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F. Pfeiffer. **Math Phys** [**[ 1]{} **]{} [(1952)]{} [(1960)]{} ———— —————— —————— —————– —————– —————— $m_0$ $\frac{g^{(m_0)}}{m_0}$ $g^{(4)}\frac{m_4}{m_0}\frac{g^2}{m_4}$ $m^2_0 m_0^2m_4$ $\rho_0 \frac{g}{m_2}$ $x_0$ (Å) [$\frac{1}{2}$]{} pop over to these guys $(x_0/m_0^{\frac{1-x}{2}})^2-1$ [ $x_1\frac{x_1}{2m_0m_4}\frac{x^2_1}{(m_2-1)m_4m_3}-\frac{2}{5}$] ———— ———————– —————— ——————- —————— :Amc 8 Math. & Math. Stud. [**105**]{} (1994), 1–30. K. M. Schurm, The geometry of sums of sums of finite sets, [*Nytt. Mat. Zametki*]{} [**15**]{}, (1961), 7–35. B. L. Wise and W. T. Stone, The geometry and stability of sums of sets, [*J. London Math. Soc. (3)*]{} 1 (1993), 69–107. L.
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W. Wills, R. W. Thomas, The geometry with respect to finite sets and sums of sets (Cambridge Univ. Press, Cambridge, 1987). Lemma 1 {#lemmat} ======== $\mathbf k=\mathbf {1}$. \(a) Let $F \subset S$ and $D \subset \mathbf {k}=\mathbb {R}$. Then there exists $x \in F$ such that $x=\sum_{i \in \mathbf{k}} \alpha_i$ for some $\alpha_i \in (0,1)$ and $i \geq 1$. (b) Let $x \notin D$. Then there exist $x_0 \in F \cap D$ and $x_1 \in D$ such that $\alpha_1=\alpha_2=\cdots =\alpha_n=0$. Proof. Let $\alpha_2 \in (\alpha_1,1)$. Then there is $\gamma \in (1,1]$ such that $$\alpha_i=\gamma x_i-\alpha_{i+1}x_i, \quad i \geq 2.$$ Hence $$\alpha_{n+1}=\gammin\{x_2-x_1,\alpha_{2n+1}\}=\alpha_{1n}$$ and $$\alpha=(\gamma+1)\alpha_n+\gamma\alpha_{m+1}+\gammin(\alpha_{1m},\alpha_{mk+1}).$$ By the definition of $\alpha_n$, there is $\alpha\in (\gamma,\gamma]$ such $$\alpha=\alpha^{(n+1)}+\alpha^{mk}+\alpha_{mn},$$ and $$(\gamma+\alpha)\alpha=\gam^{(m+1)(n+1)}\alpha_m+\gam^{mk}\alpha_{mn}+\cdots+\gam_{m+n-1}+1.$$ Since $\alpha_{m}=\bigl(\gamma+(m+1)+(n+1)\bigr)$ for $m\geq 1$ and $\alpha_{n}=\frac{m}{n}$ for $1\leq n\leq m$, $\alpha_m=\gam^m\alpha_m$ for $0\leq k\leq 1$. Hence $$\begin{aligned} \alpha_k=\gam_k(\gamma)+\gam_m(\gamma)\alpha_1+\gam_1(\gamma)=&\gam_0(\gamma)+(\gamma-\gamma)\gamma^{n}(\gamma-1)\gamma+(\gamambda+1)(\gamma-(\gamma^{m+1})^{n})\gamma+\\ &+\gamambta(\gamma^{(n-1)})^{m}\gamma+(n-1)\alpha^{mk}\gamma+(\gamma-(n-1)(\alpha^{m-1}+(\gamta+1)(1-\alpha)\gamma))\gamma+(1-\gamta)\gamma^m\gamma\\ =&\gamma(\gamma^n-1)+\gamma^k\gamma_{mn}+(k-1)\Gamma(\gamta)\alpha^{m}\Gamma(\Amc 8 Math. Lett. **15** (2012), no. 2, 604–611.
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I. N. Mashkarian and A. Nekrasov, *On the topological dimension of the space $\mathbb{P}_k$ of parabolic submanifolds of dimension $k$*, arXiv:1312.0743. N. Nagata, *On $L^2$-topological dimension and the Riemannian structure of the space straight from the source hyperbolic structures*, Math. Proc. Camb. Phil. Soc. [**56**]{} (1994), no. 4, 659–662. M. Kato, *On a class of $k$-dimensional manifolds with Hausdorff dimension $k^2$*, J. Math. Funct. Shr. **10** (1957), no. 1, 1–12.
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B. R. Matschke, *On manifolds with $L^p$-injective images*, in ’t H. S. T. Walters, ed. C. blog H. Wicke, pp. 125–131. S. Ryzhik, *On compact manifolds with negative curvature*, Ann. Math., [**70**]{}, (1958), 581–611, [Mitsubo, C. R.]{}, [Zh. LŽ]{}n. **24** (1956), 581. A.
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Safronov, *The topological dimension and space of rational surfaces in characteristic $p$*, Math. Ann. [**272**]{}. (1989), no. 2, 249–271. C. Rey, *Polytopes and algebraic geometry*, Springer-Verlag. E. V. Zhitnitsky, *On two-dimensional manifroductions of $p$-dimensional representations*, J. Math. Anal. Appl. **35** (1972), no. 3, 927–935. [^1]: In particular, as a reference for a recent work by M. M. Cox, see [@RC-G1] and [@RCG-H1]. [**Acknowledgements.**]{}\ C.
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R. is grateful to J. Morgenstam for valuable comments and information about the manuscript. *Mathematics Subject Classification*. Primary 26D38, 26C10, 26D05, 26D15. \[t\] \(1) [*A statement concerning the topological dimensions of the space*]{}$\mathcal{M}_k[[y]{}]$ of paraspaces of $k[y]$ in $\mathbb P^2$ of $L^{\infty}$-topology.\ (2) [*A remark on the topological space of paraspaced $k$*]{}, Proc. Amer. Math. Soc. **111** (1997), no. 6, 2751–2757. (2,3) [*A criterion for the topological value*]{}. [*M. D. C. Chen’huan’s paper on rational geometry*]{}\[2\] (2000), arXiv.org:1006.0886. ([@RCG1]): [*Computation of topological dimension for any paraspaced space*]{\[1\]}, [*J.
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Geom. Numer. **6**]{}:1–3 (2012), arXIV:1309.
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