xi_{k}.\]
+
+By Q-superlinear convergence,
+
+\[\lim_{k\rightarrow\infty} \frac{\xi_{k+1}}{\xi_{k}^{q}} =Q_{q},q>1,0<Q_{q}<\infty.\]
+
+# Taking logarithms, define
+
+\[(\mathrm{A}.1)d(k)=f(k+1)-qf(k)=-\ln\left( \frac{\xi_{k+1}}{\xi_{k}^{q}} \right).\]
+
+# Then
+
+\[\lim_{k\rightarrow\infty}d(k)=-\ln Q_{q}:=d,\]
+
+## so that \(\{d(k)\}\) is bounded; i.e., there exists \(M>0\text{with}|d(k)|\leq M\). Unrolling the recurrence
+
+\[f(k+1)=qf(k)+d(k)\]
+
+# yields
+
+\[f(k)=q^{k-1}f(1)+\sum_{j=1}^{k-1}q^{k-1-j}d(j).\]
+
+# Dividing by \(q^{k}\), we have
+
+\[\frac{f(k)}{q^{k}} = \frac{f(1)}{q} +\sum_{j=1}^{k-1} \frac{d(j)}{q^{j+1}} .\]
+
+# Since the tail of the series is bounded by
+
+# the limit
+
+\[\sum_{j=k}^{\infty} \frac{|d(j)|}{q^{j+1}} \leq M\sum_{j=k}^{\infty} \frac{1}{q^{j+1}} ,\]
+
+\[(\mathrm{A}.2)s=\lim_{k\rightarrow\infty} \frac{f(k)}{q^{k}} = \frac{f(1)}{q} +\sum_{j=1}^{\infty} \frac{d(j)}{q^{j+1}}\]

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+# Learning Spatially Adaptive \(\ell_{1}-\)Norms Weights for Convolutional Synthesis Regularization
+
+| Andreas Kofler | Luca Calatroni |
+| --- | --- |
+| Physikalisch-Technische Bundesanstalt (PTB) | MaLGa Center, DIBRIS, |
+| Braunschweig and Berlin, Germany | Universita di Genova, |
+
+andreas.kofler@ptb.de
+
+| Christoph Kolbitsch | Kostas Papafitsoros |
+| --- | --- |
+| Physikalisch-Technische Bundesanstalt (PTB) | School of Mathematical Sciences |
+| Braunschweig and Berlin, Germany | Queen Mary University of London |
+| christoph.kolbitsch@ptb.de | London, UK |
+
+Abstract—We propose an unrolled algorithm approach for learning spatially adaptive parameter maps in the framework of convolutional synthesis-based \(\ell_{1}\) regularization. More precisely, we consider a family of pre-trained convolutional filters and estimate deeply parametrized spatially varying parameters applied to the sparse feature maps by means of unrolling a FISTA algorithm to solve the underlying sparse estimation problem. The proposed approach is evaluated for image reconstruction of low-field MRI and compared to spatially adaptive and non-adaptive analysis-type procedures relying on Total Variation regularization and to a well-established model-based deep learning approach. We show that the proposed approach produces visually and quantitatively comparable results with the latter approaches and at the same time remains highly interpretable. In particular, the inferred parameter maps quantify the local contribution of each filter in the reconstruction, which provides valuable insight into the algorithm mechanism and could potentially be used to discard unsuited filters. Index Terms—Neural Networks, Convolutional Dictionary Learning, Sparsity, Adaptive Regularization, Low-Field MRI
+
+## I. INTRODUCTION
+
+Convolutional Sparse Coding (CSC) and Convolutional Dictionary Learning (CDL) approaches [1] rely on the assumption that the signal/image of interest \(\mathbf{x}\in\mathbb{R}^{N}\) (or \(\mathbb{C}^{N})\) is well-approximated by a linear combination of convolutions of normalized filters \(d_{k}\in\mathbb{R}^{k_{f}\times k_{f}}\) with sparse feature maps \(\mathbf{s}_{k}\in\mathbb{R}^{N}\) (or \(\mathbb{C}^{N})\text{for}k=1,\ldots,K\), that is
+
+\[\mathbf{x}\approx\sum_{k=1}^{K}d_{k}*\mathbf{s}_{k},\text{where}\forall k\mathbf{s}_{k}\text{is sparse}.(1)\]
+
+The project (22HLT02 A4IM) has received funding from the European Partnership on Metrology, co-financed from the European Union’s Horizon Europe Research and Innovation Programme and by the Participating States. LC acknowledges the financial support of the European Research Council (grant MALIN, 101117133).
+
+MMS, Istituto Italiano ` di Tecnologia, Genoa, Italy luca.calatroni@unige.it
+
+k.papafitsoros@qmul.ac.uk
+
+By enforcing sparsity by means of \(\ell_{1}-\text{regularization}\), given a set of signals \(\{\mathbf{x}_{l}\}_{l=1}^{L}\text{and some}\lambda>0\), the CDL problem can be thus typically formulated as:
+
+\[\begin{aligned} \min_{\{d_{k}\},\{\mathbf{s}_{k}\}} \frac{1}{2} \sum_{l=1}^{L}\Vert\mathbf{x}_{l}-\sum_{k=1}^{K}d_{k}*\mathbf{s}_{k,l}\Vert_{2}^{2}+\lambda\sum_{l=1}^{L}\sum_{k=1}^{K}\Vert\mathbf{s}_{k,l}\Vert_{(^{1}2)} \\ \text{such that}\Vert d_{k}\Vert_{2}&=1,\forall k=1,\ldots,K. \end{aligned}\]
+
+In the context of image reconstruction, given possibly incomplete observed data \(\mathbf{y}\in V\) modeled as the noisy output of a linear forward operator A : \(V\rightarrow W\) , CDL and CSC are employed as variational regularization methods, see e.g. [2]:
+
+\[\begin{aligned} \min_{\mathbf{x},\{d_{k}\},\{\mathbf{s}_{k}\}} \frac{1}{2} \Vert\mathbf{Ax}-\mathbf{y}\Vert_{2}^{2}+ \\ \frac{\alpha}{2} \sum_{k=1}^{K}\Vert\mathbf{x}-d_{k}*\mathbf{s}_{k}\Vert_{2}^{2}+\lambda\sum_{k=1}^{K}\Vert\mathbf{s}_{k}\Vert_{1}^{(3)} \\ \text{uch that}\Vert d_{k}\Vert_{2}&=1,\forall k=1,\ldots,K, \end{aligned}\]
+
+where \(\alpha>0\) enforces the synthesis constraint. Note that in
+
+(3) it is implicitly assumed that the underlying sparsifying model is unknown so that the elements \(\mathbf{x},\{d_{k}\},\{\mathbf{s}_{k}\},k= 1\ldots,K\) are reconstructed jointly, typically by alternating the minimization between solving (2) and computing the current update of desired image depending on the estimated dictionary. These methods are thus often referred to as blind Compressed Sensing approaches, see, e.g. [3] for MRI applications. Problem (3) is a non-convex optimization problem whose solution is, typically, computationally demanding, and further requires careful tuning of the regularization parameters \(\lambda,\alpha\), initializations \(\mathbf{x}^{0},\{d_{k}^{0}\},\{\mathbf{s}_{k}^{0}\}\) and, depending on the algorithms considered, possibly also of other parameters coupling the image update and the CSC and CDL, see, e.g. [4] for ADMM schemes. Furthermore, the regularization parameter \(\lambda\) dictates the strength of the imposed regularization in terms of the sparsity of the feature maps only globally and independently

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+4 3
+
+2
+
+where \(e_{\gamma,\rho}(x_{1},t_{1},x_{2},t_{2})=-\infty\) for any other choice of \(x_{1},x_{2}\). Then the metric \(e\) arising from planting measures along a finite or countable set of curves is the smallest metric which is bounded below by satisfying \(e\geq d\), and for each curve \(\gamma_{i}\) and planted measure \(\rho_{i},e\geq e_{\gamma_{i},\rho_{i}}\). All metrics not of this form have infinite rate. For each such planted measure, the contribution to the rate function is \(I_{i}= \int\rho \frac{3}{i} dt\), so that the rate function \(I\) of a metric \(e\) is the sum over all planted measures \(I=\sum_{i}I_{i}\). Our goal in this paper is to find the lowest rate metric for which a geodesic \(F\) from \((0,0)\) to \((0,1)\) passes through the point \((1,a)\). Our metrics will generally have a measure planted along just one curve, which will be the geodesic from \((0,0)\) to \((0,1)\). Any measure planted outside of this geodesic will not increase its length, but may increase the length of paths other than \(F\) , which only reduces the possibility of \(F\) being a geodesic, all while increasing the rate. We can then define \(e\text{or}e(F,\rho)\) as the metric arising from planting \(\rho\) along \(F\) , which will be our geodesic. We will in fact prove a result about \(e\): Suppose \(\mathcal{M}_{\varepsilon}\) is a metric sampled from the conditional law of \(\mathcal{L}_{\varepsilon}\), given that the rightmost geodesic \(\gamma\) of \(e\) from \((0,0)\) to \((0,1)\) satisfies \(\gamma(a)\geq1\).
+
+Theorem 1.3. As \(\varepsilon\rightarrow0,\mathcal{M}_{\varepsilon}\rightarrow e(F,\rho)\) in probability with respect to uniform convergence on bounded sets, with \(F\) as in Theorem 1.2 and \(\rho\) as given by:
+
+\[\rho(t)= \begin{cases} \frac{1}{a^{2}(3- \sqrt{8a} )^{2}} , & ift\leq2a \\ \frac{2}{at(3- \sqrt{8a} )^{2}} , & if2a\leq t\leq1. \end{cases}\]
+
+## 2 Solution of a relaxed problem
+
+One possible candidate for \(F\) is two straight line segments, from \((0,0)\) to \((1,a)\) and from \((1,a)\) to \((0,1),\text{with}\rho\) just sufficient to compensate \(d\), such that \(e(0,0,F(t),t)=0,\forall t\in[0,1]\). However, for \(a< \frac{1}{2}\), this will not be a geodesic: instead, there will be straight lines from \((0,0)\) to a point beyond \((1,a)\) whose Dirichlet length is greater than the length of \(F\) to that point. As a result, the geodesic will skip the steeper line segment to \((1,a)\) and take such a straight line as a shortcut to \(F\) , and then follow part of the other line segment to \((0,1)\). Similarly, if \(a> \frac{1}{2}\), it will follow part of the line segment from \((0,0)\) to \((1,a)\) and then shortcut the steep segment to proceed directly to \((0,1)\). This suggests that we should try to solve for a metric where the segment from \((0,0)\) to \((1,a)\) is not skipped by such a shortcut. In other words, we should ask what is the lowest rate metric for which there is a geodesic candidate \(F\) , so that the distance from \((0,0)\) along \(F\) to any point on \(F\) is greater than the distance of a shortcut from \((0,0)\) to that point. Inspired by this intuition, we will first solve an intermediate optimization problem: fix \(F_{0}\text{and}\rho_{0}\) (and therefore \(e(F_{0},\rho_{0}))\text{for}t\in[0,t_{1}]\). Consider pairs \((F,\rho)\) that agree with \((F_{0},\rho_{0})\text{on}[0,t_{1}]\). Which pair minimizes the rate of the

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+The maps \(L_{I}\rightarrow L_{J}\) here are induced by the ideal inclusions \(O_{\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)}(-D(I))֒\rightarrow O_{\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)}\). We call this screen the universal screen on \(\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\). Definition 4.2. Let \(X_{n}\) : Set \(\rightarrow\mathbf{Sch}^{op}\) be the functor which sends a scheme \(H\) to the set of screens on \(H\) up to isomorphism of the screen data. Similarly let \(X(I_{1},\ldots,I_{k})\) denote the subfunctor sending \(H\) to the set of screens on \(H\) up to isomorphism which satisfy the \(I_{r}-\)vanishing property for each \(1\leq r\leq k\). Proposition 4.3. The Fulton-MacPherson compactification \(\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\) represents the functor \(X_{n}\). Furthermore, the natural isomorphism \(\hom(-,\text{FM}_{n}^{\mathrm{a}\text{lg}}(X))\rightarrow X_{n}\) is given by sending the element \(f:H\rightarrow\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\) to the pullback to \(H\) of the universal screen on \(\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\) described above. Similarly \(D(I_{1},\ldots,I_{n})\subseteq\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\) represents \(X(I_{1},\ldots,I_{k})\). The natural isomorphism is given in the same way here as above.
+
+Proof. This is theorem 4 in [FM94].
+
+### 4.2 Definition and Properties of \(T_{d,n}\)
+
+In this section we recall some properties of the \(T_{d,n}\) spaces introduced by Chen, Gibney, and Krashen [CGK09]. We also introduce a functor of points for these spaces and use this to define an operad with spaces \(\{T_{d,n}\}_{n\in\mathbb{N}}\). If \(\text{pr}_{i}:X^{n}\rightarrow X\) denotes the projection to the \(i\):th component it is clear from definitions that the composition
+
+\[D([n])֒\rightarrow\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\longrightarrow ^{\rho}X^{n}-^{\mathrm{p}}-^{\mathrm{r}}\rightarrow^{i}X\]
+
+is independent of the integer \(i\). We denote this morphism by \(\pi:D([n])\rightarrow X\). In [CGK09], Chen, Gibney, and Krashen show that for a fixed \(n\), any smooth scheme \(X\) of dimension \(d\), and any point \(x\in X\), the fibers \(\pi^{-1}(x)\) are isomorphic. We define \(T_{d,n}\text{as}\pi^{-1}(0)\) for the map \(\pi:D([n])\rightarrow\mathbb{A}^{d}\) but by the above we could equally well have taken any other smooth variety instead of \(X=\mathbb{A}^{d}\). Remark. Just like the topological case notice that while we have up until this point indexed coordinates with \([n]\) we are free to use any finite index set \(I\text{and FM}_{I}^{\mathrm{a}\text{lg}}(X)\text{and}T_{d,I}\) together with all related maps and divisors are defined in the analogous way. The variety \(T_{d,n}\) is smooth and comes equipped with a smooth closed subscheme \(T_{d,n}(I)\) for every non empty \(I\subseteq[n].\text{For}I\in P_{2}(n)\), i.e. \(I/=[n]\text{and}|I|\geq2\), we define \(T_{d,n}(I)\) as the pullback of \(D(I)\subseteq\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\) to \(T_{d,n}\). For other \(I\) (i.e. \(I\) is \([n]\) or a one point set) we take \(T_{d,n}(I):=T_{d,n}\). Similarly, for a collection of (distinct) subsets \(I_{1},\ldots,I_{k}\subseteq[n]\) we define \(T_{d,n}(I_{1},\ldots,I_{k}):=\bigcap_{i}T_{d,n}(I_{i}).\text{For}I/=[n]\text{and}|I|\geq2\) the subscheme \(T_{d,n}(I)\) is a smooth divisor of \(T_{d,n}\) and the union of all these divisors is a smooth normal crossings divisor. Additionally, the intersection \(T_{d,n}(I_{1},\ldots,I_{k})\) is non empty if and only if \((I_{1},\ldots,I_{k})\) is nested. All this essentially follows from the analogous results for \(\text{FM}_{n}^{\mathrm{a}\text{lg}}(X)\) recalled in Section 4.1. Details can be found in [CGK09]. Remark. The reason we include definitions of \(T_{d,n}(\{i\})\text{and}T_{d,n}([n])\) is that this will simplify the notation for some later results. The first goal of this section will be to give a functor of points description of the schemes \(T_{d,n}(I_{1},\ldots,I_{k})\). For this I will introduce the following Definition 4.4. For a scheme \(H\), an integer \(d\), and a finite, non-empty, set \(I\) with at least two element we let \(F^{H,d}\)
+
+denote the quasi-coherent sheaf of modules defined by
+
+\(I\)
+
+Here \(\Bigg\langle\)
+
+\[\{t_{ij}^{k}\}_{i,j\in I}^{1\leq k\leq d}\]
+
+\[F_{I}^{H,d}:=\left\langle\{t_{ij}^{k}\}_{i,j\in I}^{1\leq k\leq d}\right\rangle\Big/\sim.\]
+
+\(\Bigg\rangle\) denotes the free module with generators \(t_{ij}^{k}\text{and}\sim\) is generated by sections of the form \(t_{ij}^{k}+t_{jl}^{k}-t_{il}^{k}\). We will also denote this by \(F_{I}^{d}\text{or}F_{I}\text{if}H\text{or}H,d\) are clear from context.

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+Corollary 11.7. Suppose that \(f^{\prime}=\sum_{\tau}p_{\tau,!}(f^{\tau})\text{and}\tilde{f}\) is a nice transfer. Then we have the identity
+
+\[\frac{1}{2^{1-|S|}L(0,\eta)} \sum_{\tau\in\mathcal{V}_{n}(E/F)}\sum_{\Pi}J_{\Pi}^{\tau,\tau_{0}}(f^{\tau})=\sum_{\pi}I_{\pi}(\tilde{f}),\]
+
+where the left-hand sum ranges over irreducible cuspidal representations \(\Pi\) of \(\text{GL}_{n}(\mathbb{A}_{E})\) with central character \(\omega^{\prime}\) and the right-hand sum ranges over irreducible cuspidal automorphic representations of \(\text{GL}_{n}(\mathbb{A}_{F})\) with central character \(\omega or\omega\eta\).
+
+To obtain the necessary refined comparison, we make another use of [Ram18, Theorem A]. We introduce the following terminology to simplify a few statements.
+
+Definition 11.8. Fix a character \(\omega\) of \(Z(\mathbb{A}_{F})\) and set \(\omega^{\prime}=\omega\circ\text{Nm}_{E/F}\) . Fix a split place \(v_{1}\) and a supercuspidal representation \(\pi_{v_{1}}\) of \(\text{GL}_{n}(F_{v_{1}})\). We say that the functions
+
+\[\tilde{f}\in C_{c}^{\infty}(\text{GL}_{n}(\mathbb{A}_{F})\times\mathbb{A}_{F,n})\text{and}\{f^{\tau}\}_{\tau\in\mathcal{V}_{n}(E/F)}\subset C_{c}^{\infty}(\text{GL}_{n}(\mathbb{A}_{E}))\]
+
+are efficient transfers for \((\omega,\pi_{v_{1}})\) if they give a \(\omega-\)nice transfer such that \(f_{v_{1}}\in C_{c}^{\infty}(\text{GL}_{n}(F_{v_{1}}))_{\Omega}\) where \(\Omega\) is the block of \(\pi_{v_{1}}(f_{v_{1}}\) is said to be essentially a matrix coefficient of \(\pi_{v_{1}}\) in this case) and that \(f\tau_{nv_{1}}\) is related to \(f_{v_{1}}\) as in Proposition 9.12; note that this implies that \(f_{v_{1}}^{\tau}\) is essentially a matrix coefficient of the base change \(\Pi_{v_{1}}\simeq\pi_{v_{1}}\otimes\pi_{v_{1}}\) of \(\pi_{v_{1}}\).
+
+Making use of the strong multiplicity one theorem [Ram18, Theorem A], we may separate the terms in the preceding sum. For a cuspidal automorphic representation \(\Pi\) of \(\text{GL}_{n}(\mathbb{A}_{E})\), we denote by \(\mathcal{B}(\Pi)\) the (finite) set of cuspidal automorphic representations \(\pi\) of \(\text{GL}_{n}(\mathbb{A}_{F})\) such that \(\Pi=\text{BC}_{E}(\pi)\). We have the following comparison of relative characters.
+
+Proposition 11.9. For almost all split places \(v\), we fix an irreducible unramified representation \(\pi_{v}^{0}\). For a fixed split place \(v_{1}\), we fix a supercuspidal representation \(\pi_{v_{1}}\) of \(\text{GL}_{n}(F_{v_{1}})\). Then there exists at most one cuspidal automorphic representation \(\Pi\) of \(\text{GL}_{n}(\mathbb{A}_{E})\) such that if \(\tilde{f}\text{and}\{f^{\tau}\}_{\tau\in\mathcal{V}_{n}(E/F)}\) are efficient transfers for \((\omega,\pi_{v_{1}})\), then
+
+\[\frac{1}{2^{1-|S|}L(0,\eta)} \sum_{\tau\in\mathcal{V}_{n}(E/F)}J_{\Pi}^{\tau,\tau_{0}}(f^{\tau})=\sum_{\pi\in\mathcal{B}(\Pi)}I_{\pi}(\tilde{f}),\]
+
+where the sum on the right runs over all cuspidal automorphic representations \(\pi\) of \(\text{GL}_{n}(\mathbb{A}_{F})\) such that
+
+\((1)\pi_{v}\cong \pi_{v}^{0}\) for almost all split \(v\),
+
+\((2)\pi_{v_{1}}\) is our fixed supercuspidal representation,
+
+and where \(\Pi=\text{BC}_{E}(\pi)\) is the base change of any \(\pi\) appearing in the sum.
+
+Remark 11.10. Note that the comparison above is vacuous unless \(\pi=\bigotimes_{v}\pi_{v}\) is \(\text{GL}_{n}-\)elliptic at some split place \(v_{2}\) (this is built into the requirement that \(\tilde{f}\text{and}\{f^{\tau}\}_{\tau\in\mathcal{V}_{n}(E/F)}\) are efficient transfers). By Lemma 11.16, it suffices for \(\pi_{v_{2}}\) to be supercuspidal.

