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2c9fe20ecf2879ab19779733393b7c94d0a82b06 | subsection | 9 | 16 | Evolutionary fluctuation response relationship | Here, the mutation rate is given by the probability that a path in the network is added or deleted at each generation.As mentioned, for a given network, there are fluctuations in the abundances of each chemical.
We took the phenotype variable x=log(n_{i_s}), since the distribution of n_{i} is approximately lognormal, w... | {
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0a994859c7c69ab8def575d86d4d0e63830ac01d | subsection | 10 | 16 | Theoretical discussion | Is it possible to formulate a phenomenological theory to support the relationship observed in numerical (and partially in in vivo) experiments presented in §3.1?Here we consider the distribution both in phenotype x and genotype a. Through the evolutionary process, the genotype changes from its dominant type a=a_0, and ... | {
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a166e546453773eeb2b907f5f3def0194d43ed05 | subsection | 11 | 16 | Theoretical discussion | First, we considered the average \overline{x}_a over the distribution P(x,a) for a given fixed a, and then considered the distribution of \overline{x}_a according to the distribution p(a),
noting that
\overline{x}_a \equiv \int x P(x,a) dx=X_0+C(a-a_0).
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c77f14f87bad2b19aae56bc0bd27c2bf43ba04b8 | subsection | 12 | 16 | Theoretical discussion | Here, the phenotype at each generation is within a small range, and the deviation of V_{ig} from V_g is not so large. Indeed, the estimate of the critical mutation rate for the error catastrophe is not accurate enough to distinguish between the two. Thus, the above theoretical estimate for the error catastrophe is cons... | {
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ea27b16c86bad75bb88fb2852a3ee012de936693 | subsection | 13 | 16 | Consistency between cell replication and reproduction of multicellular organisms | We briefly discuss cell differentiation, i.e., diversification into a discrete set of cell types through development and robustness in the population distribution of each cell type through development.
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df0ec91392395f9399a3b4a9e4592863a384d188 | subsection | 14 | 16 | Consistency between cell replication and reproduction of multicellular organisms | Cells having other networks without any oscillatory dynamics in chemical concentrations often divide faster. However, as the number of cells grows, the speed of division for such non-oscillating cells is drastically reduced, while for cells with chaotic dynamics and differentiation, the speed is not so reduced. This is... | {
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abffa5e3e123d3405191308dbc19c407e590cb26 | subsection | 15 | 16 | Conclusion | Here we reviewed three problems in biology from the viewpoint of `consistency between different levels'. First, as a result of consistency between molecule replication and cell reproduction, chemical reaction dynamics are shown to be at a critical state, and a power law distribution of chemical abundances (gene express... | {
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8cf21eee5a4bcb683858859002d1a8d8aa4921ae | abstract | 0 | 30 | Abstract | We study 12 parameter families of two qubit density matrices, arising from a
special class of two-fermion systems with four single particle states or
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b97c2b2a02d725041e581b72a159391f38e163fd | subsection | 1 | 30 | Introduction | Entanglement is the basic resource of quantum information
processing. As such it has to be quantified and its
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dc2461f4d9b855d615ba1879de2b714f84e10b40 | subsection | 2 | 30 | Introduction | Hence as an extra constraint we impose
an antisymmetry condition on the amplitudes of\vert \Psi \rangle =\sum _{ijkl=0}^1{\Psi }_{ijkl}\vert ijkl\rangle ,as{\Psi }_{ijkl}=-{\Psi }_{klij},i.e. we impose antisymmetry in the first and second pairs of indices.An alternative (and more physical) way is the one of imposing su... | {
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bc65a0236bce1eb6e86b161b94826f9c38aa8a8b | subsection | 3 | 30 | The density matrix | Let us parametrize the 6 amplitudes of our normalized four qubit state
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53dbd89dc6b3b75c075eee4c4715bb63dce9007e | subsection | 4 | 30 | The density matrix | Notice, that \mathbf {x}, \mathbf {y}
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47,
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617,
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20,
831,
186,
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272... | [
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0.108886... |
43d8cd866dac4834ebc0d1963e27456d48c9d6bc | subsection | 5 | 30 | The density matrix | A special
rotation from \mathbf {x} to \mathbf {x^{\prime }} ({\bf x}^{\prime }\ne -{\bf x}) can be written asU(\mathbf {\hat{u}}, \alpha )^\dagger (\mathbf {x}{{\sigma }}) U(\mathbf {\hat{u}}, \alpha ) = \mathbf {x^{\prime }}{{\sigma }},\\
U(\mathbf {\hat{u}}, \alpha ) = \frac{1}{\sqrt{2\mathbf {x}^2(\mathbf {x}^2 + ... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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... |
4e60f5fb5a0e67581681093d68e65bbf13e2bf7e | subsection | 6 | 30 | The density matrix | ()), the