outputs/arxiv_math/2503.09519_pg7_pg1_repeat1.md

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+[^10]: -10
+
+[^10]: -10
+
+4000
+
+6000
+
+8000
+
+10000
+
+(b)
+
+| 10 -15 |  |  |  |  |  | 10 -15 |  |
+| --- | --- | --- | --- | --- | --- | --- | --- |
+|  |  |  |  |  |  | 10 -20 |  |
+| 10 -20 |  |  |  |  |  |  |  |
+|  |  |  |  |  |  | 10 -25 |  |
+| 10 -25 |  |  |  |  |  | 10 -30 |  |
+|  |  |  |  |  |  | 10 -35 |  |
+|  | 100 | 200 | 300 (a) | 400 | 500 |  | 2000 |
+
+Figure 6: The values of \(\Delta_{p}(t;1/2,2)\). The top curve corresponds to \(p=8\) implemented in double precision, the middle curve in gray color (the bottom curve) correspond to \(p=8\) and (respectively, \(p=12)\) impemented in quadruple precision.
+
+was chosen because \(\zeta(s)\) does not grow too rapidly as \(t\rightarrow+\infty\) in this strip, unlike in any strip where \(\text{Re}(s)<1/2\). The results of these computations are presented in Figure 6. The top graph shows the error \(\Delta_{8}(t;1/2,2),\text{where}\zeta_{8}(s)\) was implemented in double precision (the benchmark values \(\zeta(s)\) were computed in higher precision). We observe that rounding errors dominate the approximation error for \(t>200\). These rounding errors primarily arise when computing \(\chi(s)\) and evaluating the values of \(n^{-s}\text{and}n^{s-1}\text{for}s=\sigma+\mathrm{i}t\) with even moderatly large \(t\). Indeed, it is easy to verify numerically that when computing the value of \(2^{\mathrm{i}t}\) in double precision for \(t\) of the order 102, precision of the result will be around \(10^{-14},\text{and when}t\) increases to 103, the precision drops to about \(10^{-13}\). This suggests that we lose one decimal digit of precision every time \(t\) increases by a factor of ten. Precision is also lost due to cancellation errors when we add many terms in the main sum in (3), though this effect likely plays a lesser role compared to rounding errors. The middle graph in Figure 6 (shown in gray) displays the errors \(\Delta_{8}(t;1/2,2)\), where this time \(\zeta_{8}(s)\) was implemented in quadruple precision. We observe that rounding errors do not affect the results in this range of \(t\). The quadruple precision implementation of \(\zeta_{8}(s)\) produces errors smaller than \(10^{-13}\) for \(t>250\) and smaller than \(10^{-15}\text{for}t>2000\) in the strip \(1/2\leq\sigma\leq2\). Of course, when \(t\) becomes very large (of the order 10[^10] or greater), rounding errors will eventually become noticeable. The bottom graph on Figure 6 shows the errors \(\Delta_{12}(t;1/2,2)\) for the quadruple precision implementation of \(\zeta_{12}(s)\). These errors are significantly smaller: we find that \(\Delta_{12}(t;1/2,2)<10^{-25}\text{for}t>5000\). The effects of rounding errors become clearly noticeable for \(t>2000\), though they do not pose a significant issue in this range, as the maximum approximation error remains larger than the rounding errors. However, when \(t\) reacher 105 or greater, rounding errors will dominate the approximation error, and the graph will resemble the top graph (though with a much smaller overall error, of order \(10^{-27})\). Now that we have (hopefully) convinced the reader of the accuracy of our approximations, it is time to discuss how they were derived and to present an algorithm for computing the coefficients \(\omega_{p,j}\text{and}\lambda_{p,j}\).

outputs/arxiv_math/2503.09528_pg16_pg1_repeat1.md

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+<table>
+<tr>
+<th>0.5</th>
+<th></th>
+<th></th>
+<th></th>
+<th></th>
+<th></th>
+<th></th>
+<th></th>
+</tr>
+<tr>
+<td rowspan="4">-0.5</td>
+<td></td>
+<td>1.0</td>
+<td></td>
+<td></td>
+<td></td>
+<td>1.0</td>
+<td></td>
+</tr>
+<tr>
+<td>1.0 1.5 2.0 2.5</td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>0.5</td>
+<td></td>
+<td></td>
+<td></td>
+<td>0.5</td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>-0.5</td>
+<td>0.5</td>
+<td>1.0 1.5 2.0 2.5</td>
+<td></td>
+<td></td>
+<td>0.5 1.0 1.5 2.0 2.5</td>
+</tr>
+<tr>
+<td rowspan="5">-1.0</td>
+<td></td>
+<td>-0.5</td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td>-0.5</td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>-1.0</td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td>-1.0</td>
+<td></td>
+</tr>
+<tr>
+<td>(a) Missing 0</td>
+<td></td>
+<td>(b)</td>
+<td>Missing 1</td>
+<td></td>
+<td></td>
+<td>(c) Missing 2</td>
+</tr>
+<tr>
+<td></td>
+<td>1.0</td>
+<td></td>
+<td></td>
+<td>1.0</td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>0.5</td>
+<td></td>
+<td></td>
+<td>0.5</td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>-0.5 0.5</td>
+<td>1.0 1.5</td>
+<td>2.0 2.5</td>
+<td>-0.5 0.5</td>
+<td>1.0</td>
+<td>1.5</td>
+<td>2.0 2.5</td>
+</tr>
+<tr>
+<td></td>
+<td>-0.5</td>
+<td></td>
+<td></td>
+<td>-0.5</td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>-1.0</td>
+<td></td>
+<td></td>
+<td>-1.0</td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td>(d) Missing 1 + i</td>
+<td></td>
+<td>(e) Missing 2 + i</td>
+<td></td>
+<td></td>
+<td></td>
+</tr>
+</table>
+
+Figure 3. Base \(b=1+2i\) with all possible one missing digit.
+
+-  of Hausdorff the metric. With one missing digit, \(C_{b,D}\) is \(1/N^{1/2}(b)-\) close to \(C_{b,D_{b}}.\) Notice that \(C_{b,D}\) is self-similar whose contraction ratio is \(1/N^{1/2}(b)\). From here, we see that the thickness of \(C_{b,D}\) is at least \(c_{d}N^{1/2}(b)\) for some constant \(c_{d}>0.\) The thickness estimate holds for all \(C_{b,D,j}\) because of the self-similarity. However, we cannot directly use Theorem 11 to obtain intersections between different missing digit sets without checking the conditions (2,3). Consider different integers \(b_{1},\ldots,b_{k}\in\mathbb{Z}[i]\) with large enough norms so that Theorem 11 applies. Consider the logarithmic map
+
+\[\log:a\in\mathbb{Z}[i]\rightarrow(\log|a|,\arg a)\in(0,\infty)\times\mathbb{T}.\]
+
+The first component is the standard log for positive numbers. The second component is the argument of \(a\) as a complex number.3 For simplicity, let us first look at the two integers \(b_{1},b_{2}.\) We claim that there are infinitely many positive rational integer pairs \((l_{1},l_{2})\) such that \(b_{1}^{l_{1}}/b_{2}^{l_{2}}\) is close to one. In fact, consider the set
+
+\[P=\{b_{1}^{l_{1}}/b_{2}^{l_{2}}:(l_{1},l_{2})\in\mathbb{Z}_{>0}^{2}\}\subset\mathbb{C}.\]
+
+3In general, such a logarithmic map can be defined in each Galois embedding individually.

outputs/arxiv_math/2503.09536_pg7_pg1_repeat1.md

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+We also denote the induced pairing by \(\langle\cdot,\cdot\rangle_{A}\), which satisfies
+
+\[(2.24)\langle m,\phi\rangle_{A}=\sum_{i=1}^{k}a_{i}(\phi(q_{i})-\phi(p_{i})),m=\sum_{i=1}^{k}a_{i}(\delta_{q_{i}}-\delta_{p_{i}})\in A_{0}(X,e),\phi\in\text{Lip}_{\mathrm{b}}(X),\]
+
+understanding that \(\phi(e)=0\). Proposition 2.11. Let \(X\subset\mathbb{R}^{n}\) be a closed set and \(e\not\in X\). Then,
+
+(a) \(A(X,e)\cong Æ(X)\) via \(m\mapsto m\)
+
+\(X\text{for}m\in A_{0}(X,e)\), extending by density,
+
+\[(\mathrm{b})\text{Lip}_{\mathrm{b}}(X)\simeq Æ(X)^{*}\text{via}\phi\mapsto(m\mapsto\langle m,\phi\rangle_{Æ}).\]
+
+Proof. By [Wea18, Thm. 3.3, Cor. 3.4], the pairing (2.24) extends to an isometric isomorphism \(A(X,e)^{\sim^{*}}=\text{Lip}_{\mathrm{b}}(X)\). In particular this implies that for \(m\in Æ_{0}(X,e)\),
+
+\[\Vert mX\Vert_{\text{Lip}_{\mathrm{b}}(X)^{*}}=\sup\{|\langle m,\phi\rangle_{A}|:\Vert\phi\Vert_{\text{Lip}_{\mathrm{b}}(X)}\leq1\}=\Vert m\Vert_{A}.\] (2.25)
+
+\[\begin{aligned} X&\in Æ_{0}(X) \\ \langle mX,\phi\rangle_{Æ} \end{aligned}\]
+
+Hence it follows that \(A(X,e)\cong Æ(X)\) by sending each \(m\in A_{0}(X,e)\) to \(m\) and extending by density, thereby proving (a). Since we also have \(\langle m,\phi\rangle_{A}=\langle m\) for all \(m\in A_{0}(X,e)\text{and}\phi\in\text{Lip}_{\mathrm{b}}(X)\), it follows that \(Æ(X)^{\sim^{*}}=\text{Lip}_{\mathrm{b}}(X)\) via the pairing \(\langle\cdot,\cdot\rangle_{Æ}\), establishing (b).
+
+- 
+
+Example 2.12. If \(X\subset\mathbb{R}^{n}\) is closed, we have \(\mathcal{M}(X)\subset Æ(X)\), by noting for that each \(\mu\in\mathcal{M}(X)\), the mapping \(\phi\mapsto\int_{X}\phi d\mu\) is well-defined and weakly\(-^{*}\text{continuous on Lip}_{\mathrm{b}}(X)\). However this space is strictly larger in general; if \(a\in X\) is an accumulation point of \(X\), then we can find a sequence \((a_{k})_{k}\subset X\) converging to \(a\text{such that}a_{k}\neq a\text{for all}k\). By passing to a subsequence if necessary, assume that \(\sum_{k}|a_{k}-a|<\infty.\text{Then}m=\sum_{k=1}^{\infty}(\delta_{a_{k}}-\delta_{a})\in Æ(X)\) by noting the series converges absolutely in Lip\((_{\mathrm{b}}X)^{*}\). Lemma 2.13. Let \(X\subset\mathbb{R}^{n}\) be any set, and \(\phi_{k},\phi\in\text{Lip}_{\mathrm{b}}(X).\text{Then},\text{as}k\rightarrow\infty\),
+
+\[(2.26)\phi_{k}⇀^{*}\phi\text{in Lip}_{\mathrm{b}}(X)\Longleftrightarrow \begin{cases} \phi_{k}\rightarrow\phi\text{uniformly on bounded subsets of}X, \\ \sup_{k}\Vert\phi_{k}\Vert_{\text{Lip}_{\mathrm{b}}(X)}<\infty. \end{cases}\]
+
+Proof. Using the identification Lip\((_{\mathrm{b}}X)\cong \text{Lip}_{\mathrm{b}}( \overline{X} )\), we can assume without loss of generality that \(X\) is closed. If \(\phi_{k}⇀^{*}\phi\text{weakly}^{*}\text{in Lip}_{\mathrm{b}}(X)\), by the Banach-Steinhaus theorem, we have \(\Vert\phi_{k}\Vert_{\text{Lip}_{\mathrm{b}}(X)}\) is uniformly bounded in \(k\). Then by applying the Arzelà-Ascoli theorem, there is a subsequence \(\phi_{k_{j}}\) which converges uniformly to \(\phi\text{on}X\cap B_{M}(0)\) for each \(M\in\mathbb{N}\), and hence \(\phi_{k_{j}}\rightarrow\phi\) uniformly on bounded subsets of \(X\). Since the limit is unique, this convergence also holds for the entire sequence \(\phi_{k}\). Conversely since Lip\((_{\mathrm{b}}X)\) is the dual of a separable space, the weak∗-topology is compact and metrisable on norm-bounded subsets (see e.g. [Bre11, Thm. 3.16, 3.28]). Therefore \(\phi_{k}\) admits a weakly∗-convergent subsequence, but since this limit is uniquely determined as \(\phi\) using the uniform convergence, the entire sequence \(\phi_{k}\) converges weakly∗ to \(\phi\).
+
+- 
+
+We will often use Lemma 2.13 with \(X=[0,1]\), noting that Lip\((_{\mathrm{b}}[0,1])=\mathrm{W}^{1,\infty}((0,1))\). For general open sets \(U\) however, we have a slightly different characterisation for weak∗ convergence in \(\mathrm{W}^{1,\infty}(U)\). Lemma 2.14. Let \(U\subset\mathbb{R}^{n}\) be an open set. Then if \(\phi_{k},\phi\in\mathrm{W}^{1,\infty}(U),\text{as}k\rightarrow\infty\),
+
+\[\phi_{k}⇀^{*}\phi\text{in W}^{1,\infty}(U)\Longleftrightarrow \begin{cases} \phi_{k}\rightarrow\phi\text{pointwise}, \\ \sup_{k}\Vert\phi_{k}\Vert_{\mathrm{W}^{1,\infty}(U)}<\infty. \end{cases}\] (2.27)
+
+In addition, the space \(\mathrm{C}_{\mathrm{b}}^{1}(U)\) is sequentially weakly∗ dense in \(\mathrm{W}^{1,\infty}(U)\). We note that \(\mathrm{C}_{\mathrm{b}}^{1}(U)\not\subset \text{Lip}_{\mathrm{b}}(U)\) in general, so we do not get an analogous density statement there. Also the below proof shows in fact that \(\phi_{k}\rightarrow\phi\) locally uniformly in \(U\) in (2.27), however pointwise convergence will suffice for our purposes.

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+Let \(u\text{and}v\) be primitive elements such that \(t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\in\langle u\rangle\text{and}t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}\in\langle v\rangle\). Applying Proposition 5.1 to \(y\) and using Lemma 2.2, we obtain that
+
+\[\begin{aligned} \mathbb{E}_{u}\otimes\mathbb{E}_{v}(y) \\ &=c_{e,e}^{2}e\otimes e+c_{t_{1}^{n},e}^{2}t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes e+c_{e,t_{2}^{m}}^{2}e\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}+c_{(t_{1}^{n},t_{2}^{m})}^{2}t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1} \\ &=0+0+0+c_{(t_{1}^{n},t_{2}^{m})}^{2}t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}\in\mathcal{A}. \end{aligned}\]
+
+Since \(c_{(t_{1}^{n},t_{2}^{m})}\neq 0\), we get that \(t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}\in\mathcal{A}\). Now, replacing \(h_{1},h_{2}\) by \(h_{1}^{2}\) and \(h_{2}^{2}\) respectively, and using the same argument (noting that \(\langle t_{i},h_{i}^{2}\rangle\cong \mathbb{F}_{2}\) for each \(i=1\) and 2), we get that
+
+Consequently,
+
+\[t_{1}^{n}h_{1}^{2}t_{1}^{n}h_{1}^{-2}\otimes t_{2}^{m}h_{2}^{2}t_{2}^{m}h_{2}^{-2}\in\mathcal{A}.\]
+
+\[\begin{aligned} (h_{1}\otimes h_{2})(h_{1}t_{1}^{-n}h_{1}^{-1}t_{1}^{n}\otimes h_{2}t_{2}^{-m}h_{2}^{-1}t_{2}^{m})(h_{1}^{-1}\otimes h_{2}^{-1}) \\ &=h_{1}^{2}t_{1}^{-n}h_{1}^{-1}t_{1}^{n}h_{1}^{-1}\otimes h_{2}^{2}t_{2}^{-m}h_{2}^{-1}t_{2}^{m}h_{2}^{-1} \\ &=h_{1}^{2}t_{1}^{-n}h_{1}^{-2}h_{1}t_{1}^{n}h_{1}^{-1}\otimes h_{2}^{2}t_{2}^{-m}h_{2}^{-2}h_{2}t_{2}^{m}h_{2}^{-1} \\ &=(t_{1}^{n}h_{1}^{2}t_{1}^{n}h_{1}^{-2}\otimes t_{2}^{m}h_{2}^{2}t_{2}^{m}h_{2}^{-2})^{-1}\left(t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}\right)\in\mathcal{A}. \end{aligned}\]
+
+Therefore, \(h_{1}t_{1}^{-n}h_{1}^{-1}t_{1}^{n}\otimes h_{2}t_{2}^{-m}h_{2}^{-1}t_{2}^{m}\in\mathcal{A}\). Hence,
+
+\[t_{1}^{2n}\otimes t_{2}^{2m}=(t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1})(h_{1}t_{1}^{-n}h_{1}^{-1}t_{1}^{n}\otimes h_{2}t_{2}^{-m}h_{2}^{-1}t_{2}^{m})\in\mathcal{A}.\]
+
+Therefore, \(\mathbb{E}_{t_{1}}\otimes\mathbb{E}_{t_{2}}(a)(h_{1}t_{1}^{2n}h_{1}^{-1}\otimes h_{2}t_{2}^{2m}h_{2}^{-1})\in\mathcal{A}\). Observe that
+
+\[\mathbb{E}_{t_{1}}\otimes\mathbb{E}_{t_{2}}(a)(h_{1}t_{1}^{2n}h_{1}^{-1}\otimes h_{2}t_{2}^{2m}h_{2}^{-1})=\sum_{k,l\in\mathbb{Z}}c_{t_{1}^{k},t_{2}^{l}}t_{1}^{k}h_{1}t_{1}^{2n}h_{1}^{-1}\otimes t_{2}^{l}h_{2}t_{2}^{2m}h_{2}^{-1}.\]
+
+Let \(u^{\prime}\text{and}v^{\prime}\) be primitive elements such that \(t_{1}^{n}h_{1}t_{1}^{2n}h_{1}^{-1}\in\langle u^{\prime}\rangle\text{and}t_{2}^{m}h_{2}t_{2}^{2m}h_{2}^{-1}\in\langle v^{\prime}\rangle\). Using Proposition 5.1, and Lemma 2.2, we see that
+
+\[t_{1}^{n}h_{1}t_{1}^{2n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{2m}h_{2}^{-1}= \frac{1}{c_{t_{1}^{n},t_{2}^{m}}^{2}} \mathbb{E}_{u^{\prime}}\otimes\mathbb{E}_{v^{\prime}}\left(\mathbb{E}_{t_{1}}\otimes\mathbb{E}_{t_{2}}(a)(h_{1}t_{1}^{2n}h_{1}^{-1}\otimes h_{2}t_{2}^{2m}h_{2}^{-1})\right)\in\mathcal{A}.\]
+
+We recall that \(t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}\in\mathcal{A}\). Therefore,
+
+\[\begin{aligned} (t_{1}^{n}h_{1}t_{1}^{2n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{2m}h_{2}^{-1})(t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1})^{-1} \\ &=t_{1}^{n}h_{1}t_{1}^{2n}h_{1}^{-1}h_{1}t_{1}^{-n}h_{1}^{-1}t_{1}^{-n}\otimes t_{2}^{m}h_{2}t_{2}^{2m}h_{2}^{-1}h_{2}t_{2}^{-m}h_{2}^{-1}t_{2}^{-m} \\ &=t_{1}^{n}h_{1}t_{1}^{n}h_{1}^{-1}t_{1}^{-n}\otimes t_{2}^{m}h_{2}t_{2}^{m}h_{2}^{-1}t_{2}^{-m} \\ &=(t_{1}^{n}h_{1}\otimes t_{2}^{m}h_{2})(t_{1}^{n}\otimes t_{2}^{m})(t_{1}^{n}h_{1}\otimes t_{2}^{m}h_{2})^{-1}\in\mathcal{A}. \end{aligned}\]
+
+- 
+
+Since \(\mathcal{A}\) is invariant, it follows that \((g,h)=t_{1}^{n}\otimes t_{2}^{m}\in\mathcal{A}\). The claim follows. Since property-AP is preserved under extensions, \(\Gamma\) has property-AP, and we can argue as before to conclude the proof. Remark 5.5. Our proof will show that \(\Gamma=\Gamma_{1}\times\Gamma_{2}\times\ldots\times\Gamma_{n}\) has \(C^{*}-\text{ISR}\) property if every \(\Gamma_{i}\) is a torsion-free non-amenable acylindrically hyperbolic group with property-AP.