transformation above rotates the third component of \mathbf {w} into
zeroU^\dagger _\mathbf {x}(\mathbf {w}{{\sigma }})U_\mathbf {x} = \mathbf {w^{\prime }}{{\sigma }},\\
\mathbf {w^{\prime }} = \mathbf {w} - \frac{\mathbf {w}\mathbf {x^{\prime }}}{r^2+\mathbf {x}\mathbf {x^{\prime }}}(\mathbf {x} + \mathbf {... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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63749052d0b0d7d9f7475cd136d3e5124acbb591 | subsection | 7 | 30 | The density matrix | Hence\mathbf {w^{\prime }}^2 = \mathbf {w}^2, \qquad \mathbf {z^{\prime }}^2 = \mathbf {z}^2,\\
\Vert \mathbf {w^{\prime }}\Vert ^2 = \Vert \mathbf {w}\Vert ^2, \qquad \Vert \mathbf {z^{\prime }}\Vert ^2 = \Vert \mathbf {z}\Vert ^2.and\eta ^{\prime } = \etaare invariant under local U(2)\times U(2) transformations. | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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099bca7e302d209df2296ac945c04f2f495b6e28 | subsection | 8 | 30 | The density matrix | (The
entanglement measure \eta is also invariant under the larger
group of U(4) transformations.)Now by employing the local U(2)\times U(2) transformations
U_{\bf x}\otimes V_{\bf y}, our density matrix can be cast to
the form,\varrho ^{\prime } =
\left(U_\mathbf {x}\otimes V_\mathbf {y}\right)^\dagger \varrho \left(U_... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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6b8a757ca2f1b14dfbe2fa11e0ba86c9151d84d8 | subsection | 9 | 30 | The density matrix | Having
obtained the canonical form of our reduced density matrix
\varrho , now we turn to the calculation of the corresponding
entanglement measures. | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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33c5a7bb4d66ff96822eeae0a12fc278c806d266 | subsection | 10 | 30 | Concurrence | In this section we calculate the Wootters-concurrence
of our density matrix \varrho defined in Eqs. (REF ) -
(). This quantity is defined as{\cal C}={\rm max}\lbrace 0,\lambda _1-\lambda _2-\lambda _3-\lambda _4\rbracewhere \lambda _1\ge \lambda _2\ge \lambda _3\ge \lambda _4
are the square roots of the eigenvalues of... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 114,
"openalex_id": "",
"raw": "W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998), S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).",
"source_ref_id": "d5975063b0f86775531cbcc2ddc4c2086150629f",
"start": 0
... | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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0cc06e77a2048dd15152fc058ea0530b613a63b8 | subsection | 11 | 30 | Concurrence | \end{split}The eigenvalues of the blocks (\tilde{\alpha }_0I+\mathbf {\tilde{{\alpha }}}{{\sigma }}) and
(\tilde{\beta }_0I+\mathbf {\tilde{{\beta }}}{{\sigma }}) are
\tilde{\alpha }_0 \pm \sqrt{ \tilde{{\alpha }}^2 } and
\tilde{\beta }_0 \pm \sqrt{ \tilde{{\beta }}^2 } ,
respectively. Now, we can express these with th... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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c36834d18f19fb73c3461253e8ae08c99ef19cb6 | subsection | 12 | 30 | Concurrence | Straightforward
calculation shows, that:\begin{split}
\alpha _1^2+\alpha _2^2 &=
2\Vert \mathbf {w}^{\prime }\Vert ^2\Vert \mathbf {z}^{\prime }\Vert ^2 +
\mathbf {w}^{\prime 2}\overline{\mathbf {z}}^{\prime 2} + \overline{\mathbf {w}}^{\prime 2}\mathbf {z}^{\prime 2}-2rs,\\
\beta _1^2+\beta _2^2 &=
2\Vert \mathbf {w}... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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e234a95248c6f68aa0a8559179a2534ccb5f6909 | subsection | 13 | 30 | Concurrence | \Biggr \rbrace
\end{split}The biggest one of these is \lambda _{max} = \frac{1}{4}\left(
\sqrt{1-\gamma _-^2} + \sqrt{1-\gamma _-^2-\eta ^2} \right) and
after subtracting the others from it, we get finally the nice
formula for the concurrence\mathcal {C}(\varrho ) = \max \left\lbrace 0,
\frac{1}{2}\left( \sqrt{1-\gamm... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 760,
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"raw": "P. Lévay, Sz. Nagy, J. Pipek, Phys. Rev. A72, 022302 (2005).",
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"start": 519
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} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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4ac8ba9a5915bfa2ae28c0861905cb865539387b | subsection | 14 | 30 | Negativity | Another entanglement-measure which we can calculate for \varrho
is the negativity. It is related to the notion of partial
transpose and the criterion of Peres. It is defined by
the smallest eigenvalue of the partially transposed density
matrix, as follows, \mathcal {N}(\varrho ) = \max \left\lbrace 0, -2 \mu _{min} \r... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 160,
"openalex_id": "",
"raw": "A. Peres, Phys. Rev. Lett. 77, 1413 (1996).",
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"start": 84
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"end": 33... | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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e5173992c8d06274eecc3608649de6ac35c54bf3 | subsection | 15 | 30 | Comparsion of concurrence and negativity | For a 2-qubit density matrix
we can write the following inequalities between the concurrence and the negativity\sqrt{ (1-\mathcal {C})^2 + \mathcal {C}^2 } - (1-\mathcal {C})
\le \mathcal {N}\le \mathcal {C},which are known from a paper of Audenaert et. al.