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+## 2. Stochastic control problem
+
+As already mentioned in the introduction, the turnpike phenomenon is a significant principle in optimal control theory. In fact, it establishes that during a specific interior subinterval of a given time horizon, the dynamic control strategy can be approximated with arbitrary precision by the solution of a (generally less complex) static optimization problem. This insight greatly reduces the computational effort typically associated with dynamic control strategies. In our framework, we start by considering the continuous trajectories of the interacting agents. We denote by \(x_{k}(t)\in\mathbb{R}^{d}\) the position of agent \(k\) at time \(t\), with \(k=1,\ldots,N\), and the time interval is defined as \([t_{0},T]\). The dynamics of the agents are described by the following stochastic differential equation (SDE)
+
+\[dx_{k}(t)=\left( \frac{1}{N} \sum_{\ell=1}^{N}p(x_{k}(t),x_{\ell}(t))(x_{\ell}(t)-x_{k}(t))+u_{k}(t)\right)dt+\sigma dW_{k}(t),(1)\]
+
+where the term \(\sigma dW_{k}(t)\) represents the stochastic noise component, with \(W_{k}(t)\) being standard \(d-\)dimensional independent Brownian motions. The initial condition for each agent \(k\) is given by \(x_{k}(t_{0})=x_{k}^{0}\in\mathbb{R}^{d},\text{where}x_{k}^{0}\) is a stochastic variable, accounting for randomness in the initial state distribution. The interaction between agents is mediated by a nonlinear interaction kernel \(p(x_{k}(t),x_{\ell}(t))\) which describes the forces governing their behavior. For further examples and theoretical background, we refer to [7]. The system operates within a well-defined probabilistic framework. Let \((\Omega,\mathcal{F},\mathbb{P})\) be a complete probability space, where \(\Omega\) represents the sample space, \(\mathcal{F}\) is the \(\sigma-\)algebra of measurable events, and \(\mathbb{P}\) is the probability measure. The stochastic processes \(W_{k}(t)\), are adapted to the filtration \(\mathbb{F}=(\mathcal{F}_{t})_{t\geq t_{0}}\), which represents the natural filtration generated by the Brownian motions. This filtration captures all the information available up to time \(t\), ensuring that the evolution of the stochastic processes depends on the past without anticipating future information. The dynamics of the agents are thus driven by both deterministic controls \(u_{k}(t)\) and the stochastic influences represented by the Brownian motions \(W_{k}(t)\). The control \(u_{k}(t)\) is applied individually to each agent, with the aim of minimizing a cost functional over a finite time horizon. The control objective is to drive the system towards a desired state \(\bar{x}\in\mathbb{R}^{d}\) while minimizing the associated cost. The latter (to be minimized with respect to the control term) incorporates a tracking term, which measures the deviation of each agent from the desired state \(\bar{x}\), and a regularization term to penalize the magnitude of the control. Specifically, the cost functional, which depends on \(\boldsymbol{x}=(x_{1},\ldots,x_{N}),\boldsymbol{u}=(u_{1},\ldots,u_{N})\), is given by
+
+\[\begin{aligned} \mathcal{C}(\boldsymbol{x},\boldsymbol{u})&=\int_{\Omega}\int_{t_{0}}^{T} \frac{1}{N} \sum_{k=1}^{N}\left(\Vert x_{k}(t,\omega)-\bar{x}\Vert^{2}+\gamma\Vert u_{k}(t)\Vert^{2}\right)dtd\mathbb{P}(\omega) \\ &=\mathbb{E}\left[ \frac{1}{N} \int_{t_{0}}\sum_{k=1}^{TN}\left(\Vert x_{k}(t)-\bar{x}\Vert^{2}+\gamma\Vert u_{k}(t)\Vert^{2}\right)dt\right]. \end{aligned}\] (2)

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+Cutoff describes a phase transition: as we run the family of Markov chains on \(X_{n}\), the total variation distance is almost equal to 1, and then suddenly it drops and approaches zero as \(n\) goes to infinity. The formal definition of cutoff is given below.
+
+Definition 2. A family of Markov chains on \(X_{n}\) is said to have cutoff at time \(t_{n}\) with window \(w_{n}=o(t_{n})\) if and only if
+
+\[\lim_{c\rightarrow\infty n}\lim_{\rightarrow\infty}d_{x}^{(n)}(t_{n}-cw_{n})=1\text{and}\lim_{c\rightarrow\infty n}\lim_{\rightarrow\infty}d_{x}^{(n)}(t_{n}+cw_{n})=0,\]
+
+where \(d_{x}^{(n)}(t)\) denotes the total variation distance of the \(n-\)th Markov chain after \(t\) steps starting at \(x\).
+
+Given a Markov chain exhibiting cutoff, one can ask for more precise control on the exact distance from stationarity. This is known as the limit profile with respect to the sequences \((t_{n})\text{and}(w_{n})\), defined as:
+
+\[\Phi_{x}(c):=\lim_{n\rightarrow\infty}d_{x}^{(n)}(t_{n}+cw_{n}),\text{for all}c\in\mathbb{R},\] (1)
+
+if this limit exists. The limit profile is known for only a few Markov chains, such as the riffle shuffle [2], the asymmetric exclusion process on the segment [4], the simple exclusion process on the cycle [14], and the simple random walk on Ramanujan graphs [16], etc. Teyssier [25] determined the limit profile for random transpositions. Using representation theory of the symmetric group \(S_{n}\), he used Fourier transform arguments for studying limit profiles that work for random walks on groups using a generating set that is a conjugacy class. The same limit profile is proved to be true for the \(k-\)cycle card shuffle [18], under the assumption that \(k=o(n)\) and for star transposition [17]. Olesker–Taylor and Schmid also studied the limit profile for the Bernoulli–Laplace chain [21]. In Section 6 we give a more extensive presentation of the above results. Furthermore, if \(P\) is aperiodic, irreducible and reversible on \(X\), then the spectrum of \(P\) satisfies
+
+\[-1<\beta_{|X|}\leq\ldots\leq\beta_{2}<\beta_{1}=1,\]
+
+and there is an orthonormal eigenbasis \(\{f_{j}:X_{n}\rightarrow\mathbb{R}\}\). The main assumption on \(P\) is that there is a continuous (or bounded) function \(g\text{on}\mathbb{R}\) such that
+
+\[\limsup_{n\rightarrow\infty}w_{n}^{2}\sum_{i=2}^{|X_{n}|}f_{i}(x)^{2}(1-\beta_{i})^{2}e^{-2(t_{n}+cw_{n})(1-\beta_{i})}\leq g(c),\] (2)
+
+for all \(c\in\mathbb{R}\).
+
+Theorem 3. Let \(P_{n}\) be the transition matrix of a reversible Markov chain on \(X_{n}\) that exhibits cutoff at \(t_{n}\) with window \(w_{n}\) and satisfies (2). Assume that \(\Phi_{x}\), the total variation limit profile with respect to the sequences \((t_{n})\text{and}(w_{n})\) at \(x\), exists. Then \(\Phi_{x}\) is continuous for every
+
+\[x\in X.\]

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+Notice firstly that \(\alpha_{j}+\gamma_{j}\) separates the surface \(S\), meaning that \(\alpha_{j}+\gamma_{j}=0\in H_{1}(S)\), or in other words,
+
+\[-\alpha_{j}=\gamma_{j}=\tau_{*}(\alpha_{j}),\] (4.3)
+
+where \(\tau_{*}\) denotes the automorphism of \(H_{1}(S)\) induced by \(\tau\) , i.e. \(\tau_{*}=\Psi(\tau)\). Notice secondly that \(\beta_{j}\) was sent to its inverse, or in other words,
+
+\[-\beta_{j}=\tau_{*}(\beta_{j}).\] (4.4)
+
+We’ve shown that equations (4.3) and (4.4) hold for all odd \(j\) inside \(\{1,...,n\}\). Now, by braid and disjointness relations, we have that
+
+\[cbabcb=cbacbc=cbcabc=bcbabc,\]
+
+but then, since \(cbabcb=\tau\) has order 2, we know \(cbabcb\) is equal to its own inverse, and thus
+
+\[cbabcb=(bcbabc)^{-1}=c^{-1}b^{-1}a^{-1}b^{-1}c^{-1}b^{-1}.\]
+
+This shows that every right Dehn twist in \((cab)^{2}\) can be replaced with a left Dehn twist and vice versa, and nothing will change. Therefore, if \(j\in\{1,...,n\}\) is even, \((cab)^{2}\) will affect the curve \(\alpha_{j}\text{and}\beta_{j}\) in the exact same manner as what is depicted in Figure 6 and Figure 7. We conclude that equations (4.3) and (4.4) hold for all \(j\in\{1,...,n\}\) and hence, \(\Psi(\tau)=\tau_{*}=-\text{Id}_{2n}\in\text{Sp}_{2n}(\mathbb{Z})\), as required. The proof above can be utilized to prove a slightly more precise corollary which will come in handy in the next section. For the following result, we let \(\lambda\) denote the homeomorphism of \(S\) given by the following rotation by 180°:
+
+\(\square\)
+
+Figure 8
+
+![](figure)
+
+![](figure)
+
+Corollary 4.2 (Corollary of the previous proof). Let \(S\) be a closed surface of genus \(n\) and let \([\lambda]\in\text{Mod}(S)\) denote the mapping class of \(\lambda\). We have that \([\lambda]=(cab)^{2}\in\Theta_{n}^{1}\).
+
+Proof. Consider the following simple closed curves in \(S\), which satisfy the assumptions required for the Alexander method [1, Section 2.3]:
+
+Figure 9
+
+![](figure)
+
+![](figure)

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+With these lemmas at hand, we now prove our main theorem by an inductive argument. In particular, we show that the following two families of Gaussian processes, introduced in Section 4, remain non-degenerate throughout training:
+
+\[\begin{aligned} \{\widehat{Z}^{W_{0}^{l}\delta x_{s}^{l-1}(\xi_{i})}\}_{i\in[m],s\in[t],2\leq l\leq L}, \\ \{\widehat{Z}^{W_{0}^{l\top}dh_{s}^{l}(\xi_{i})}\}_{i\in[m],s\in[t],2\leq l\leq L}. \end{aligned}\] (C.1) (C.2)
+
+Recall that a Gaussian process is non-degenerate if its covariance matrix \(C\) at any finite collection of points satisfies \(\det(C)\neq 0\) (Adler and Taylor, 2009). Using the filtration framework introduced in Section 4, our proof follows the natural flow of computation in the network, proceeding layer by layer and separately handling forward and backward passes. We break this into four key steps, each building upon the results of previous steps:
+
+- Step 1: prove non-degeneracy for the features in the first hidden layer \(\widehat{Z}^{W_{0}^{2}\delta x_{s}^{1}(\xi_{i})}\). This forms our base case as it only depends on the input data and network initialization, providing the foundation for our inductive argument.
+
+- Step 2: prove non-degeneracy for the features in remaining layers \(\widehat{Z}^{W_{0}^{l}\delta x_{s}^{l-1}(\xi_{i})},3\leq l\leq L\). This step leverages the non-degeneracy established in Step 1 and shows how it propagates through deeper layers of the network.
+
+- Step 3: prove non-degeneracy for the gradients in the last layer \(\widehat{Z}^{W_{0}^{L\top}dh_{s}^{L}(\xi_{i})}\). Here we transition from analyzing forward features to backward gradients, showing how the established feature properties ensure meaningful gradient flow.
+
+- Step 4: prove non-degeneracy for the gradients in remaining layers \(\widehat{Z}^{W_{0}^{l\top}dh_{s}^{l}(\xi_{i})},2\leq l\leq L-1\). Finally, we complete our analysis by showing how gradient non-degeneracy propagates backward through the network, ensuring effective training dynamics at all layers.
+
+The proof proceeds by induction on the time step \(t\), where at each step we verify these properties hold across all layers. This structure allows us to carefully track how the non-degeneracy property is maintained as information flows both forward and backward through the network during training. This systematic proof structure allows us to establish the global property of non-degeneracy by carefully tracking local changes at each layer and time step. We now proceed with the detailed proof.
+
+Proof of Theorem 4.5. Considering Trajectory Until Error Signals Vanish. Throughout this proof, we focus on the training trajectory up to the time when all error signals \(\mathring{\chi}_{t,i}\) become zero. This is because once the error signals vanish, there are no further parameter updates, and the training dynamics remain static thereafter. Our analysis ensures that up to this point, the Gaussian processes governing the feature and gradient updates remain non-degenerate, thereby maintaining the linear independence of features across all layers. Connecting \(\widehat{Z}^{W\delta x}\) to \(h^{l}\mathbf{and}x^{l}\). Recall from Section 3 that each pre-activation \(h^{l}(\xi)\) and post-activation \(x^{l}(\xi)\) can be decomposed into a primary Gaussian increment plus lower-order (history-dependent) terms in the infinite-width limit: Because these additional terms do not alter the essential covariance structure when conditioned on past information (they vanish or become deterministic in the limit), the linear (in)dependence

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+## 2 The Mandelstam Region
+
+A symmetric \(n\times n\) matrix \(S=[s_{ij}]\) of \(\text{rank}r\) is said to be a Mandelstam matrix if
+
+- the diagonal entries are non-negative, \(s_{ii}\geq0\text{for}i=1,\ldots,n\); and
+
+- it has precisely one positive eigenvalue and \(r-1\) negative eigenvalues.
+
+We denote the set of all Mandelstam matrices of \(\text{rank}r\) by \(\mathcal{M}_{n,r}\). This is a semialgebraic set in \(\mathbb{R}(_{2}^{n+1})\), the space of all symmetric matrices. The following is the Mandelstam analogue of the familiar characterization of positive semidefinite matrices in terms of principal minors. Lemma 2.1. A symmetric \(n\times n\) matrix \(S\) is Mandelstam if and only if
+
+\[(-1)^{|I|-1}\det(S_{I})\geq0\text{for all}I\subseteq[n],\] (4)
+
+where \(\det(S_{I})\) are the principal minors of \(S\).
+
+Proof. This follows from the general results in [6]. We refer to Baker’s exposition in [3]. The key step is Cauchy’s interlacing theorem [12]. This states that the eigenvalues of \(S_{I}\) interlace the eigenvalues of \(S_{J}\text{whenever}I\subset J\). Hence, if \(S_{I}\) has at most one positive eigenvalue then so does \(S_{J}\). But \(S_{J}\) cannot have all negative eigenvalues because its trace is non-negative.
+
+The name of our matrices refers to the physicist Stanley Mandelstam (1928–2016) who is credited for introducing the variables \(s_{ij}\) in the context of scattering amplitudes. In [14] the role of \(\mathcal{M}_{n,r}\) as a kinematic space is recognized. A term more familiar to mathematicians might be “Lorentzian matrices.” These encode Lorentzian quadratic forms [5, 6]. We here use the term Lorentzian matrix for a Mandelstam matrix whose entries \(s_{ij}\) are all non-negative. Mandelstam matrices arise as Gram matrices of momentum vectors in \(\mathbb{R}^{1+d}\) with the
+
+Lorentzian inner product. A non-zero momentum vector is any vector \(p\in\mathbb{R}^{1+d}\) of the form
+
+\[p=\lambda(1,x_{1},\ldots,x_{d}),\] (5)
+
+for some scalar \(\lambda\neq 0,\text{and}x=(x_{1},\ldots,x_{d})\) in the closed unit ball \(\mathbb{B}^{d}=\{x\in\mathbb{R}^{d}:||x||\leq1\}\).
+
+Given \(n\) momentum vectors, \(p^{(i)}\), their Gram matrix \(S=[s_{ij}]\) has entries \(s_{ij}=p^{(i)}\cdot p^{(j)}\). This is the matrix in (1). The entries of \(S\) may now be written as
+
+\[s_{ij}=\lambda_{i}\lambda_{j}\left(1-\langle x^{(i)},x^{(j)}\rangle\right)\] (6)
+
+Here · is the Lorentz inner product on \(\mathbb{R}^{1+d}\text{and}\langle,\rangle\) is the Euclidean inner product on \(\mathbb{R}^{d}\).
+
+Lemma 2.2. A symmetric \(n\times n\) matrix \(S\) is Mandelstam, i.e. \(S\) lies in the region \(\mathcal{M}_{n,\leq1+d}\), if and only if it is the Gram matrix of \(n\) momentum vectors in \((1+d)-\)dimensional spacetime.
+
+Proof. Assume that \(S\) has no zero rows or columns. For the only-if direction, take a Mandelstam matrix \(S\). By Lemma 2.1 and diagonalization of symmetric matrices, it can be factorized as in (1). Namely, we write \(S=MDM^{T}\) , where \(D=\text{diag}(1,-1,-1,\ldots,-1)\). Let the

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+and then observing that the canonical function of \(Y\) writes as \(k(x)=\lambda e^{Ax-B|x|}\), implying
+
+\(\tau_{+}^{-A}=\tau_{-}^{A}=\lambda\delta_{B}\). The Dirac integration to which (5.25) reduces, produces
+
+\[k_{\rho}(x)=\lambda e^{(A^{2}-B^{2})x/2}=\lambda e^{-\theta_{+}\theta_{-}x/2}.\] (5.27)
+
+Therefore, the bilateral gamma process can be represented in law as \(Y_{t}=W_{T_{t}}+(\theta_{-}-\theta_{+})t/2\) where \(\mathcal{L}(T_{1})\sim\Gamma(\lambda,\theta_{+}\theta_{-}/2)\). This has been known and used at least since Madan et al. (1998). Example 10. Subordinated representation of \(\text{CTS}_{\alpha}\) processes. Madan and Yor (2008) through a series of propositions covering a substantial part of the article, explain how to represent a \(\text{CTS}_{\alpha}(\mathbb{R})\) process as a subordinated Brownian motion with drift. Based on the discussion so far, the main Proposition 2 therein trivialize. First of all notice that the assumptions in Theorem 6 are met also in the case of \(\text{CTS}_{\alpha}(\mathbb{R})\) processes with symmetric spherical part. Namely, assuming \(\lambda_{+}=\lambda_{-}\) and letting again \(A=(\theta_{-}-\theta_{+})/2,B=(\theta_{-}+\theta_{-})/2\), leads to \(k(x)= x^{-\alpha}e^{Ax-B|x|}\), so that using the Thorin measure found in Subsection 4.3 we deduce the required
+
+density relationship \(\tau_{+}^{-A}(y)=\tau_{-}^{A}(y)=\lambda(y-B)^{\alpha-1}(\Gamma(\alpha))^{-1}1_{\{y\geq B\}}\). It then follows from
+
+Corollary 6, and with the substitution \(w=(s-B) \sqrt{x}\), that
+
+\[\begin{aligned} k_{\rho}(x)&=\lambda \frac{e^{xA^{2}/2}}{\Gamma(\alpha)} \int_{x/^{B}2}^{\infty}e^{- \frac{s^{2}x}{2}}(s-B)^{\alpha-1}ds=\lambda \frac{e^{(A^{2}-B^{2})x/2}}{\Gamma(\alpha)x^{\alpha/2}} \int_{0}^{\infty}e^{- \frac{w2}{2} -wB \sqrt{x}}w^{\alpha-1}dw \\ &=\lambda \frac{e^{-\theta_{+}\theta_{-}}}{x^{\alpha/2}} H_{-\alpha}\left( \frac{\theta_{+}+\theta_{-}}{2} \sqrt{x} \right) \end{aligned}\] (5.28)
+
+where \(H_{a},a<0\), is the Hermite function, given by
+
+\[H_{a}(z)= \frac{1}{\Gamma(-a)} \int_{0}^{\infty}e^{-x^{2}/2-xz}x^{-a-1}dx,z>0.\] (5.29)
+
+The corresponding Lévy density coincides with that determined in Madan and Yor (2008), Proposition 2.
+
+Example 11. Subordinated representation of generalized\(-z\) processes. Let \(\mu\in\text{GZD}_{G}(\mathbb{R})\) with
+
+Lévy measure of the form (4.20), i.e. \(c(\pm1)=c_{\pm}>0,\sigma(\pm)=\sigma>0\text{and}\lambda(du)=\lambda(\delta_{1}(du)+ \delta_{-1}(du)),\lambda>0\). We can rewrite again \(k(x)=e^{Ax/\sigma-B|x|/\sigma}/(1-e^{-|x|/\sigma}),A=(c_{-}-c_{+})/2\),
+
+\(B=(c_{-}+c_{+})/2\) so we are under the assumption of Theorem 6. Based on (4.22) and (5.25), integrating term by term the series results in
+
+\[\begin{aligned} k_{\rho}(x)&=\lambda e^{\overline{2\sigma^{2}}}\sum_{k=0}^{\infty_{\underline{A^{2}x}}}\exp\left(-x \frac{(k+B)^{2}}{2\sigma^{2}} \right)=\lambda\sum_{k=0}^{\infty}\exp\left(-x \frac{k^{2}+2Bk+B^{2}-A^{2}}{2\sigma^{2}} \right) \\ &=\lambda\sum_{k=0}^{\infty}\exp\left(-x \frac{(k+c_{+})(k+c_{-})}{2\sigma^{2}} \right). \end{aligned}\] (5.30)
+
+The subordinator is then of the form of an infinite GGC convolution of gamma processes
+
+\(G^{k}=(G_{t}^{k})_{t\geq0},k\geq0,\mathcal{L}(G_{1}^{k})\sim\Gamma\left(\lambda, \frac{(k+c_{+})(k+c_{-})}{2\sigma^{2}} \right)\) whose limit in law is not otherwise known.

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+with global minima at index \(i_{0}\) there are exactly \(\sum_{i=1}^{i_{0}-1}(k-1)k^{i-1}\) descents by Theorem 5.3. The number of inversions can be expressed through the values of the permutation before the minimum: \(I(k,n,w)=^{k}\)
+
+\[\tfrac{(^{n}k-1)}{4} \sum_{i=1}^{n}(k^{n-i}-k^{n-i-(c_{w})_{i}})= \tfrac{k^{n}(k-1)}{4} \sum_{i=1}^{i_{0}-1}\]
+
+\((_{i=1}k^{n-i}-k^{n-i-w_{i}+1})\). Now, we describe some bounds on the number of descents and inversion in the stable configuration depending on the permutation. We start with permutations with an increasing tail.
+
+Proposition 6.1. Given permutation \(w\), if there exists an \(i_{0}\in\mathbb{N}\) such that for all indices \(i\geq i_{0}\), we \(\big)\sum^{i_{0}-1}\)
+
+\[k_{i=1}^{i_{0}-1i-1}\left(_{2}^{k^{n-i}}\right)\]
+
+have \(w_{i}<w_{i+1},thenC_{k,n,w}\) has at most \(\sum_{j=1}^{i_{0}-1}(k-1)k^{j-1}\) descents and at most \(\big(_{2}^{k}\) inversions.
+
+Proof. Because \(w_{i}<w_{i+1}\) for all integers \(i\text{such that}i\geq i_{0}\), we find that \((c_{w})_{i}=0\text{for all}i\geq i_{0}\).
+
+Therefore, the support of \(c_{w}\) is a subset of \([i_{0}-1]\). Thus, by Theorem 5.3, we have that there are at
+
+most \(\sum_{j=1}^{i_{0}-1}(k-1)k^{j-1}\) descents in \(C_{k,n,w}\). Because \((c_{w})_{i}=0\text{for all}i\geq i_{0}\), we have
+
+\[I(k,n,w)= \begin{pmatrix} k \\ 2 \end{pmatrix} \sum_{i=1}^{i_{0}-1}k^{i-1} \begin{pmatrix} k^{(c_{w})_{i}} \\ 2 \end{pmatrix} k^{2n-2i-2(c_{w})_{i}}= \frac{k-1}{4} \sum_{i=1}^{i_{0}-1}(k^{2n-i}-k^{2n-i-(c_{w})_{i}}).\]
+
+Since for each \(i\), we have \((c_{w})_{i}\leq n-i\) by definition of Lehmer code, we obtain
+
+\[I(k,n,w)\leq \frac{k^{n}(k-1)}{4} \sum_{i=1}^{i_{0}-1}(k^{n-i}-k^{n-i-(n-i)})= \frac{k(k-1)}{4} \sum_{i=1}^{i_{0}-1}k^{n-1}(k^{n-i}-1)= \begin{pmatrix} k \\ 2 \end{pmatrix} \sum_{i=1}^{i_{0}-1}k^{i-1} \begin{pmatrix} k^{n-i} \\ 2 \end{pmatrix} .\]
+
+One can observe that the upper bounds on the number of inversions and descents in Proposition 6.1 are tight.
+
+Example 24. Consider any positive integers \(i_{0},n,k\text{such that}i_{0}<n\text{and}k\geq2\). Let \(w\) be a
+
+permutation in \(S_{n}\) defined by \(w_{i}=n+1-i\text{for}i\in[i_{0}-1]\text{and}w_{i}=i-i_{0}+1\text{for}i\in\{i_{0},i_{0}+1,\ldots,n\}\). This is a special case of a valley permutation, where the increasing part consists of smaller numbers than the decreasing part. We obtain that the Lehmer code of this permutation is \((c_{w})_{j}=n-i\) for each \(i\in[i_{0}-1]\text{and}(c_{w})_{i}=0\text{for}i\in\{i+1,i+2,\ldots,n\}\). Therefore we obtain from Theorem 4.1 that the number of inversions in the stable configuration \(C_{k,n,w}\) resulting from firing strategy \(F_{w}\) is \(k_{i=1}^{i-1}\left(^{k^{n-i}}\right)\) with increasing tail starting at \(i_{0}\).
+
+\[\left(^{k}\right)\sum_{2i=1}^{i_{0}-1}\]
+
+. This is exactly the upper bound on the number of inversions in \(C_{k,n,w}\text{for}w\)
+
+2
+
+Also observe that Theorem 5.3 and the fact that \(\text{supp}(c_{w})=[i_{0}-1]\) imply that \(C_{k,n,w}\) has exactly
+
+\[\sum_{i=1}^{i_{0}-1}\]
+
+\((_{i=1}k-1)k^{k-1}\) descents. This is equal to the upper bound on the number of descents in \(C_{k,n,w}\) from Proposition 6.1.
+
+On a similar note, we calculate the lower bound for the number of inversions and descents in \(C_{k,n,w}\) in the case where \(w\) has a decreasing tail.

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+## 6 Numerical Computations
+
+In this section, we state numerical computations that show several phenomena discussed in the introduction. In particular, we will observe a suppression of Ostwald ripening, splitting scenarios and instabilities of flat fronts and growing particles. The numerical simulations also support the sharp interface asymptotics, as we get a good agreement between phase field computations and exact solutions of the sharp interface problem. All our numerical simulations are for the quartic potential (2.7) and the interpolation functions (2.8), (2.9) and (2.10), for a fixed value \(r_{c}\in(0,1]\).
+
+### 6.1 Finite element method
+
+We assume that \(\Omega\) is a polyhedral domain and let \(\mathcal{T}_{h}\) be a regular triangulation of \(\Omega\) into disjoint open simplices. Associated with \(\mathcal{T}_{h}\) is the piecewise linear finite element space
+
+\[S^{h}=\{\zeta\in C^{0}( \overline{\Omega} ):\zeta|_{o}\in P_{1}(o)\forall o\in\mathcal{T}_{h}\},\]
+
+where we denote by \(P_{1}(o)\) the set of all affine linear functions on \(o\), see [13]. Let \((\cdot,\cdot)^{h}\) be the usual mass lumped \(L^{2}-\)inner product on \(\Omega\) associated with \(\mathcal{T}_{h}\), and let \(\pi^{h}:C^{0}( \overline{\Omega} )\rightarrow S^{h}\) be the standard interpolation operator. In addition, let \(\tau\) denote a chosen uniform time step size. Then our finite element approximation of (2.1) is given as follows. Let \(\phi_{h}^{0}\in S^{h}\), e.g., \(\phi_{h}^{0}=\pi^{h}\varphi_{0}\text{if}\varphi_{0}\in C^{0}( \overline{\Omega} ).\text{Then},\text{for}n\geq0\), find \((\phi_{h}^{n+1},\mu_{h}^{n+1})\in S^{h}\times S^{h}\) such that
+
+\[\frac{1}{\tau} (\phi_{h}^{n+1}-\phi_{h}^{n},\chi_{h})^{h}+(m(\phi_{h}^{n})\nabla\mu_{h}^{n+1},\nabla\chi_{h})^{h}=(S_{\varepsilon}(\phi_{h}^{n}),\chi_{h})^{h}\forall\chi_{h}\in S^{h},(6.1\mathrm{a})\]
+
+\[\beta\varepsilon(\nabla\phi_{h}^{n+1},\nabla\eta_{h})+ \frac{\beta}{\varepsilon} (\psi^{\prime}(\phi_{h}^{n+1}),\eta_{h})^{h}-(\mu_{h}^{n+1},\eta_{h})^{h}=0\forall\eta_{h}\in S^{h}.(6.1\mathrm{b})\]
+
+We implemented the scheme (6.1) with the help of the finite element toolbox ALBERTA, see [29]. To increase computational efficiency, we employ adaptive meshes, which have a finer mesh size \(h_{f}\) within the diffuse interfacial regions and a coarser mesh size \(h_{c}\) away from them. In particular, we use the strategy from [5, 7], and refine an element \(o\) if \(\eta_{o}=|\max_{o}|\phi_{h}^{n}|-1|>0.5\), unless it is already of size \(h_{f}\), and similarly coarsen an element \(o\text{if}\eta_{o}<0.1\), unless it is already of size \(h_{c}\). For simplicity we assume from now on that \(\Omega=\prod_{i=1}^{d}(0,L_{i})\subset\mathbb{R}^{d},\text{with}L_{1}\geq L_{2}\geq\cdots \geq L_{d}\), and then let \(h_{f}= \frac{L_{d}}{N_{f}} ,h_{c}= \frac{L_{d}}{N_{c}}\) for two chosen integer parameters \(N_{f}>N_{c}\). Here, unless otherwise specified, for the computations with a phase field parameter \(\varepsilon=(2^{k}\pi)^{-1},k\in\mathbb{N}\), we choose \(N_{f}=8N_{c}=2^{3+k}L_{d}\). This ensures that the interfacial regions are accurately resolved, while using a relatively coarser mesh in the pure regions. For the time discretization we choose \(\tau=10^{-3}\), unless stated otherwise. The nonlinear system of equations arising at each time level of (6.1) are solved with the help of Newton’s method. The resulting linear systems at each iteration in two spatial dimensions are solved by direct factorization using the package UMFPACK, see [15], and in three spatial dimensions with a \(V\) -cycle multigrid solver using a block Gauss–Seidel smoother. For the initial data \(\varphi_{0}\) we in general choose a diffuse interface representation of a desired sharp interface, with signed distance function \(d_{0}: \overline{\Omega} \rightarrow\mathbb{R}\). In particular, unless otherwise stated, we let
+
+\[\varphi_{0}(x)=\tanh\left( \frac{d_{0}(x)}{\varepsilon \sqrt{2}} \right).\] (6.2)
+
+For the definition of \(S_{2}\) in (2.2), recall (2.4), we need to specify \(K_{\pm}\text{and}L\). For convenience, for the numerical simulations to follow, we will define the relations \(\rho_{\pm}= \frac{K_{\pm}}{2\beta}\) ,

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+Clearly, if \(D_{i}^{\prime}\) contains both vertices of \(Y,D_{i}\) contains vertices on either halfspace of \(H_{k}\text{in}C_{0}\). Since \(D_{i}\cap C_{0}\) is convex in \(C_{0},D_{i}\) contains an edge dual to \(H_{k}\), thus \(D_{i}^{\prime}\cap C=C\) is the edge \(\overline{e}\) dual to \(H_{k}\text{in}Y\) : the intersection is convex in \(C\). Otherwise, \(D_{i}^{\prime}\) contains only one vertex of \(Y\) , and the intersection \(D_{i}^{\prime}\cap C\) is this vertex, which is convex in \(C\). Moreover, in all cases, \(\iota_{0}\circ c(D_{i}^{\prime})=\iota_{0}\circ c_{0}(D_{i})=\mathbb{S}_{\Delta_{i}}\). This proves that \(D_{i}^{\prime},c\text{and}\iota_{0}\) satisfy Assumptions \((a)\text{and}(b)\) of Proposition 5.6, \((1.)\). Since \(H_{k}\not\in K\), edges of \(X\) dual to \(H_{k}\) are not in \(C_{k}\). Choose such an edge \(e\) and \(p\) a geodesic path in \(C_{k}\) with the same endpoints. The projection of \(p\text{in}Y\) joins the two vertices of \(Y\) , thus some edge of \(p\) does not project to a loop in \(Y\) . Let \(K_{k}\in K\) be a hyperplane dual to such an edge. By Lemma 2.15, every hyperplane of \(X\) transverse to \(H_{k}\) is transverse to \(K_{k}\). In particular, \(H_{k}\text{and}K_{k}\) are not transverse. Symmetrically, choose an edge \(f\text{in}X\) dual to \(K_{k}\text{and}q\) a geodesic path in \(C_{0}\) with the same endpoints. As before, some edge of \(q\) does not project to a loop in \(Y\) , thus has its endpoints in opposite halfspaces of \(H_{k}\text{in}C_{0}\). This edge of \(q\) must then be dual to \(H_{k}\). By Lemma 2.15 again, every hyperplane of \(X\) transverse to \(K_{k}\) is transverse to \(H_{k}\). By specialness of \(Y\) , there is a single edge \(\overline{f}\) dual to \(K_{k}\text{in}Y\) , and every cube spanned by the edge dual to \(H_{k}\) and other edges corresponds to a unique cube spanned by the edge dual to \(K_{k}\) and other edges and vice-versa. Therefore, there exists an involutive combinatorial automorphism \(\alpha\) of \(Y\) that exchanges \(\overline{e} \text{and} \overline{f}\) while fixing both vertices and all the other oriented edges. This proves that the collapse \(c^{\prime}:Y\rightarrow S_{1}\) of \(K_{k}\text{in}Y\) is a homotopy equivalence. Besides, \(\alpha\) induces an isomorphism \(\overline{\alpha} :S_{0}\rightarrow S_{1}\) in the sense that \(c^{\prime}\circ\alpha= \overline{\alpha} \circ c\). In particular, \(\iota_{1}= \iota_{0}\circ \overline{\alpha}^{-1} :S_{1}\rightarrow S_{0}\rightarrow\mathbb{S}\) is a combinatorial isomorphism, and \(\iota_{1}\circ c^{\prime}=\iota_{0}\circ c\circ \alpha^{-1}\). Moreover, letting \(c_{1}:X\rightarrow S_{1}\) be the collapse of \(\{H_{1},\ldots,H_{k-1},K_{k}\},c_{1}\) is a composition \(c^{\prime}\circ d\) of homotopy equivalences, hence a homotopy equivalence itself. Recall that \(\overline{e} \text{and} \overline{f}\) are the only edges of \(Y\) dual to \(H_{k}\text{and}K_{k}\) respectively. Assume \(\overline{e}\) is in \(D_{i}^{\prime}\). Then some edge \(e^{\prime}\) parallel to \(e\) is in \(D_{i}\). Let \(p^{\prime}\) be a geodesic path in \(C_{k}\) joining the endpoints of \(e^{\prime}\). By convexity of \(D_{i}\cap C_{k},p^{\prime}\) is contained in \(D_{i}\). By Corollary 2.16, every edge of \(p\) is parallel to an edge of \(p^{\prime}\). In particular, \(D_{i}\) contains some edge dual to \(K_{k} \underline{.} \text{In}Y\) , its image \(D_{i}^{\prime}\) then contains \(\overline{f}\), the only edge dual to \(K_{k}\). Symmetrically, if \(\overline{f}\) is in \(D_{i}^{\prime}\) then so is \(\overline{e}\). Therefore, the 1-skeleton of \(D_{i}^{\prime}\) is \(\alpha-\text{invariant}\), and by local convexity of \(D_{i}^{\prime}\), the whole subcomplex \(D_{i}^{\prime}\) is \(\alpha-\text{invariant}\). In particular, \(\iota_{1}\circ c^{\prime}(D_{i}^{\prime})=\iota_{0}\circ c(D_{i}^{\prime}) \underline{=} \mathbb{S}_{\Delta_{i}}\). Moreover, if \(D_{i}^{\prime}\) contains both vertices of \(Y,D_{i}^{\prime}\) contains \(\overline{e}\), hence contains \(\overline{f}\) as well. Thus, the intersection of \(D_{i}^{\prime}\) with the vertex preimage \(C^{\prime}= \overline{f}\) of \(c^{\prime}\) is convex in \(C^{\prime}\). This proves that \(D_{i}^{\prime}\), \(c^{\prime}\text{and}\iota_{1}\) satisfy Assumptions \((a^{\prime})\text{and}(b^{\prime})\) of Proposition 5.6, \((1.)\). By Proposition 5.6, \([(\iota_{0})_{*}\circ(c_{0})_{*}\circ((\iota_{1})_{*}\circ(c_{1})_{*})^{-1}]=[(\iota_{0})_{*}\circ(c)_{*}\circ((\iota_{1})_{*}\circ (c^{\prime})_{*})^{-1}]\in\mathcal{U}(A_{\Gamma};\mathcal{G})\). Thus \([(\iota_{1})_{*}\circ(c_{1})_{*}\circ((\iota_{k})_{*}\circ(c_{k})_{*})^{-1}]\in\mathcal{U}(A_{\Gamma};\mathcal{G})\) as well. Both collapses \(c_{1},c_{k}\) factor through the collapse of \(\{K_{k}\}\) and its \(\text{range}X^{\prime}\). The induction now proceeds in \(X^{\prime}\).
+
+- 
+
+Lemma 5.8. Let \(X\) be a spatial cube complex, and let \(H_{a},H_{b},H_{c},H_{d}\) be four distinct hyperplanes of \(X\). Assume that both collapses \(c_{ab}:X\rightarrow S_{ab}\) of \(\{H_{a},H_{b}\}\) and \(c_{cd}:X\rightarrow S_{cd}\) of \(\{H_{c},H_{d}\}\) are homotopy equivalences with range isomorphic to \(\mathbb{S}\). Then, up to exchanging \(H_{a}\text{and}H_{b}\), both collapses \(c_{ac}:X\rightarrow S_{ac}\) of \(\{H_{a},H_{c}\}\) and \(c_{bd}:X\rightarrow S_{bd}\) of \(\{H_{b},H_{d}\}\) are homotopy equivalences with range isomorphic to \(\mathbb{S}\).

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+3. We return to the assumptions on \(\sigma\text{and}F\) with more details in the next subsection. See Remarks & Examples 3.1 in [MMM11] for a discussion of assumptions about the cross section, and Lemma 6.1 in the same reference for the proof of statements in some cases.
+
+Formally, passing to the limit when \(\varepsilon\rightarrow0\) in equation (1.1), we obtain that the limit \(f^{0}\) is in the kernel of \(Q\) which is spanned by the equilibrium \(F\) , which means that \(f^{0}=\rho(t,x)F(v)\). Thus, it amounts to find the equation satisfied by the density \(\rho\). This question of approximation of kinetic equations by macroscopic equations has a long history, dating back to the pioneering works of E. Wigner [WB61], A. Bensoussan et al. [BLP79], as well as E.W. Larsen and J.B. Keller [LK74]. Since then, numerous papers have addressed this topic (for further references, see the papers by C. Bardos et al. [BSS84], and P. Degond et al. [DGP00]). The resulting equations fall into two categories, depending on the rate of decrease in velocity of the equilibrium \(F\) . It is well-known (see [DGP00] for instance) that when \(F\) decreases rapidly for large velocities (such as when \(F\) follows a Maxwellian distribution function), the distribution function \(f^{\varepsilon}\) converges to \(\rho(t,x)F(v)\text{as}\varepsilon\) goes to 0, for the classical scaling \(\theta(\varepsilon)=\varepsilon^{2},\text{with}\rho\) being the solution of the diffusion equation
+
+\[\partial_{t}\rho-\nabla_{x}\cdot(D\nabla_{x}\rho)=0,\] (1.5)
+
+where
+
+\[D=\int_{\mathbb{R}^{d}}(v\otimes v) \frac{F(v)}{\nu(v)} dv.\]
+
+When \(F\) decreases slowly and it is a heavy tail distribution function, satisfying
+
+\[F(v)\sim \frac{\kappa}{|v|^{d+\alpha}} \text{as}|v|\rightarrow\infty\] (1.6)
+
+for some \(\alpha>0\), the previous diffusion matrix \(D\), given by (1.5) might be infinite. In that case, the diffusion limit leading to (1.5) breaks down, which means that the choice of time scale \(\theta(\varepsilon)=\varepsilon^{2}\) was inappropriate. It has been shown in [MMM11] and [Mel10], using different methods and under various assumptions, that in such cases, the appropriate time scale involves the parameter \(\alpha\), which appears in the equilibrium. More specifically, for \(\theta(\varepsilon)=\varepsilon^{\gamma}\text{with}\gamma\) depends on \(\alpha\), the following fractional diffusion equation is obtained
+
+\[\partial_{t}\rho+\kappa(-\Delta_{x})^{\frac{\gamma}{2}}\rho=0,\] (1.7)
+
+where the fractional Laplacian appearing in the previous equation can be defined by the following singular integral
+
+\[(-\Delta_{x})^{\frac{\gamma}{2}}u(x):=c_{\alpha,d}\text{PV}\int_{\mathbb{R}^{d}} \frac{u(x)-u(y)}{|x-y|^{d+\gamma}} dy.\]
+
+Let us now discuss in more detail the contexts of the last two references cited. In [MMM11], the authors addressed the problem in the space homogeneous case (that is with \(\sigma\) independent of \(x)\). They prouved that when \(F\) satisfies (1.6) and the collision frequency \(\nu\) satisfies
+
+\[\nu(v)\sim\nu_{0}|v|^{\beta}\text{as}|v|\rightarrow\infty,\]
+
+then for \(\alpha>0\text{and}\beta<\min(\alpha;2-\alpha)\), and by taking \(\gamma:= \frac{\alpha-\beta}{1-\beta}\) , the scaling \(\theta(\varepsilon)=\varepsilon^{\gamma}\) leads to the previous fractional diffusion equation. While in [Mel10], the same problem has been addressed but for a cross section which depends on the position variable \(x\) but assuming that the collision frequency is bounded, more precisely
+
+\[0<\nu_{1}F(v)\leqslant\sigma(x,v,v^{\prime})\leqslant\nu_{2}F(v),\] (1.8)
+
+i.e. for \(\beta=0\) compared to the case of [MMM11], which gives in particular,
+
+\[0<\nu_{1}\leqslant\nu(x,v)\leqslant\nu_{2}.\] (1.9)