Our special case with fermionic correlations may give extra re... | {
"cite_spans": [
{
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"doi": "",
"end": 257,
"openalex_id": "",
"raw": "F. Verstraete, K. Audenaert, J. Dehaene, B. De Moor, J. Phys. A34, 10327 (2001). K. Audenaert, F. Verstraete, T. De Bie, B. De Moor, arXiv:quant-ph/0012074.",
"source_ref_id": "f83b1faba9cfdf9... | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
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2ca517ec167d5ae5ed79afb16b6ac3d6f97ea4df | subsection | 16 | 30 | Comparsion of concurrence and negativity | If \mathbf {w}^2=\mathbf {z}^2 then r=s and \Vert \mathbf {w}\Vert ^2=\Vert \mathbf {z}\Vert ^2 = \frac{1}{2}
are equivalent,
and if r=s then \gamma _+^2=4r^2=1,
\Vert \mathbf {w}\Vert ^4-\vert \mathbf {w}^2\vert ^2=\frac{1}{4}
and follows, that \mathbf {w}^2=0.\mathcal {C}= \mathcal {C}_{max} = \frac{1}{2} \quad \Long... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | 2,008 | en | Physics | [
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608fb3ab8ce229b822a01339c9ae937ff1b7ca29 | subsection | 17 | 30 | Comparsion of concurrence and negativity | These satisfy the first constraint of (REF ),
and from the second follows that
\cos {\alpha }=\sin {\alpha }=\frac{1}{\sqrt{2}} and
\varphi _1 = \varphi _2-\frac{\pi }{2} =:\varphi and the same for \mathbf {z}_{max}^{\prime }.\mathbf {w}_{max}^{\prime }=\frac{1}{2} e^{i\varphi } \begin{bmatrix} 1\\i\\0 \end{bmatrix},
\... | {
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} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
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1cce4b0b6034d3da25d5d486f768474ff6875c9d | subsection | 18 | 30 | Purity | The purity is measuring the degree of mixedness of a density matrix.
For our \varrho thanks to the special property of \Lambda
(see in Eq. (REF )) it can easily be calculated. We have the result\operatorname{Tr}\varrho ^2 = \frac{1}{4} (2-\eta ^2),\\
\frac{1}{4} \le \operatorname{Tr}\varrho ^2 \le \frac{1}{2}by virtue... | {
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"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
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456e1b0db2f7b1b8e7ec6eba8b8837710f6f285d | subsection | 19 | 30 | Relating different measures of entanglement | Now we would like to discuss the physical meaning of our
quantities derived in the previous section. First of all let us
notice that the{\varrho }_1={\rm Tr}_{234}(\vert \Psi \rangle \langle \Psi \vert )={\rm Tr_2}({\varrho }_{12})={\rm Tr}_2(\varrho )=\frac{1}{2}(I+{\bf x}{\sigma })\varrho _2={\rm Tr}_{134}(\vert \Psi... | {
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"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
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b9a52e28a8c4d9227d4707359256ac5213bf3f2e | subsection | 20 | 30 | Relating different measures of entanglement | Their form is\left({\varrho }_{13}\right)_{iki^{\prime }k^{\prime }}=\frac{1}{2}\left(\vert \vert {\bf z}\vert \vert ^2{\varepsilon }_{ik}{\varepsilon }_{i^{\prime }k^{\prime }}+
{\cal B}_{ik}\overline{\cal B}_{i^{\prime }k^{\prime }}\right),\qquad \left({\varrho }_{24}\right)_{jlj^{\prime }l^{\prime }}=\frac{1}{2}\lef... | {
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} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
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63831f00965a9b2601848fb0f2428972cd1baa26 | subsection | 21 | 30 | Relating different measures of entanglement | Finally these manipulations yield for {\varrho }_{24} the canonical form{\varrho }_{24}=\frac{1}{2}\begin{pmatrix}{\kappa }_0+{\kappa }_3&0&0&{\kappa }_1-i{\kappa }_2\\0&\vert \vert {\bf w}\vert \vert ^2&-\vert \vert {\bf w}\vert \vert ^2&0\\
0&-\vert \vert {\bf w}\vert \vert ^2&\vert \vert {\bf w}\vert \vert ^2&0\\
{\... | {
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d3c4b1bc205a1999c33d52089f892c7f2fbaf641 | subsection | 22 | 30 | Relating different measures of entanglement | The invariants