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+### 4.3 Proof of Upper Bound of Theorem 2
+
+We aim to prove Theorem 2 by induction \(n\). Clearly it is enough to show Theorem 2 for all induced subgraphs of \(\Lambda_{U}\). The base cases are when \(n\leq7\), which are easily verified. For the inductive step, suppose \(n\geq8\text{and}\Lambda_{U}[S]\) is an \(n-\)vertex subgraph of \(\Lambda_{U}\text{with}e\) edges. Additionally, suppose \(\Lambda_{U}[S]\) has the maximum number of edges out of all \(n-\)vertex subgraphs of \(\Lambda_{U}\). The inductive
+
+hypothesis is that all \(n^{\prime}-\)vertex subgraphs of \(\Lambda_{U},\text{where}3\leq n^{\prime}<n\), have at
+
+most \(e(n^{\prime})\) edges.
+
+## Claim 15. We may assume \(\Lambda_{U}[S]\) is 2-connected.
+
+Proof. \(\Lambda_{U}[S]\) must be connected, otherwise we could translate a connected component of \(\Lambda_{U}[S]\) to form additional edges contradicting the maximality of \(\Lambda_{U}[S]\). Suppose \(\Lambda_{U}[S]\) had a cut vertex \(v\). When we remove \(v\) from \(\Lambda_{U}[S]\) we create two connected components \(G_{1}=\Lambda_{U}[S_{1}]\text{and}G_{2}=\Lambda_{U}[S_{2}]\text{with}n_{1}\text{and}n_{2}\) vertices, and \(e_{1}\text{and}e_{2}\) edges. Case 1: \(n_{1}<4\mathbf{or}n_{2}<4\). Without loss of generality suppose \(n_{1}\leq3\). If \(n_{1}=1\), there is 1 edge removed when deleting \(S_{1}\) from \(\Lambda_{U}[S]\). Applying the inductive hypothesis to the remaining graph we obtain \(e\leq e(n-1)+1\leq e(n)\). So supposing that \(n_{1}\in\{2,3\}\), then there are at most 6 edges removed when deleting \(G_{1}\) from \(\Lambda_{U}\). Then the inductive hypothesis implies
+
+\[e\leq6(n-n_{1})-4 \sqrt{6(n-n_{1})-6} +6\leq6n- \sqrt{96n-63} \text{for}n_{1}=2,3\text{if}n\geq6.\]
+
+Case 2: \(n_{1},n_{2}\geq4\). Applying the inductive hypothesis to \(G_{1}\text{and}G_{2}\) we obtain
+
+\[e\leq_{\leq}6_{6}n_{n^{1}-}-_{6}4_{-} \sqrt{46\sqrt{n_{1}}} \frac{\overline{-6} +6n_{2}-}{6(n-5)-6} -^{4}4 \sqrt{\sqrt{6} \frac{n_{2}}{18} -_{+}6_{d}} \mathrm{e}^{+}\mathrm{g}(^{\deg}v).^{(v)}\]
+
+We can bound the degree of \(v\) by noticing the neighbourhood of \(v\) must be disconnected. It is easy to see through Menger’s theorem that the neighbourhood of \(v\text{in}\Lambda_{U}\) is 4-connected, which implies that the degree of \(v\text{in}\Lambda_{U}[S]\) is at most 8, hence
+
+\[e\leq6n-6-4 \sqrt{6(n-5)-6} -4 \sqrt{24} +8\leq6n- \sqrt{96n-63} \text{for}n\geq7.\]
+
+Claim 16. We may assume there is no line parallel to an element in \(U\) intersecting \(\Lambda\) but disjoint from \(S\), and with vertices from \(S\) on either side of the line.
+
+Proof. Suppose there is such a line \(L\). Since \(\Lambda_{U}[S]\) is 2-connected there must be at least two edges of \(\Lambda_{U}[S]\) that cross \(L.\text{If}L\) is parallel to a short edge (say parallel to \(g_{1})\), the only edges of \(\Lambda_{U}[S]\) that can cross \(L\) will be long edges perpendicular to \(L\) (in direction \(g_{1}-2g_{2})\). We shift a set of vertices on one side of the line towards \(L\) along one of the directions in \(U\). This is depicted in Figure 11, where the part of \(S\) above \(L\) is shifted by \(-g_{2}\). Notice that each

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+![](figure)
+
+![](figure)
+
+# Certificat de publicació / Certificado de publicación
+
+## La Agencia de Tecnología y Certificación Electrónica
+
+certifica que:
+
+El document amb títol / El documento con título:
+
+ANUNCIO-C9-2022-1285
+
+El resum del qual, generat amb l’algorisme SHA-256, és / Cuyo resumen o hash generado con el algoritmo SHA-256 es: wm0+ZT6id6FtQV99ZTUhO0yFeVYBBqGxr7QGMtr
+
+Es troba publicat en la URL / Se encuentra publicado en la URL: https://webges.uv.es/uvTaeWeb/DescargarAnuncio.do?codigo=ANU NCIO-C9-2022-1285
+
+En la data i hora / En la fecha y hora: 31-03-2022 14:08

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+Drahl, C., Cravatt, B.F., and Sorensen, E.J. (2005). Protein-reactive natural products. Angew. Chem. Int. Ed. 44, 5788–5809. Heyninck, K., Lahtela-Kakkonen, M., Van der Veken, P., Haegeman, G., and Vanden Berghe, W. (2014). Withaferin A inhibits NF-kappaB activation by targeting cysteine 179 in IKKb. Biochem. Pharmacol. 91, 501–509.
+
+Heyninck, K., Sabbe, L., Chirumamilla, C.S., Szarc Vel Szic, K., Vander Veken, P., Lemmens, K.J.A., Lahtela-Kakkonen, M., Naulaerts, S., Op de Beeck, K., Laukens, K., et al. (2016). Withaferin A induces heme oxygenase (HO-1) expression in endothelial cells via activation of the Keap1/Nrf2 pathway. Biochem. Pharmacol. 109, 48–61. Jessani, N., Humphrey, M., McDonald, W.H., Niessen, S., Masuda, K., Gangadharan, B., Yates, J.R., Mueller, B.M., and Cravatt, B.F. (2004). Carcinoma and stromal enzyme activity profiles associated with breast tumor growth in vivo. Proc. Natl. Acad. Sci. USA 101, 13756–13761.
+
+Jessani, N., Niessen, S., Mueller, B.M., and Cravatt, B.F. (2005). Breast cancer cell lines grown in vivo: what goes in isn’t always the same as what comes out. Cell Cycle 4, 253–255.
+
+Lee, I.-C., and Choi, B.Y. (2016). Withaferin-A—a natural anticancer agent with pleiotropic mechanisms of action. Int. J. Mol. Sci. 17, 290. Lee, J., Liu, J., Feng, X., Salazar Hernández, M.A., Mucka, P., Ibi, D., Choi, J.W., and Ozcan, U. (2016). Withaferin A is a leptin sensitizer with strong antidiabetic properties in mice. Nat. Med. 22, 1023–1032.
+
+Louie, S.M., Grossman, E.A., Crawford, L.A., Ding, L., Camarda, R., Huffman, T.R., Miyamoto, D.K., Goga, A., Weerapana, E., and Nomura, D.K. (2016). GSTP1 is a driver of triple-negative breast cancer cell metabolism and pathogenicity. Cell Chem. Biol. 23, 567–578. M, M., Zameer, F., Naidu, A., M N, N.P., Dhananjaya, B.L., and Hegdekatte, R. (2016). Evaluating the inhibitory potential of Withania somnifera on platelet aggregation and inflammation enzymes: an in vitro and in silico study. Pharm. Biol. 54, 1936–1941.
+
+Medina-Cleghorn, D., Bateman, L.A., Ford, B., Heslin, A., Fisher, K.J., Dalvie, E.D., and Nomura, D.K. (2015). Mapping proteome-wide targets of environmental chemicals using reactivity-based chemoproteomic platforms. Chem. Biol. 22, 1394–1405.
+
+Nomura, D.K., Long, J.Z., Niessen, S., Hoover, H.S., Ng, S.-W., and Cravatt, B.F. (2010). Monoacylglycerol lipase regulates a fatty acid network that promotes cancer pathogenesis. Cell 140, 49–61. Roberts, A.M., Ward, C.C., and Nomura, D.K. (2016). Activity-based protein profiling for mapping and pharmacologically interrogating proteome-wide ligandable hotspots. Curr. Opin. Biotechnol. 43, 25–33. Roberts, A.M., Miyamoto, D.K., Huffman, T.R., Bateman, L.A., Ives, A.N., Akopian, D., Heslin, M.J., Contreras, C.M., Rape, M., Skibola, C.F., et al. (2017a). Chemoproteomic screening of covalent ligands reveals UBA5 as a novel pancreatic cancer target. ACS Chem. Biol. 12, 899–904. Roberts, L.S., Yan, P., Bateman, L.A., and Nomura, D.K. (2017b). Mapping novel metabolic nodes targeted by anti-cancer drugs that impair triple-nega-tive breast cancer pathogenicity. ACS Chem. Biol. 12, 1133–1140. Sangodkar, J., Farrington, C.C., McClinch, K., Galsky, M.D., Kastrinsky, D.B., and Narla, G. (2016). All roads lead to PP2A: exploiting the therapeutic potential of this phosphatase. FEBS J. 283, 1004–1024. Tao, S., Tillotson, J., Wijeratne, E.M.K., Xu, Y., Kang, M., Wu, T., Lau, E.C., Mesa, C., Mason, D.J., Brown, R.V., et al. (2015). Withaferin A analogs that target the AAA+ chaperone p97. ACS Chem. Biol. 10, 1916–1924. Wang, C., Weerapana, E., Blewett, M.M., and Cravatt, B.F. (2014). A chemoproteomic platform to quantitatively map targets of lipid-derived electrophiles. Nat. Methods 11, 79–85. Weerapana, E., Wang, C., Simon, G.M., Richter, F., Khare, S., Dillon, M.B.D., Bachovchin, D.A., Mowen, K., Baker, D., and Cravatt, B.F. (2010). Quantitative reactivity profiling predicts functional cysteines in proteomes. Nature 468, 790–795. Xu, T., Park, S.K., Venable, J.D., Wohlschlegel, J.A., Diedrich, J.K., Cociorva, D., Lu, B., Liao, L., Hewel, J., Han, X., et al. (2015). ProLuCID: an improved SEQUEST-like algorithm with enhanced sensitivity and specificity. J. Proteomics 129, 16–24. Yu, Y., Hamza, A., Zhang, T., Gu, M., Zou, P., Newman, B., Li, Y., Gunatilaka, A.A.L., Zhan, C.-G., and Sun, D. (2010). Withaferin A targets heat shock protein 90 in pancreatic cancer cells. Biochem. Pharmacol. 79, 542–551. Zhou, Y., Li, W., Wang, M., Zhang, X., Zhang, H., Tong, X., and Xiao, Y. (2016). Competitive profiling of celastrol targets in human cervical cancer HeLa cells via quantitative chemical proteomics. Mol. Biosyst. 13, 83–91.

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+Values concerning item bank building (item linking, -calibration, equating)
+
+<table><tr><td rowspan="2"></td><td colspan="3">item bank building (evaluation)</td><td>result</td></tr><tr><td>testforms</td><td>items/form</td><td>examinees</td><td>items/bank</td></tr><tr><td>matrice test</td><td>76</td><td>12</td><td>29.728</td><td>456</td></tr><tr><td>analogy test</td><td>25</td><td>24</td><td>12.548</td><td>254</td></tr><tr><td>arithmetic test</td><td>32</td><td>18</td><td>17.008</td><td>288</td></tr></table>
+
+In a further step the adaptive 2 PL-algorithm was evaluated in combination with the three item banks mentioned above. Hence 8000 examinees (draftees) worked on the usual conventional test battery. Additionally, they received one of the three tests which corresponded to one test in the test battery. The additional test was administered as an adaptive one.
+
+The aim of adaptive testing in the testing situation is to gain maximum information with minimum effort. Hence the examinees should be given only those items that meet the person's level of functioning, so that the items will be solved in about fifty percent of the cases. If that holds, irrelevant items can be economized on, and, as one of various merits, it can be expected that adaptive testing, particularly the 2 PL, will need much less items as against conventional tests based on classical test theory. The conventional tests of our test battery consist of 20 items each with moderate reliability (\(r_{tt} = 0.74\) to \(r_{tt} = 0.88\)). Figure 1 is based on the compound result of the three tests. It shows the joint distribution of the adaptive items needed to reach a reliability of \(r_{tt} > 0.84\) for each person. This reliability is associated with a standard error of measurement of SEM = 0.39 for this sample. With the appearance of variance in the answer vector, the SEM is calculated whenever a person answers an administered item within the adaptive testing procedure. Before testing, the SEM value is fixed on a corresponding desired reliability level (in this case \(r_{tt} = 0.84\)). This cut-off value serves as stop criterion of the adaptive procedure as soon as the SEM falls below.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+ation. Variation happens readily in such markets. In contrast, selection is difficult because it is challenging to figure out which experience should be generalized from the extensive situation-specific knowledge that occurs. Which of the many experiences should be incorporated into the ongoing routines for the capabilities and which should be forgotten? The temptation is to generalize too quickly, and so to churn capabilities too often on the basis of idiosyncratic events (Gersick, 1994; Sastry, 1999). Finally, the order of implementation of dynamic capabilities is consequential. That is, dynamic capabilities are often combinations of simpler capabilities and related routines, some of which may be foundational to others and so must be learned first. Brown and Eisenhardt (1997) termed this property ‘sequenced steps.’ In their study of multiple product development processes in six firms in the computer industry, they observed that multiple product development required the combination of three simpler dynamic capabilities: single product development, probing the future and linking routines from one product development project to the next. Managers who built an effective dynamic capability to develop multiple products did so according to ‘sequence steps’ that had to be executed in the proper order. Single product development skills needed to come first to provide the platform for future products, then skills related to probing the future for new product opportunities, and finally time-pacing skills to create a product development rhythm connecting current products to future ones. Similarly, in his study of Hyundai, Kim (1998) found an appropriate sequencing of the learning of capabilities from the simpler and more predictable capabilities around manufacturing process creation to the more improvisational design ones in the development of design routines. Thus, effective implementation requires knowing both the ingredients (i.e., key commonalities of capabilities) and the recipe (i.e., order of implementation).
+
+# DISCUSSION
+
+The purpose of this paper is to explore dynamic capabilities and more generally, RBV. In addressing this agenda, we focused on the nature of dynamic capabilities, the impact of market dynamism, and their evolution. Our observations link to several research areas. Our work suggests reframing the concept of dynamic capabilities. Dynamic capabilities are not tautological, vague, and endlessly recursive as some have suggested (e.g., Priem and Butler, 2000; Williamson, 1999). Rather, they consist of many well-known processes such as alliancing, product development, and strategic decision making that have been studied extensively in their own right, apart from RBV. Their value for competitive advantage lies in their ability to alter the resource base: create, integrate, recombine, and release resources. Dynamic capabilities also exhibit commonalities across firms that are associated with superior effectiveness. So while the specifics of any given dynamic capability may be idiosyncratic to a firm (e.g., exact composition of a cross-functional product development team) and path dependent in its emergence, ‘best practice’ exists for particular dynamic capabilities across firms. These commonalities imply that dynamic capabilities are equifinal such that firms can develop these capabilities from many starting points and along different paths. They are also more homogeneous, fungible, and substitutable than is usually assumed. Overall, these observations suggest a modified conception of dynamic capabilities. Our work also suggests an expanded view of routines (Cyert and March, 1963; Nelson and Winter, 1982; Winter and Szulanski, 1999). We argue that, in moderately dynamic markets, routines in the form of dynamic capabilities are embedded in cumulative, existing knowledge. They involve analysis using existing knowledge and rules of thumb, followed by implementation. When this existing knowledge is codified, the resulting routines are often detailed and specific with predictable outcomes (Helfat, 1997; Nelson and Winter, 1982). Therefore, in moderately dynamic markets, dynamic capabilities exhibit the properties suggested in the traditional research where effective routines are efficient and robust processes (Cyert and March, 1963; Nelson and Winter, 1982). In contrast, in high-velocity markets, dynamic capabilities rely extensively on new knowledge created for specific situations. Routines are purposefully simple to allow for emergent adaptation, although not completely unstructured. Since new knowledge must be rapidly gained in each new

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+Interdisciplinary studies may have a positive impact on the development of law and facilitate its continuous adjustment to complex challenges.[^15] Chapters 8 and 9 note that the practice of international tribunals and courts consolidates customary principles and rules of fresh water law. Moreover, judicial and diplomatic means exist to assist States in the resolution of water conflicts. Dispute settlement provisions are contained in the 1992 Convention on the Protection and Use of Transboundary Watercourses and International Lakes adopted under the aegis of the United Nations Economic Commission for Europe (UNECE Water Convention) and the 1997 UN Watercourses Convention.16 Contrary to the case of the law of the sea, for example, in fresh water law there are no generally agreed upon compulsory methods producing final and binding solutions. In certain treaties or cases, however, the litigant States have agreed or agree on jurisdictional methods of settlement, through the application of conventional and customary rules of international law.17 Chapter 9 explores conflict related to the development of hydropower installations. The production of hydroelectricity is becoming increasingly attractive to States. However, hydroelectric projects involve enormous investment and have significant impacts: social, economic, environmental and others. When hydroelectric power generation projects are constructed and operated on international watercourses, disputes between riparians often arise. International courts and tribunals have had to consider the rights of use of riparian states on transboundary rivers in the context of disputes concerning hydroelectricity projects. They have dealt with the antagonism between, on the one hand, the permanence and need for stability for the construction and operation of hydroelectric projects and, on the other hand, the development of international law and the changing scientific and environmental conditions and values concerning the transboundary rivers.18
+
+## 3. SPECIFIC REGIMES OF FRESH WATER RESOURCES
+
+Section 3 continues with the analysis, focussing on specific fresh water governance regimes. Chapter 12 gives an in-depth analysis of the right to water. Several international documents have recognized this right. These include the
+
+[^15]: A Correia Lima Macedo Franca, ‘“Glocal” water management and regional
+
+organizations: multi-level law and transjuridicty’ infra Chapter 7. 16 A Tanzi, ‘Diplomacy, responsibility and accountability in transboundary water disputes’ infra Chapter 8. 17 L Caflisch, ‘Settlement of disputes concerning international watercourses’ infra Chapter 10. 18 D Azaria, ‘Fresh water and energy in international courts and tribunals: hydroelectricity installations on trasnboundary rivers’ infra Chapter 9.

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+Para a Análise Institucional toda pesquisa é intervenção, cabendo ao pesquisador explicitar como a intervenção se produziu, tomando como ponto de partida a própria implicação do pesquisador. Como vimos, o discurso da neutralidade produziu uma linha de intervenção de mão única, a intervenção como efeito inevitável do acesso à verdade. No limite, a verdade que é transformadora, sendo esta transformação desejada e benéfica, uma vez que suas condições de acesso são imparciais. Os efeitos destrutivos dessa ética conduzem a humanidade a confrontar-se com os limites do próprio planeta, de tal mote que se torna urgente explicitar como o inevitável caráter interventivo de qualquer pesquisa se processa. Portanto, é necessário um modo de se apropriar da inevitável dimensão interventiva de todo ato de pesquisar, para produzir relações menos violentas e produzir menos apropriação por parte do pesquisador. Partindo desse princípio ético, nada mais coerente do que partir do corpo do pesquisado como uma caixa de ressonância, ponto de inflexão ética de todo ato de produção de conhecimento: análise de implicação[8-10]. Sendo a análise de implicação a âncora da tripla aposta metodológica - aposta ético-metodológica: cartografar é cuidar de processos de singularização; aposta estéticometodológica: conhecer é transformar a si e o mundo (conhecer é criar); aposta político-metodológica: conhecimento é uma produção comum (do comum). A análise de implicação insere o pesquisador de modo imediato no duplo engajamento: conhecer é cuidar de si e do mundo, tendo como objeto processos singulares. Eis então nossa primeira aposta: aposta éticometodológica. A recusa à neutralidade implica num engajamento cuidadoso, pois o que afeta o pesquisador é fundamental para a cartografia dos regimes de forças de um determinado território. Os regimes de forças compostos de diferentes vetores são, antes de tudo, regimes de afetação: afetar e ser afetado. A análise de implicação não conduz a uma análise do sujeito, não se conclui no sujeito, mas parte dos afetos e afecções que mobilizam o corpo do pesquisador para mapear um plano de forças. O corpo do pesquisado opera como uma caixa de ressonância do encontro de forças, de vetores e linhas de diferentes matrizes: vetores políticos, planetários, afetivos, históricos. Se tomarmos o sujeito pesquisador como um sujeito historicamente e politicamente constituído, analisar as implicações passa essencialmente por analisar, por um lado, como essas forças históricas e políticas nos atravessam e nos constituem e por outro, como elas se atualizam no mundo atual. O exercício de análise de implicação é necessariamente um exercício de análise de si e do mundo, ou de um si no mundo e de um mundo em si. Trata-se, portanto, de um cuidado de si e do mundo como condição para acessar o plano de forças que constituem sujeitos e objetos.
+
+Ao mesmo tempo cabe notar que essa atitude é também condição para se acessar a dimensão singular da realidade. Ao contrário dos gregos, em que o cuidado era uma atitude de superação das ilusões do mundo das sensações para acessar o conhecimento universal e verdadeiro, a análise de implicação passa por um mergulho no mundo das sensações e dos regimes de afecção, não para acessar um conhecimento transcendental, mas para acessar o plano de constituição da realidade, um plano de imanência entre sujeito que conhece e mundo conhecido. A realidade, mesmo nas abordagens qualitativas, pode ser apreendida na sua dimensão de representação de estados de coisa. Interessa para essas abordagens apreender a regularidade dos fenômenos subjetivos e a normalidade das condutas numa pretensão cientificista. Capta-se desse modo somente a narrativa do instituído, de como os fenômenos são e como a realidade se apresenta. Rolnik [3] propôs uma distinção entre o olho retina e o olho vibratium, apontando que o olho retina capta o que nos é apresentado como natural, as formas constituídas, as representações. Para a cartografia interessa menos o que somos e mais o que estamos nos tornando, ou seja, a dimensão processual da realidade, em que esta variação de si é um indicador de um mundo que varia. A dimensão processual da realidade implica num exercício de ativação de um corpo vibratium ativação de um regime de sensibilidade e regimes de afetabilidade que a pesquisa instaura, uma atenção aos ritmos e intensidades. Afetamos e somos afetados durante o ato de pesquisar. O que nos afeta interessa na medida em que podemos, a partir da análise das nossas afetações, cartografar um campo problemático: análise de implicação. Acessar a realidade na sua dimensão processual ou os processos de singularização exige, portanto, método, pois do contrário apreendemos o que na realidade insiste em se afirmar como verdade, como imutável, ou seja, sua dimensão instituída. As entrevistas, grupos focais e observação podem acessar somente os sintomas institucionais, suas regularidades, as narrativas e práticas instituídas a depender do método que oriente a utilização de tais técnicas. A cartografia não se reduz a mapear estados de coisas, regularidades e verdades instituídas, embora este exercício seja indispensável para a cartografia. Para a cartografia interessam menos o singular e o diferente e mais os processos de singularização e de diferenciação. Mais do que afirmar que uma experiência local é singular, interessam os processos de singularização que se efetivam na experiência pesquisada. Daí a afirmação de que conhecer não é representar a realidade [4]. Pois a representação não acessa a dimensão processual da realidade, mas os estados de coisa, as formas instituídas. A análise de implicação é,

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+# Chapter 1
+
+# Elementary and Secondary Mathematics and Science Education
+
+Highlights.....................................................................................................................................1-4 Student Learning in Mathematics and Science.........................................................................1-4 Student Coursetaking in High School Mathematics and Science.............................................1-4 Teachers of Mathematics and Science......................................................................................1-5 Transition to Higher Education ................................................................................................1-6 Introduction..................................................................................................................................1-7 Student Learning in Mathematics and Science ............................................................................1-7 Mathematics and Science Performance in Grades 4, 8, and 12................................................1-8 Algebra Performance of Ninth Graders in 2009.....................................................................1-13 International Comparisons of Mathematics and Science Performance ..................................1-14 Student Coursetaking in High School Mathematics and Science ..............................................1-16 High School Graduation Requirements and Curriculum Standards .......................................1-17 Mathematics and Science Coursetaking in High School ........................................................1-18 Teachers of Mathematics and Science .......................................................................................1-21 Characteristics of High-Quality Teachers ..............................................................................1-22 School Factors Contributing to Teachers’ Effectiveness........................................................1-25 Transition to Higher Education..................................................................................................1-30 Completion of High School ....................................................................................................1-30 Enrollment in Postsecondary Education .................................................................................1-32 Conclusion .................................................................................................................................1-34 Notes ..........................................................................................................................................1-35 Glossary .....................................................................................................................................1-38 References..................................................................................................................................1-39
+
+### List of Sidebars
+
+Development and Content of NAEP Technology and Engineering Literacy Assessment ..........1-9 Mathematics and Science Achievement in Charter Schools......................................................1-11 Sample Items from PISA ...........................................................................................................1-16 Common Core State Standards ..................................................................................................1-18 Taking Algebra I Before High School .......................................................................................1-19 Measuring Teaching Quality......................................................................................................1-23 Teacher Attrition ........................................................................................................................1-29 State Student Tracking Systems.................................................................................................1-33

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+potentielle Erweiterung 4
+
+Johannispark
+
+potentielle Erweiterung 3
+
+Königsstein/Moosach Arme
+
+Sanierungsgebiet II " Altstadt "
+
+potentielle Erweiterung 2
+
+Volksfestplatz
+
+potentielle Erweiterung 1
+
+Bahnhof
+
+0 125 250 375 500
+
+Meter
+
+●
+
+# (Altstadt und Domberg Freising mit den frühen
+
+# Sanierungsgebiet II - Altstadt Siedlungserweiterungen) - Untersuchungsgebiet
+
+Umgriff Sanierungssatzung I "Am Wörth"
+
+![](figure)
+
+Umgriff Sanierungssatzung II "Altstadt"
+
+Ensemble Domberg und Altstadt Freising
+
+Abgrenzungsvorschlag nach dem Plan zur städtebaulichen Denkmalpflege
+
+![](figure)

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+The distributed middleware architecture of the application was based on the framework described in [10]. The physics of the scene was simulated using the Bullet Physics Library2. Simplified forms of non-convex pieces and holes were designed for the physics simulation in order to facilitate their manipulation, as the Bullet Physics Library does not handle nonconvex meshes properly.
+
+![](figure)
+
+Storage zone A
+Workspace W
+Fitters
+Non-fitters
+6 fitters complexity
+6 holes box
+Disposal zone B
+
+x3
+x3
+Fitters
+Non-fitters
+2 fitters complexity
+2 holes box[^inferred]
+
+![](figure)
+
+Storage zone A
+Workspace W
+Fitters
+Non-fitters
+6 fitters complexity
+6 holes box
+Disposal zone B
+
+x3
+x3
+Fitters
+Non-fitters
+2 fitters complexity
+2 holes box[^inferred]
+
+Fig. 2. on Top: six holes complexity task description. On Bottom: two holes complexity task description In both cases, twelve pieces were involved, six ”fitters” to place in the holed box and six ”non-fitters” to place in the disposal zone.
+
+The virtual coordinates of the flystick were linked to the physical ones by the mean of a standard proportional derivative control scheme for positions and a suboptimal control scheme with a quadratic cost for rotations, as described in [32]. Performance levels were set at
+
+2
+
+www.bulletphysics.org
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+Quality assurance procedure for ultrasound Problems with and troubleshooting ultrasound Nuclear medicine/gamma camera Computed tomography (CT) Positron emission tomography Magnetic resonance imaging (MRI) Radiation therapy Computer applications in imaging Medical network standards, DICOM, HL7 Picture archiving and communication system (PACS) Power distribution system Environmental control unit Oxygen storage and generation system
+
+# D. Test Equipment and Procedures
+
+Electrical safety analyzer Electrical safety testing Electrosurgical unit analyzer Electrosurgical unit inspection Medical ultrasound analyzer Physiologic monitor inspection Non-invasive blood pressure (NIBP) monitor analyzer NIBP inspection Invasive blood pressure inspection Cardiac output monitor inspection Pulse oximeter testing Electrocardiograph simulator/analyzer Electrocardiogram (ECG) testing Telemetry inspection Fetal monitor inspection Infusion pump analyzer Infusion pump inspection Infant incubator inspection Infant radiant warmer inspection Defibrillator analyzer Defibrillator testing External pacemaker inspection Heart-lung machine maintenance Ventilator inspection and maintenance Anesthesia machine maintenance Gas analyzer Pressure meter
+
+Flowmeter X-ray unit testing Centrifuge preventive maintenance Microscope maintenance Water bath inspection Slide stainer inspection Water bath operation and maintenance Laboratory incubator operation and maintenance Slide stainer operation and maintenance Diluter operation and maintenance Cell counter inspection Colorimeter operation and maintenance Spectrophotometer operation and maintenance
+
+# E. Troubleshooting and Repair
+
+Hospital grade plug Schematic, wiring diagram Circuit extraction Relay troubleshooting Ventilator operation and troubleshooting Compressor operation and repair Alternating pressure pump operation and troubleshooting Heating pad pump operation and troubleshooting Oxygen concentrator operation and troubleshooting Low-volume suction machine operation and troubleshooting Electrocardiograph operation and troubleshooting Defibrillator troubleshooting Physiologic monitor troubleshooting NIBP troubleshooting Invasive blood pressure troubleshooting Pulse oximeter operation and troubleshooting Infusion pump operation and troubleshooting Fetal monitor operation and troubleshooting Electrosurgical unit operation and troubleshooting Sterilizer operation and troubleshooting Diagnostic ultrasound troubleshooting Dental x-ray operation and troubleshooting X-ray unit operation and troubleshooting Blood gas analyzer operation and troubleshooting Centrifuge operation and troubleshooting Centrifuge repair

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+# CERTIFICATE
+
+This is to certify the thesis entitled "EFFECT OF PHYSICAL AND COMBINATION OF PHYSICAL AND YOGIC EXERCISES ON PERFORMANCE AND PERFORMANCE RELATED FACTORS OF BASKETBALL PLAYERS" submitted by M.M.G. HARI HARA RAMACHANDHIRAN for the award of the degree of Doctor of Philosophy in the Department of Physical Education, Bharathidasan University, Tiruchirappalli, is a bonafide record of research work carried out under my guidance and supervision. It is further certified that to the best of my knowedge the work reported here does not form part of any other thesis or dissertation for the award of any other degree or diploma.
+
+Place: Tiruchirappalli Date: ) 5 - o 5 - o S
+
+f-BTvA/Nra*TjC(G«»C-
+
+![](figure)
+
+![](figure)
+
+Qu. K. CHANDRASEKARAK.LTi.i^^
+
+LECTURER ^ V\e^d '/<:.
+
+©2PARTOENT OF PHYSICAL EDUCAYEC! BHARATHIDASAN UNIVERSIHj TIRUCHIRAPALLI-620Q2iC:,
+
+![](figure)
+
+(Dr. P. MARIAYYAH)
+
+m. ?. PARIAYYAf:. n»A=
+
+PROFESSOR AND HEAD, ESPARRIENT OF PHYSICAL EDUCATICj EHARATHIDASAN UNIVERSITY^ ?IRUCHIRAPPALU - 6 2 0 0?!6^

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+## ● Anita Gramigna y Juan C. González-Faraco Ferrara (Italia) y Huelva (España)
+
+Recibido: 26-12-08 / Revisado: 28-02-09 Aceptado: 23-05-09 / Publicado: 01-10-09
+
+DOI:10.3916/c33-2009-03-007
+
+# Videojugando se aprende: renovar la teoría del conocimiento y la educación
+
+### Learning with videogames: Ideas for a Renewal of the Theory of Knowledge and Education
+
+#### RESUMEN
+
+En este artículo se reflexiona sobre la relación del juego con el desarrollo cognitivo y los procesos educativos en un tiempo de profundos cambios tecnológicos. En concreto, sobre el papel de los videojuegos en la construcción del conocimiento y la formación de los niños. Profundizando en la lógica de los videojuegos y en la relación compleja que los usuarios entablan con ellos, pueden extraerse interesantes ideas para repensar la teoría del conocimiento y la educación. El texto ana-liza primeramente el papel de los videojuegos en el mundo lúdico, para internarse posteriormente en sus modelos narrativos propios. Finalmente se valoran sus virtualidades para el aprendizaje y en general para la educación, sin olvidar sus riesgos. Se concluye en la valorización cognitiva y pedagógica del pensamiento conectivo que pueden desarrollar los videojuegos.
+
+#### ABSTRACT
+
+In this paper we reflect on the relationship of games, cognitive development and educational processes during a time of profound technological change. Specifically, we consider the role of videogames in knowledge construction and in children’s education. By studying both the logic of video games and how players understand them, we can develop interesting ideas for re-thinking theories on knowledge and education. Primarily, this text analyzes the role of video games in the world of play in order to delve into its unique narrative models. Finally, we evaluate their virtual features for learning and education in general, then focusing on the risks entailed. We conclude with a cognitive-pedagogical evaluation of the connective thought that videogames can develop.
+
+#### DESCRIPTORES / KEY WORDS
+
+Videojuego, multimedialidad, aprendizaje, educación, construcción del conocimiento, pensamien- to lúdico. Videogames, multimedia, learning, education, knowledge construction, recreational thought.
+
+❖ Dra. Anita Gramigna es profesora del Dipartimento di Scienze Umane de la Facoltà di Lettere e Filosofia de la Università degli Studi di Ferrara (Italia) (grt@unife.it).
+
+❖ Dr. Juan Carlos González Faraco es profesor del Departamento de Educación de la Facultad de Ciencias de la Educación de la Universidad de Huelva (España) (faraco@uhu.es).

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+# August 2021
+
+![](figure)
+
+![](figure)
+
+## 10 Please Note: Due to COVID-19 restrictions, public meetings may be held through Video Teleconference and current schedules may change.
+
+###### MONDAY
+
+###### TUESDAY
+
+2 3:30 4:00 Verrocchio: BH View: Matthew Sculptor/Painter Specktor* 3 5:00 7/26 Health & Safety* 7:30 Greystone: Doug MacDonald 8:00 7/21 Architectural* 9:30 Sci&Movies: Incredible Hulk 10:00 Cañon: Music Carolyn*
+
+2:30 City Council Study Session* 4:30 The Syrinx Quintet 5:00 7/23 Audit & Finance Cmte.* 6:00 Cañon: Music Carolyn* 7:00 City Council Formal Mtg.* 9:30 DaVinci: To Know How To See 10:30 Verrocchio: Sculptor/Painter
+
+9 3:30 4:00 5:00 Simply Cañon: Mayor’s Bill Science: Cabinet* Rotella* 3D Medical 10 6:00 7/21 Architectural* 7:30 Greystone: Dimitry Olevsky 8:00 7/27 Recreation & Parks* 10:00 Verrocchio: Sculptor/Painter 10:30 Asian Life: Summer
+
+10:00 Arts & Culture* 3:30 Asian Life: Summer 4:00 8/4 Metro Purple Line Mtg. 5:00 8/5 Traffic & Parking* 7:00 Verrocchio: Sculptor/Painter 7:30 Simply Science: 3D Medical 8:00 8/5 Design Review* 10:00 8/9 Mayor’s Cabinet Mtg.*
+
+16
+
+3:30 Sci&Movies: Warp Drive 4:00 Cañon: Music Carolyn 5:00 8/11 Small Business Mtg.* 6:00 8/6 Audit & Finance Cmte.* 7:00 Asian Life: Summer 7:30 Greystone: Doug MacDonald 8:00 8/12 Public Works* 10:30 Greystone: Mei Hotta
+
+# 23
+
+3:30 STEM: Linguistic Diversity 4:00 Health & Safety* 6:00 Cañon: Bill Rotella* 7:00 Pets 90210: Pet Adoption 7:30 Simply Science: 3D Medical 8:00 8/18 Architectural* 10:00 8/19 Property Owners Cmte.* 3:30 Sci&Movies: Plants 4:00 8/19 Next BH Cmte.* 5:00 8/24 Recreation & Parks* 6:30 Cañon: Country Nation* 7:30 STEM: Linguistic Diversity 8:00 8/23 Health & Safety* 10:00 Verrocchio: Sculptor/Painter 10:30 Greystone: Riner Scivally
+
+# 30
+
+2:30 City Council Study Session*
+
+17
+
+4:30 Simply Science: 3D Medical 5:00 Cañon: Country Nation 6:00 8/9 Mayor’s Cabinet Mtg.* 7:00 City Council Formal Mtg.* 9:30 Sci&Movies: Warp Drive 10:00 8/11 Small Business Mtg.*
+
+## 24 2:00 Recreation & Parks*
+
+4:30 Greystone: Dimitry Olevsky 5:00 8/23 Health & Safety* 6:30 Sci&Movies: Warp Drive 7:00 8/23 Sunshine Task Force* 8:00 8/19 Human Relations* 10:30 STEM: Linguistic Diversity 3:30 Greystone: Parsa Peykar 4:00 Cañon: Singular Nature* 5:00 8/26 Planning* 7:30 Asian Life: Summer 8:00 8/4 Metro Purple Line Mtg. 9:00 Kariné Poghosyan: Piano* 10:00 Pets 90210: Pet Adoption 10:30 Sci&Movies: Plants
+
+# 31
+
+###### WEDNESDAY
+
+## 4 12:00 Metro Purple Line Mtg.*
+
+3:00 8/3 Study Session* 5:30 Verrocchio: Sculptor/Painter 6:00 Rent Stabilization* 8:00 Cañon: Music Carolyn* 9:00 BH View: Matthew Specktor* 10:00 7/26 Sunshine Task Force*
+
+11
+
+9:00 Small Business Task Force* 3:30 Greystone: Doug MacDonald 4:00 Verrocchio: Sculptor/Painter 4:30 Branding & Licensing Cmte.* 6:00 Cañon: Bill Rotella* 7:00 8/9 Mayor’s Cabinet Mtg.* 8:00 8/5 Traffic & Parking* 10:00 8/6 Audit & Finance Cmte.*
+
+# 12
+
+###### THURSDAY
+
+# 5
+
+9:00 Traffic & Parking* 1:30 Design Review* 3:30 Greystone: Mei Hotta 4:00 8/3 Formal Meeting* 7:00 Cañon: Bill Rotella* 8:00 8/3 Study Session* 10:00 7/23 Audit & Finance Cmte.* 8:00 Public Works* 1:30 Planning* 3:30 Greystone: Dimitry Olevsky 4:00 8/9 Mayor’s Cabinet Mtg.* 5:00 8/5 Design Review* 7:00 Cañon: Country Nation* 8:00 8/4 Rent Stabilization* 10:30 Simply Science: 3D Medical
+
+19
+
+18
+
+1:30 Architectural* 3:30 Meet the Author: Barbara Dee 4:00 8/11 Branding & Licensing* 5:00 8/17 Study Session* 7:00 Greystone: Riner Scivally 7:30 Verrocchio: Sculptor/Painter 8:00 8/17 Formal Meeting* 10:30 Greystone: Cara Lopez Lee
+
+9:00 Human Relations*
+
+3:30 Pets 90210: Pet Adoption
+
+5:30 Sci&Movies: Warp Drive 6:00 Asian Life: Summer 6:30 Next Beverly Hills Cmte.* 8:00 8/17 Study Session*
+
+10:00 8/11 Branding & Licensing*
+
+# 25
+
+26
+
+3:30 Simply Science: 3D Medical 4:00 8/12 Planning* 7:00 STEM: Linguistic Diversity 7:30 Greystone: Doug MacDonald 8:00 8/24 Recreation & Parks* 10:00 Greystone: Tobolowsky 10:30 Pets 90210: Pet Adoption 12:00 Metro Purple Line Mtg.* 3:30 Pets 90210: Pet Adoption Sept.
+
+1:30 Planning*
+
+5:30 Greystone: Peter J. Harris 6:00 Pets 90210: Pet Adoption 6:30 Asian Life: Summer 7:00 Cañon: Singular Nature* 8:00 8/10 Arts & Culture* 10:00 8/23 Sunshine Task Force*
+
+2
+
+# 1
+
+9:00 Traffic & Parking* 1:30 Design Review* 4:00 Simply Science: 3D Medical 4:30 Sci&Movies: Plants 5:00 8/18 Architectural* 7:00 Cañon: Joyce Partise* 8:00 9/1 Rent Stabilization* 10:00 8/11 Small Business Mtg.*
+
+Sept.
+
+4:00 8/19 Property Owners Cmte.* 5:00 8/23 Sunshine Task Force* 6:00 Rent Stabilization* 9:00 Sci&Movies: Plants 9:30 Cañon: Rod Lightning 10:30 Greystone: Doug MacDonald
+
+BHTV ON TIME WARNER CABLE CHANNEL 10 IN BEVERLY HILLS and WEBCAST AT BEVERLYHILLS.ORG
+
+4:30 Property Owners Task Force*
+
+# Weekday Schedule
+
+###### FRIDAY
+
+# 6
+
+3:00 8/4 Rent Stabilization* 5:00 8/4 Metro Purple Line Mtg. 6:00 DaVinci: To Know How To See 7:00 Verrocchio: Sculptor/Painter 7:30 Asian Life: Summer 8:00 8/3 Formal Meeting* 10:00 Cañon: Bill Rotella*
+
+13
+
+12:00 8/11 Branding & Licensing* 1:00 8/11 Small Business Mtg.* 2:00 8/9 Mayor’s Cabinet Mtg.* 3:00 8/12 Public Works* 6:30 Simply Science: 3D Medical 7:00 8/4 Metro Purple Line Mtg. 8:00 8/12 Planning* 10:00 BH View: Matthew Specktor*
+
+20
+
+3:00 8/10 Arts & Culture* 4:30 Verrocchio: Sculptor/Painter 5:00 8/17 Formal Meeting* 7:00 Greystone: Doug MacDonald 7:30 Sci&Movies: Warp Drive 8:00 8/19 Property Owners Cmte.* 9:00 8/19 Next BH Cmte.* 10:00 Cañon: Joyce Partise
+
+27
+
+3:00 8/19 Human Relations* 5:30 8/19 Next BH Cmte.* 6:30 Meet the Author: Hena Khan 7:00 STEM: Linguistic Diversity 7:30 Greystone: Cara Lopez Lee 8:00 8/26 Planning* 10:30 Sci&Movies: Warp Drive
+
+4:00 9/2 Traffic & Parking*
+
+Sept.
+
+3 3:00 Cañon: Music Carolyn 6:30 Greystone: Mei Hotta 7:00 Sci&Movies: Plants 7:30 Simply Science: 3D Medical 8:00 9/2 Design Review* 9:30 Pets 90210: Pet Adoption 10:00 8/23 Sunshine Task Force*

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+![](figure)
+
+![](figure)
+
+INSTRUCTIONS: Please provide information concerning your health for participation in 4-H Events for the current year. If you are a person with a disability and desire any assistive devices, services, or other accommodations to participate in activity, please contact your local Extension office during business hours at least 7 days prior to the event to discuss accommodations. PLEASE PRINT ALL INFORMATION. (NOTE: Both sides of this form must be completed.)
+
+COUNTY ____________________________________________________________________
+
+# IDENTIFICATION
+
+NAME ______________________________________________________________________ FEMALE □ MALE □
+
+Last
+
+First
+
+MI
+
+MAILING ADDRESS _______________________________________ CELL PHONE (_______) ___________________
+
+CITY __________________________________________________________ STATE______ ZIP__________________
+
+HOME PHONE (______) ___________________ BIRTHDATE _______________ EMAIL________________________
+
+# EMERGENCY CONTACT
+
+NAME _______________________________________________________ CELL PHONE (_____) ________________
+
+ADDRESS ____________________________________________________ HOME PHONE (_____) ________________
+
+RELATIONSHIP___________________________________ WORK PHONE (_____) ________________
+
+# PHYSICIAN/INSURANCE INFORMATION
+
+NAME OF PHYSICIAN _______________________________________________ PHONE (_____) ________________
+
+MEDICAL/HOSPITAL INSURANCE ____________________________________________________________________
+
+Carrier
+
+Policy ID #
+
+# MEDIA RELEASE
+
+The Virginia Polytechnic Institute and State University/College of Agriculture and Life Sciences (CALS) periodically uses electronic and traditional media (e.g., photographs, video, audio footage, testimonials) for publicity and educational purposes. By my signature on this form, I acknowledge receipt of this document and give permission to the College of Agriculture and Life Sciences and its designee to use such reproductions for educational and publicity purposes in perpetuity without further consideration from me. I understand that I will need to notify Virginia Tech/College of Agriculture and Life Sciences if any changes to my situation occur that will impact this media release permission.
+
+PLEASE INITIAL
+
+YES__________
+
+NO___________

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+la actualizan, posibilitando así, la constitución de una colectividad, un ‘nosotros’”. (TATSCH, 2013, p. 67)[^3]
+
+En lo que respecta a los cánones nacionales envueltos en la traducción a analizar y sus representaciones del gaucho, es interesante observar que “la literatura gauchesca rioplatense contribuyó en la invención teórica de una nueva región cultural”, entre cuyos textos apropiados destaca Martín Fierro, el cual “puede ser considerado uno de los textos fundadores de la gauchesca brasileña” (MASSINA, 2002, p. 103-104). En otras palabras, es de estimarse cierto vínculo entre la traducción del poema de Hernández y la forma en que la cultura receptora se ve representada a sí misma a través de los gauchos allí retratados. Entre los rasgos más sobresalientes de dicha caracterización, interesa especialmente reflexionar acerca del atributo heroico de Fierro, dado que en el poema de Hernández existe una voluntad de escamotear dicho rasgo (BORGES-GUERRERO, 1999). Contemporáneamente a la publicación de Martín Fierro, la década de 1870 dio a la luz dos textos literarios que inmortalizaron al gaucho en la literatura brasileña: O gaúcho, de José de Alencar, y O vaqueano, de Apolinário Porto Alegre. El aspecto de mayor relevancia en el que las referidas obras brasileñas coinciden es la construcción de una imagen idealizada del gaucho y de la sociedad a la que pertenecía. Ya fuera que se ocultara el trabajo del peón de estancias, de los inmigrantes europeos y del esclavo en las charqueadas, así como la devastación de la economía, familias y estancias durante la revolución; o que se presentara a gauchos, patrones, esclavos, indios y soldados farropillas como héroes que conviven en un mundo sin jerarquías; como resultado se exalta la figura del guacho y se instaura el mito de la democracia de la pampa, dispositivos discursivos estratégicos para legitimar el poder a las elites locales, en una comunidad integrada gracias a una identidad regional, a su vez proyectada como guardiana de las fronteras nacionales y, por tanto, participante de una identidad mayor (DE CAMPOS, 2008). La literatura regionalista de tema gauchesco seguiría la misma tendencia durante su edad de oro. De acuerdo con Ligia Chiappini, dentro de esta tradición literaria “el personaje no solo es reducido a un tipo social, sino a un tipo social mitificado: el gaucho y su antípoda –el antigaucho, el antihéroe” (LEITE, 1978, p. 39)[^4]. Estas obras, que sirven como referencia para la
+
+[^3]: Texto fuente: “Fala a partir da coletividade, do lugar social ocupado pelo grupo que representa como pessoa autorizada (...). É responsável pelo que é dito a partir dos lugares da memória, distribuindo os saberes e os dizeres que atualizam a memória, possibilitando assim, a constituição de uma coletividade, um ‘nós’”.
+
+[^4]: Texto fuente: “O personagem não é só reduzido a tipo social, mas a um tipo social mitificado: o gaúcho heroi e seu antípoda – o antigaúcho, o anti-herói”.

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+maar niet over met hoeveel zij zullen zijn.[^4] Strikt genomen kunnen quota de aanwezigheid van bepaalde maatschappelijke groepen in de politiek enkel bevorderen zonder een formele verbintenis te bevatten in termen van de electorale uitkomst. Zij kunnen de kans van bepaalde kandidaten om verkozen te worden echter sterk verhogen. Het argument van deze bijdrage is dat per wet opgelegde quota net zoals voorbehouden zetels een bepaald aantal mannelijke respectievelijk vrouwelijke verkozenen kunnen opleggen en dit onafhankelijk van de verkiezingsuitslag, ook al spelen zij louter in op de inputzijde van het electorale proces. De doorslaggevende factor in dit laatste geval is de definitie van de quota in kwestie, meer bepaald de wijze waarop de formulering van quota inspeelt op de modaliteiten van het kiesstelsel waarin zij toegepast worden. Het argument is dat hoe meer een quotawet de modaliteiten van een kiesstelsel uitbuit ten bate van een genderevenwicht in de politiek, hoe meer deze maatregel een uitslag ‘garandeert’, ook al hebben die quota betrekking op de inputzijde van het electorale proces. Echter, enkel quotawetten en niet door partijen opgelegde quota kunnen een dergelijke impact hebben. Quotawetten gelden tenslotte voor alle partijen in dezelfde mate. Als een quotawet zodanig geformuleerd is dat zij het aantal mannelijke en vrouwelijke gekozenen bepaalt, heeft de verkiezingsuitslag enkel nog invloed op de numerieke verhoudingen tussen de verschillende partijen. Gezien partijquota van partij tot partij kunnen verschillen heeft de verkiezingsuitslag in dat geval een grotere invloed op het totaal aantal mannelijke en vrouwelijke gekozenen. Partijquota zouden enkel de dwingende impact van wettelijke quota hebben indien alle partijen zichzelf dezelfde regels opleggen en deze op identieke wijze naleven. Deze bijdrage sluit aan bij recent onderzoek naar de interactie tussen de modaliteiten van kiesstelsels en genderquota.[^5] De achterliggende hypothese is dat dergelijke maatregelen enkel effect hebben indien zij nauwkeurig zijn afgestemd op het kiesstelsel waarin zij toegepast worden. Deze bijdrage wil aantonen dat wettelijk opgelegde genderquota een effect kunnen afdwingen door de formulering van de quotamaatregelen nauwkeurig af te stemmen op de precieze modaliteiten van het
+
+[^4]: Tenzij men uitgaat van de weinig realistische hypothese dat zodanig veel stemmen op Vlaamse kandidaten uitgebracht worden dat zij recht zouden hebben op een groter aantal zetels dan wettelijk voor hun is voorbehouden.
+
+[^5]: BAUDINO, C., The Parity Reform in France: Promises and Pitfalls. Onuitgegeven paper, 2002. DESCHOUWER, K. & P. MEIER, Tussen wens en werkelijkheid: reële en mogelijke impact van quo- tawetten op kieslijsten. Wetenschappelijke monografie 9. Brussel, Ministerie van de Vlaamse Ge- meenschap, 2002. HTUN, M.N. & JONES, M.P., Engendering the Right to Participate in Decision-making: Electoral Quotas and Women’s Leadership in Latin America, pp. 32-56 in N. Craske & M. Molyneux (eds.), Gender and the Politics of Rights and Democracy in Latin Ameri- ca. New York, Palgrave, 2002. JONES, M., Gender Quotas, Electoral Laws, and the Election of Women: Lessons from the Argentine Provinces, in Comparative Political Studies, vol. 31, 1998, 3-21. MARQUES-PEREIRA, B. & P. NOLASCO (eds.), La représentation politique des femmes en Amérique latine. Paris, L’Harmattan, 2001. MOSSUZ-LAVAU, J., Political Parity between Men and Women in France: History and First Balance. Paper delivered at the Conference on Women’s Quotas in Urban Local Government: A Cross-national Comparison, New Delhi, Institute of Social Sciences, February 2003.

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+![](figure)
+
+![](figure)
+
+# Cont... (distribution time)
+
+### In P2P architecture:
+
+- In P2P, when a peer receives some file data, it can use its own upload capacity to redistribute the data to other peers.
+
+Calculating the distribution time for the P2P architecture is somewhat more complicated. A simple expression for the minimal distribution time:
+
+- Step1: At the beginning, only the server has the file.
+
+- To get this file into the community of peers, minimum distribution time is at least F/us.
+
+- Step2: the peer with the lowest download rate cannot obtain all F bits of the file in less than F/dmin seconds.
+
+- Step3: the total upload capacity of the system is, utotal = us + u1 + ... + uN.
+
+  - The system must deliver (upload) F bits to each of the N peers.
+
+  - This cannot be done at a rate faster than utotal.
+
+  - So, the minimum distribution time is also at least NF/(us + u1 + ... + uN).
+
+![](figure)
+
+- Finally, the minimum distribution time for P2P

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+In accordance with the invention, this object is attained by means of a novel yarn guide for the traversing feeding of a yarn to a rotatingly driven winding bobbin for producing a cheese in a winding head of a textile machine. In accordance with the present invention, the yarn guide is directly connected with at least one mechanical energy storage device which is permanently coupled with the yarn guide during the entire traversing movement for aiding the reversal of the direction of movement. The potential energy of the mechanical energy storage device continuously increases in the direction toward the dead points of the traversed area. As a result, the mechanical energy storage device together with the yarn guide constitute an essentially harmoniously oscillating mechanical system.
+
+The invention is advantageously further developed by the below-described additional features.
+
+The direct connection of the yarn guide with a mechanical energy storage device results in that no indirect transfer of forces is necessary at all. The only mass creating a moment of inertia is constituted by the yarn guide itself. By means of this the dynamics of the system are clearly improved over the prior art. The forces and moments of inertia in the area of the considerable angular accelerations at the reversing points are very small. It is possible to achieve a very rapid reversal of the direction of movement in the area of the end areas of the traversing movement without a considerably energy outlay.
+
+In this connection it should be pointed out that in accordance with the present invention the entire oscillating body consisting of the yarn guide finger, the shaft supporting it, as well as the coil used as drive mechanism and its frame, are called the yarn guide.
+
+It is possible to change the traversing width exclusively by means of the drive mechanism, i.e. by means of control or regulation. No mechanical adjustment at all is required for this if the energy storage device is designed for a maximum yarn guide stroke. Since with an increased stroke the torque required in the area of the reversal of the movement direction is also increased, the greater potential energy of the energy storage device being created in case of an increased stroke is useful. The noise level is lowered by the jolt-free operation of the yarn guide, and its service life is increased. Moreover, the regulating quality of the control is improved and the precise keeping of the traversing width is simplified.
+
+The employment of a torsion spring, in particular a helical spring, is particularly advantageous in connection with the yarn guide of the invention. With an oscillation amplitude of, for example 60 to 70E, applicable here, such a helical spring also has the same progression of the characteristic curve of the spring to the left and right of the zero point. However, two helical springs can also be employed, which can then additionally be used for the electrical current supply for a coil of a drive mechanism of the yarn guide.
+
+If the helical springs have opposite winding directions, the same spring torque progression is achieved in both oscillation directions in every case. However, as already mentioned, this advantage gains importance only if the angle of the deflection of the yarn guide out of the position of rest exceeds a threshold value, because in that case the progression of the characteristic curve of the spring is different during opening and closing of the spring.
+
+While the energy storage device reduces the area under the graph of the drive torque square up to a third of the value which would be necessary without such an energy storage device anyway, setting the resonance frequency of the oscillating system to the upper limit of the required oscil-
+
+lation frequency leads to the support of the oscillating motion, and therefore the relief of the electromechanical drive mechanism, being especially intense during extreme situations.
+
+It is assured by the sturdiness of the arrestment of the energy storage device, that the energy storage device aids the oscillating motion in both directions to the same extent.
+
+There is, of course, also the possibility to make the selection of the springs from the start in such a way, that their characteristic spring curves corresponds as much as possible to the desired oscillation behavior. In this connection it is even desirable that the characteristic spring curve does not extend in a straight line, but instead progressively increases after the reversing point, as already mentioned. The dimensioning of the spring should be selected for optimization in such a way that the integral under the driving torque square becomes the smallest. Because of this, the required power consumption of the drive mechanism for the yarn guide is the lowest.
+
+The flexibility of the oscillation system can be improved still further if, in addition, the suspension of the at least one spring during the oscillation movement of the yarn guide is adjusted in a controlled manner. In a range of a lesser oscillation frequency, for example 5 Hz, the spring assistance in the reversing area of the yarn guide is required not at all or only to a small extent. The energy for bending the spring or, if an additional spring is used, the two springs, must be supplied by the electrical drive mechanism of the yarn guide in the remaining oscillation range outside of the oscillation dead center points, in which no angular acceleration is present. This energy can be reduced in that the suspension of the at least one spring for all practical purposes oscillates synchronously. This means that at very low oscillation frequencies the suspension could also oscillate with the same amplitude. However, in this case the drive mechanism for the suspension must be included when determining the energy balance. It can therefore be advantageous, depending on the actual conditions, to let the suspension oscillate along at the same frequency, but with reduced amplitude. In that case the bending of the spring takes place only in the lower range of the characteristic curve of the spring.
+
+With increasing frequency, the requirement for assisting the movement reversal increases. For this reason a relief of the spring in the area of the reversing points is disadvantageous. The potential spring energy stored in the area of the reversing points can be clearly increased by appropriate phase-shifting between the oscillating movement of the suspension and of the yarn guide. However, in this way merely a far-reaching effect on a fixed frequency would be achieved, which could also be achieved with a stronger spring or a plurality of springs.
+
+But the flexibility of the system offers the possibility of relieving the spring over the larger part of the stroke by an oscillation of the suspension in the same direction, and to clearly increase the spring force only directly prior to reaching the reversing point by swiftly reversing the a oscillating movement of the suspension into the opposite direction.
+
+A "weak spring" is created in this way over the larger portion of the stroke length, and a very strong spring in the reversing points. This exactly corresponds to the desired assistance of the oscillating movement of the yarn guide. However, in the course of this, the system is not disadvantageously affected by an abrupt introduction of a spring force or adjustment of a spring constant, as in the prior art already
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+![](figure)
+
+![](figure)
+
+numerous industrial type structures associated with the water industry. Here the valley widens and the river basin affords room for pools and wetland areas supporting a range of plant and fish life. Approaching the valley head there is a gradual change to wilderness and a sense of being in the heart of the Bowland fells.
+
+● 5.
+
+![](figure)
+
+(53.97486; -2.52965)
+
+This is the furthest point of the route. After taking in the scenery, retrace your steps back to the bridge then follow the `Short Route` description back to the village.

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+![](figure)
+
+![](figure)
+
+![](figure)
+
+![](figure)
+
+# Midwifery
+
+### Corrigendum
+
+## Corrigendum to “Midwifery continuity of carer in an area of high socioeconomic disadvantage in London: A retrospective analysis of Albany Midwifery Practice outcomes using routine data (1997–2009)” [Midwifery 48 (2017) 1–10]
+
+Caroline S.E. Homer, RM, MMed(ClinEpi), PhD Professora,⁎, Nicky Leap, DMid Adjunct Professor of Midwiferya, Nadine Edwards, PhD Independent Researchera,b,c, Jane Sandall, PhD, CBE Professor, NIHR Senior Investigatorb,c
+
+a Centre for Midwifery, Child and Family Health, Faculty of Health, University of Technology Sydney, PO Box 123, Broadway, Sydney, New South Wales 2007, Australia b Division of Women's Health, Faculty of Life Sciences & Medicine, King's College London, United Kingdom c Women's Health Academic Centre, St. Thomas' Hospital, London SE1 7EH, United Kingdom
+
+The authors regret that on page 8 we inadvertently referred to NICE as providing policy and policy direction. This was incorrect. NICE provides national clinical guidance based on systematic reviews. The sentence should read: A large study comparing perinatal outcomes by planned place of birth (Birthplace in England Collaborative Group, 2011) supported the clinical guidance to offer women with low risk pregnancies a choice of birth settings, including birth at home (NICE, 2014). The authors would like to apologise for any inconvenience caused.
+
+DOI of original article: http://dx.doi.org/10.1016/j.midw.2017.02.009 ⁎ Corresponding author. E-mail address: caroline.homer@uts.edu.au (C.S.E. Homer).

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+límites de exposición o se presenta irritación, puede requerirse ventilación y evacuación.
+
+## Consideraciones generales sobre higiene
+
+Manipular de acuerdo con las buenas prácticas de higiene y seguridad industrial. No comer, beber o fumar mientras se manipula este producto. Lavar las manos antes de los recesos e inmediatamente después de manipular el producto. Úsense guantes adecuados y protección para los ojos/la cara. Evitar el contacto con la piel, los ojos o la ropa.
+
+## Propiedades físicas y químicas
+
+Estado físico Aspecto Olor Color Umbral olfativo
+
+Punto de inflamación Tasa de evaporación Inflamabilidad (sólido, gas)
+
+## 9. PROPIEDADES FÍSICAS Y QUÍMICAS
+
+Pasta / gel Cobre Bronce Petróleo No hay información disponible No hay datos disponibles
+
+## Valores
+
+Neutral > 315 °C
+
+## Observaciones Método
+
+No se conocen
+
+No se conocen No se conocen No se conocen
+
+No se conocen No se conocen
+
+Propiedad pH Punto de fusión o congelación Punto de ebullición y rango de ebullición< 316 °C > 221 °C No hay datos disponibles No hay datos disponibles Límite de inflamabilidad en el aire 7% 0.9% <0.01 kPa @ 20°C >5 (air = 1) 1.17 Insoluble No hay datos disponibles Not Available >315 °C / >500 °F No hay datos disponibles No hay datos disponibles No hay datos disponibles No hay información disponible No hay información disponible
+
+Límite superior de inflamabilidad Límite inferior de inflamabilidad Presión de vapor Densidad de vapor Densidad relativa Solubilidad en agua Solubilidad(es) Coeficiente de reparto: n-octanol/agua Temperatura de autoinflamación Temperatura de descomposición Viscosidad cinemática Viscosidad dinámica Propiedades explosivas Propiedades comburentes
+
+Otras informaciones Punto de ablandamiento Peso molecular Contenido de COV (%) Densidad del líquido Densidad aparente Tamaño de partícula Distribución granulométrica
+
+## Reactividad
+
+## Estabilidad química
+
+## Posibilidad de reacciones peligrosas Polimerización peligrosa
+
+No hay información disponible No hay información disponible Ninguno(a) No hay información disponible No hay información disponible No hay información disponible No hay información disponible
+
+No se conocen
+
+No se conocen No se conocen No se conocen No se conocen
+
+## 10. ESTABILIDAD Y REACTIVIDAD
+
+No hay información disponible.
+
+Estable en condiciones normales.
+
+Ninguno durante el procesado normal.
+
+No ocurre polimerización peligrosa. _____________________________________________________________________________________________
+
+![](figure)

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+![](figure)
+
+![](figure)
+
+Figura 1. Comparación del uso de los principales recursos electrónicos disponibles para los usuarios del Hospital de Fuenlabrada (eneromayo 2005)
+
+![](figure)
+
+![](figure)
+
+Figura 2. Media de utilización de UTD: 548 consultas/mes
+
+gran diferencia con respecto a otros1 (figura 1).
+
+- Su frecuencia de uso va en aumento (figura 2); de hecho se ha llegado a picos de casi 1.000 consultas mensuales cuando el produc- to empezó a estar disponible, con una media en los últimos meses de entre 600 y 700 consultas.
+
+- Las materias más consultadas son enfermedades infecciosas y gastroenterología (figura 3). El he-cho de que temáticas como ginecología y obstetricia o pediatría no aparezcan en puestos más elevados de uso, siendo numerosos los facultativos de estas especialidades en el hospital, orienta a la biblioteca en el sentido de organizar campañas de difusión y formación específicas para estos servicios.
+
+- Tan sólo el 0,5% de las consultas se encaminaron a los apartados What’s new in..., lo cual demuestra que el uso de UTD es primordialmente clínico, más que de actualización, aunque sin embargo tiene una alta capacidad para ello, como el propio nombre del produc- to sugiere.
+
+- De los datos provisionales de la encuesta distribuida entre los usuarios se pueden avanzar algunos datos:
+
+1. Amigabilidad de su sistema
+
+de consulta (con una nota media de 3’35 sobre 5).
+
+2. La sensación de éxito en la búsqueda de información es alta (con una nota media de 3’14 sobre 5; ningún usuario valoró negativamente esta variable).
+
+![](figure)
+
+![](figure)
+
+Figura 3. Temática de las consultas realizadas a UpToDate (marzo 2004 a junio 2005)
+
+3. Se realiza un gran número de búsquedas en presencia del paciente. Entre los recursos electrónicos disponibles desde las consultas médicas es el más utilizado (39% de los que alguna vez han utilizado recursos electrónicos estando con un paciente). Además, mientras que con otros recursos electrónicos accesibles on site los usuarios se que-jan de la no accesibilidad desde el exterior, en el caso de UTD no ocurre; el personal médico no lo echa tanto de menos como el acceso a ciertas revistas, libros, etc., electrónicos.
+
+## «Es un recurso que merece la pena suscribir en hospitales en su versión online, ya que los facultativos pueden consultarlo desde su puesto de trabajo»
+
+- El buscador de contenidos no permite las combinaciones booleanas aunque sí un “estrechamiento” de los resultados (Fox; Moawad, 2003). No obstante, consultada la empresa distribuidora, tienen entre las mejoras previstas del producto incluir su utilización para 2006.
+
+- A la hora de interpretar las estadísticas de uso se observa una no coincidencia entre la indización de algunos temas y la identificación de especialidades. Por ejemplo: la revisión más visitada durante los

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+#   ! <   7 ! .    @AB9   !    # !   !   ! "   C: 4 
+
+### 7  !    # 7   ! 
+
+Mit diesem letzten Abschnitt eines dreiteiligen Überblicks über die Entwicklung der drei Fischbestände, die für die Fischerei in der Ostsee nach den Fangmengen das Hauptergebnis erbringen, soll nach dem Dorsch und dem Hering nun auch die Sprotte behandelt werden. Es sei auch in diesem Zusammenhang noch einmal auf das Ziel der 1973 verabschiedeten „Danziger Konvention zum Schutz der lebenden Ressourcen der Ostsee“ hingewiesen, das in Weiterentwicklung und Anpassung an die heutigen Bedingungen in einer nachhaltigen Nutzung der Fischbestände der Ostsee besteht.
+
+###   4      7
+
+Für die Sprotte der Ostsee haben sich die zuständigen Wissenschaftler im Internationalen Rat für Meeresforschung (ICES) 1990 darauf verständigt, von einem relativ einheitlichen Bestand auszugehen und Abschätzungen dieses Bestandes nicht mehr, wie vorher üblich, in Untergebiete aufzuteilen. Dieses ist zumindest für die eigentliche Ostsee sinnvoll, die den Bereich im Südwesten von der Arkonasee (Gebiet 24) bis zum Norden an der Grenze zur Ålandsee umfasst (Gebiet 29). Die Sprotte der Beltsee entspricht vom Körperbau eher der des Kattegatts/Skagerraks und stellt somit eine Übergangsform zur Nordseesprotte dar. Die Sprotte ist, wie
+
+![](figure)
+
+![](figure)
+
+Abbildung 1: Hauptlaichplätze der Sprotte in der Ostsee (verändert nach Parmanne et al. 1993). Main spawning area of Baltic Sea sprat (adapted from Parmanne et al.1998).
+
+der Dorsch, ein Portionslaicher. Das bedeutet, dass sie ihre Eier über einen größeren Zeitraum (der Einzelfisch über ca. 2 Monate) verteilt in Portionen in das Wasser abgibt, wo diese sich dann frei schwebend zu Larven entwickeln. Daraus ergibt sich eine weitere Gemeinsamkeit mit ihrem Haupträuber, dem Dorsch: die Eientwicklung ist vom Salzgehalt des Wassers abhängig. Aller-dings schweben und entwickeln sich Sprotteneier noch bei deutlich geringerem Salzgehalt als der Laich des Dorsch. Deshalb laichen Sprotten erfolgreich auch noch bis hinein in den Bottnischen Meerbusen und im Finnischen Meerbusen, ganz im Gegensatz zum Dorsch, des-sen potentiell nördlichstes Laichgebiet etwa im mittleren Gotlandbecken liegt.
+
+## Fish stocks of the Baltic Sea and their development since 1970 and conclusions for sustainable exploitation – Part 3: Sprat
+
+The sprat of the Baltic Sea is not as short-lived as in other Seas probably because fish predator species are restricted mainly on cod and salmon. Sea bird populations are much smaller and marine mammals are rare in the Baltic Sea. The sprat stock biomass is fluctuating strongly. The fluctuation is mainly influenced by the stock recruitment and is also dependent on the strength of the cod stock. After a strong decrease during the 1980ies sprat catches increased again from 1992 onwards and reached a peak with over half a million tonnes in 1997. At about the same time the character of the Baltic Sea sprat fishery changed from catches mainly for human consumption to catches mainly for industrial purposes initiated by the fishery of Sweden. The recent record high catches of sprat have been possible only due to the low level of the cod stock of the main Baltic Sea Basins over some years. A sprat fishery on such a high catch level might cause conflicts with a recovering cod stock in future.

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+# Bell Tower Millfield Street [1]
+
+By: Huettner, Alfred F. (Alfred Francis), b. 1884
+
+Image not found https://hpsrepository.asu.edu/bitstream/handle/10776/2140/Huettner_033.web.jpg
+
+![](figure)
+
+![](figure)
+
+## Subject
+
+Woods Hole (Mass.) [2]
+
+## Topic
+
+Places [3] People [4]
+
+## Publisher
+
+Marine Biological Laboratory Archives
+
+## Rights
+
+Licensed as Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported. http://creativecommons.org/licenses/by-nc-sa/3.0/
+
+## Format
+
+Photographs [5]
+
+## Last Modified
+
+Wednesday, July 4, 2018 - 04:40
+
+## DC Date Accessioned
+
+Tuesday, May 15, 2012 - 13:23
+
+## DC Date Available
+
+Tuesday, May 15, 2012 - 13:23
+
+## DC Date Created

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+THE FACULTY OF SOCIAL SCIENCES of the University of
+
+the West Indies is pleased to be associated with the Institute of Caribbean Studies at the UWI Mona, in presenting the Conference on Caribbean Culture, in honour of distinguished Caribbean Scholar, P rofessor Rex Nettleford. The conference will be opened on Sunday, March 3, 1996 by the Rt. Hon. Michael Manley, P.C., and closes with an address by the honoree on Wednesday, March 6, at 8 pm.
+
+# PROG RAMMEcDRAFT)
+
+|  | Monday, March 4th |
+| --- | --- |
+| 9am | Plenary Social Science Lecture |
+|  | Theatre (SSLT) |
+| 10:00 am | Coffee Break |
+| 10:30. 12:30 | Panels' |
+| 12:30 · 2 pm | Lunch |
+| 2 · 3.30 pm | Panels' |
+| 4-6 pm | Panels' |
+
+| 9am | Plenary Social Science Lecture |
+| --- | --- |
+|  | Theatre (SSLT) |
+| 10:00 am | Coffee Break |
+| 10:30 . 12:30 | Panels' |
+| 12:30 · 2 pm | Lunch |
+| 2 · 3.30 pm | Panels' |
+| 4-6 pm | Panels' |
+
+<table>
+<tr>
+<th colspan="3">Wednesday March 6th.</th>
+</tr>
+<tr>
+<td>9am</td>
+<td>Plenary</td>
+<td>Social Science Lecture</td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td>Theatre (SSLT)</td>
+</tr>
+<tr>
+<td>10:00 am</td>
+<td>Coffee Break</td>
+<td></td>
+</tr>
+<tr>
+<td>10:30 . 12:30</td>
+<td>Panels'</td>
+<td></td>
+</tr>
+<tr>
+<td>12:30-2 pm</td>
+<td>Lunch</td>
+<td></td>
+</tr>
+<tr>
+<td>2 · 3.30 pm</td>
+<td>Panels'</td>
+<td></td>
+</tr>
+<tr>
+<td>5:30pm</td>
+<td>Panel</td>
+<td></td>
+</tr>
+<tr>
+<td>8 pm</td>
+<td>Closing address</td>
+<td>Hon. Rex Nettleford</td>
+</tr>
+<tr>
+<td></td>
+<td>Venue</td>
+<td>Phillip Sherlock Centre</td>
+</tr>
+</table>
+
+<table>
+<tr>
+<th colspan="3">Wednesday March 6th.</th>
+<th>Wyndham Kingst o n</th>
+</tr>
+<tr>
+<td>9am</td>
+<td>Plenary</td>
+<td>Social Science Lecture</td>
+<td>Single/double occupancy us $80.00</td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td>Theatre (SSLT)</td>
+<td>All prices include tax and service charges</td>
+</tr>
+<tr>
+<td>10:00 am</td>
+<td>Coffee Break</td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td>10:30 . 12:30</td>
+<td>Panels'</td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td></td>
+<td>Conference attendees traveling from the U.S. via Air Jamaica are</td>
+</tr>
+<tr>
+<td>12:30-2 pm</td>
+<td>Lunch</td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td></td>
+<td>eligible for special fares. To inquire about these rates or make</td>
+</tr>
+<tr>
+<td>2 · 3.30 pm</td>
+<td>Panels'</td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td></td>
+<td>reservations, please advise your Air Jamaica representative to use</td>
+</tr>
+<tr>
+<td>5:30pm</td>
+<td>Panel</td>
+<td></td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td></td>
+<td></td>
+<td>record 'WCULTURE:'.</td>
+</tr>
+<tr>
+<td>8 pm</td>
+<td>Closing address</td>
+<td>Hon. Rex Nettleford</td>
+<td></td>
+</tr>
+<tr>
+<td></td>
+<td>Venue</td>
+<td>Phillip Sherlock Centre</td>
+<td></td>
+</tr>
+</table>
+
+• Three panels will be running simultaneou sly.
+
+## CoiwEREI'!CE LAI'KlUAGE
+
+English
+
+## f.:VEI'W'Kl PROGRAM
+
+Performance by NDTC Cocktail Reception
+
+## AccomtODA1101'1 DJ:TAILS
+
+## The Jamaica Pegasus
+
+George Lamming
+
+<table>
+<tr>
+<th>Standard rooms (standard or double occupancy)</th>
+<th>us $ 85. 00</th>
+</tr>
+<tr>
+<td>Executive rooms (standard or double occupancy)</td>
+<td>us $130. 00</td>
+</tr>
+<tr>
+<td>Junior suites</td>
+<td></td>
+</tr>
+<tr>
+<td>(standard or double occupancy)</td>
+<td>us $175. 00</td>
+</tr>
+<tr>
+<td>f.:dge Hill Luxury Apartments</td>
+<td></td>
+</tr>
+<tr>
+<td>Studio units</td>
+<td>us $ 55. 00</td>
+</tr>
+<tr>
+<td>Mayfair Hotel</td>
+<td></td>
+</tr>
+<tr>
+<td>Single Standard Room</td>
+<td>us $51. 25</td>
+</tr>
+<tr>
+<td>Single Superior Room</td>
+<td>us$61.25</td>
+</tr>
+<tr>
+<td>Double Superior Room</td>
+<td>us $51.25</td>
+</tr>
+<tr>
+<td>I Bedroom Suite</td>
+<td>us $88.76</td>
+</tr>
+<tr>
+<td>Double Superior Suite</td>
+<td>US $125.00</td>
+</tr>
+<tr>
+<td>Senio r Co mmo n Ro o m. UWI Mo na</td>
+<td></td>
+</tr>
+<tr>
+<td colspan="2">(Very limited accommodation, prices subject to change)</td>
+</tr>
+<tr>
+<td>Single occupancy</td>
+<td>US $35.00 per night</td>
+</tr>
+<tr>
+<td>Double occupancy</td>
+<td>US $50. 00 per night</td>
+</tr>
+</table>
+
+| Standard rooms (standard or double occupancy) | us \$ 85. 00 |
+| --- | --- |
+| Executive rooms (standard or double occupancy) | us \$130. 00 |
+| Junior suites |  |
+| (standard or double occupancy) | us \$175. 00 |

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+بهداشتی درمانی شهر کاشان و ارایه آزمون افسردگی بک به همکاران مربوطه, از آنها خواسته شد تا به کلیه مادरانی که شرایط ورود به این مطالعه را دارند آزمون بک را ارایه کرده و افرادی را که افسردگی خفیف دارند انتخاب و به صورت تصادفی و یک در میان در دو گروه ۱ و ۲ قرار دهند. به گروه ۱ در مورد انجام ورزش‌های بعد از زایمان آموزش داده و هر هفته از طریق تلفن اجرای تمرین‌ها را پیگیری کرده و در صورت عدم ادامه ورزش, از مطالعه حذف می‌شوند, همچنین یک جزوه آموزشی در مورد ورزش‌های بعد از زایمان نیز در اختیار این گروه قرار می‌گرفت, ولی به گروه دوم هیچ آموزش یا جزوه آموزشی در این زمینه ارایه نشد. بعد از اتمام هفته ششم بعد از زایمان و همزمان با مراقبت‌های معمول, مجدداً ابزار بک به هر دو گروه ارایه شد و میزان افسردگی در دو گروه تعیین شد. در ضمن پژوهشگر از اینکه کدام گروه ورزش کرده‌اند, اطلاع نداشت و لازم به ذکر است که در گروه‌های سنی مختلف تعداد نمونه یکسانی در هر دو گروه انتخاب شد تا اثر سن به عنوان مداخله گر کنترل شود. برای تجزیه و تحلیل داده‌ها از روش‌های آماری توصیفی و آزمون t استفاده شد.
+
+یافته‌ها
+
+یافته‌های حاصل از این پژوهش نشان داد که از میان مادरانی که از افسردگی خفیف بعد از زایمان رنج می‌برند, ۶۰ درصد در گروه سنی ۲۰ تا ۳۰ سال بودند. ۵5 درصد خانه دار, ۴4 درصد دارای تحصیلات دیپلم و زیر دیپلم, ۵2 درصد دارای تحصیلات دانشگاهی بودند. 64 درصد سابقه دو بار زایمان داشتند (جدول 1). شش هفته بعد از زایمان 71 درصد از گروه تجربی در مقابل 42 درصد از گروه مقایسه بهبود یافته بودند, 8 درصد از گروه تجربی در مقابل 19 درصد از گروه مقایسه به مشورت با روانپزشک نیاز داشتند یعنی بیماریشان تشدید یافته بود. آزمون t اختلاف آماری معنی داری در دو گروه نشان داد و به عبارتی دیگر ورزش در بهبود افسردگی خفیف بعد از زایمان تأثیر مثبت داشته است. (جدول شماره 2) (p<0/05)
+
+در تحلیل انجام شده با آزمون مجذور خی بین متغیرهای متعدد مثل سن مادر, تحصیلات مادر و تعداد زایمان, ارتباط معنی دار مشاهده گردید. افسردگی خفیف در گروه سنی زیر 20 سال تفاوت معنی دار با سایر گروه‌های سنی دارد. همین طور افسردگی در مادران با تحصیلات دیپلم و زیر دیپلم و دارای 4 فرزند تفاوت معنی دار با سایر گروه‌ها نشان داد. هم چنین در این مطالعه متغیرهای دیگری مثل خواسته یا ناخواسته بودن حاملگی, سابقه‌ای قبلی مرگ نوزاد, شغل مادر, سابقه‌ای سقط و نوع تغذیه نوزاد, ارتباط معنی داری مشاهده نشد (جدول شماره 3).
+
+اگرچه دارو درمانی و انواع درمان‌های شناختی در مورد اختلالات افسردگی اساسی, موثر واقع شده است ولی در مورد سطوح پایین تر این اختلال, کمتر موثر بوده است (16). همچنین این درمان‌ها وقت‌گیر, گران و دارای عوارض جانبی هستند (17, 18). بنابراین نیاز به توسعه درمان‌های غیر دارویی در سطوح مختلف این اختلال ضروری به نظر می‌رسد. امروزه ورزش و فعالیت جسمی به عنوان یک روش درمانی, بسیار مورد توجه قرار گرفته است (19). در پژوهشی مشابه در شهر اردبیل بر تأثیر ورزش در بهبود افسردگی مادران تأکید شده است (20). نتایج همکاران در تحقیقات خود به مقایسه فعالیت‌های جسمانی و غیر جسمانی بر ابعاد مختلف سلامتی و خلق افراد پرداختند. آنها اظهار کردند که افراد شرکت کننده در فعالیت‌های جسمانی پیشرفت بهتری در موارد مربوط به سلامتی و خلق نشان می‌دهند (21). بنابراین, از آنجایی که یکی از اقدامات موثر در جهت کاهش اثرات سوء ناشی از افسردگی پس از زایمان بر روابط مادر و نوزاد و پیشگیری از ایجاد اختلال در روابط با همسر و خانواده تمرینات ورزشی است, به نظر رسید ضروری است که تأثیر ورزش بر افسردگی پس از زایمان در زنان مراجعه کننده به مراکز بهداشتی درمانی کاشان مورد بررسی قرار گیرد.
+
+مواد و روش‌ها
+
+این پژوهش یک کارآزمایی دوسو کور می‌باشد. همکاران مستقر در مراکز بهداشتی درمانی از تقسیم نمونه به 2 گروه 25 نفری مورد (ورزشی) و شاهد (غیر ورزشی) آگاهی نداشتند و پیگیری انجام یا عدم انجام ورزش توسط پژوهشگر صورت گرفت. آزمودنی‌های گروه ورزشی از وجود گروه غیر ورزشی اطلاعی نداشتند. به طریق مشابه آزمودنی‌های گروه غیر ورزشی نیز از وجود گروه ورزشی آگاهی نداشتند. همکاران پژوهشگر هنگام ارایه مجدد ابزار بک پس از انجام مطالعه از 2 گروه بودن آزمودنی‌ها و انجام و عدم انجام ورزش توسط نمونه‌ها آگاهی نداشتند. جمعیت مورد مطالعه کلیه مادरانی بودند که زایمان طبیعی کرده و در هفته دوم بعد از زایمان (از روز هفتم تا چهاردهم) به مراکز بهداشتی درمانی کاشان مراجعه نموده بودند. به کلیه افراد جمعیت مورد مطالعه, آزمون افسردگی بک ارایه گردید و در صورتی که نمره این تست (11-16) یا افسردگی خفیف را نشان می‌داد, این افراد به صورت یک در میان در دو گروه مطالعه قرار می‌گرفتند. حجم نمونه در این پژوهش 50 نفر بود که به صورت تصادفی ساده, 25 نفر در گروه تجربی و 25 نفر در گروه مقایسه قرار گرفتند. موارد حذف نمونه شامل سابقه افسردگی, مصرف داروهای آرامبخش, زایمان چند قلو و نداشتن تلفن برای تماس بود. با مراجعه به مراکز
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+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+# holtchemistrystudyguidestoichiometryanswer