H and L are given by the expressionsH=\Psi _0\Psi _{15}-\Psi _1\Psi _{14}-\Psi _2\Psi _{13}+\Psi _3\Psi _{12}-\Psi _4\Psi _{11}
+\Psi _5\Psi _{10}+\Psi _6\Psi _9-\Psi _7\Psi _8,andL={\rm Det}\begin{pmatrix}\Psi _0&\Psi _1&\Psi _2&\Psi _3\\
\Psi _4&\Psi _5&\Psi _6&\Psi _7\\
\Psi _8&\Psi _9&\Psi _{10}&\Psi... | {
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"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
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98fe0431e394ce20cc46d0b4e7db7f7364a297d4 | subsection | 23 | 30 | Relating different measures of entanglement | (13) one can check that{\cal C}^2_{13}=s^2+\frac{1}{2}(\eta ^2+\sigma ^2)-2\vert \vert {\bf z}^2\vert \vert \vert {\bf w}^2\vert ,\quad {\cal C}^2_{24}=r^2+\frac{1}{2}(\eta ^2+\sigma ^2)-2\vert \vert {\bf w}^2\vert \vert \vert {\bf z}^2\vert .\quadHence we have the inequality{\cal C}^2_{13}+{\cal C}^2_{24}\le s^2+r^2+{... | {
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} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
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d3f12a229f943f37366a4776085fd40256e5128b | subsection | 24 | 30 | Relating different measures of entanglement | (84-85) are positive as they should be, hence the generalized Coffman-Kundu-Wootters inequalities of distributed entanglement,
hold{\cal C}^2_{12}+{\cal C}^2_{13}+{\cal C}^2_{14}\le {\cal C}^2_{1(234)}
\qquad {\cal C}^2_{12}+{\cal C}^2_{23}+{\cal C}^2_{24}\le {\cal C}^2_{2(134)}.For separable states we have {\cal C}_{... | {
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"János Pipek"
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17edf1588c5178ec09b2b5cd33c5078fff308549 | subsection | 25 | 30 | Bures metric | As we have emphasized our density matrix {\varrho } can be
regarded as a reduced density matrix of a two-particle system on
(\mathbf {C}^2\otimes \mathbf {C}^2)\wedge (\mathbf {C}^2\otimes \mathbf {C}^2),
meaning\varrho = {\Psi }{\Psi }^\dagger ,where {\Psi } is the 4\times 4 antisymmetric matrix occurring
in Eq. (REF ... | {
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"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
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6c208a73f0a3086d9a5201f75eae43735fc038d5 | subsection | 26 | 30 | Bures metric | (REF ), \varrho ^{-1} can be calculated easily\varrho ^{-1} = \frac{4}{\eta ^2} \left( \mathbf {1}- \Lambda \right),hence:G = \frac{1}{2} d\varrho \varrho ^{-1} = \frac{1}{2\eta ^2}\left(d\Lambda - d\Lambda \Lambda \right),and the Bures-metric:ds^2_B = \frac{1}{4\eta ^2}\operatorname{Tr}\left(d\Lambda d\Lambda - d\Lamb... | {
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"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
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dc4f1ef5560d69b49409cb9911dc49c18b0fcf90 | subsection | 27 | 30 | Bures metric | However, using the
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"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
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8537857c8ff6dd563e170a4e2a3d13e6aa014521 | subsection | 28 | 30 | Conclusions | In this paper we investigated the structure of a
12 parameter family of two-qubit density matrices with fermionic
purifications. Our starting point was a four-qubit state with a
special antisymmetry constraint imposed on its amplitudes. Such
states are elements of the space
(\mathbf {C}^2\otimes \mathbf {C}^2)\wedge (\... | {
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} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
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ce474fd61452e454b99ace9f035bf4303d4c6ec3 | subsection | 29 | 30 | Upper bound of negativity | In this fermionic-correlated case, defined by equations
(REF ), (REF ), () and (REF ),
we can prove the following inequality:Theorem:
For all entangled \varrho :\mathcal {N}(\varrho ) \le \frac{1}{2} \left( \sqrt{ 2 - (1-2\mathcal {C}(\varrho ))^2 } -1 \right).Proof:
Insert Eqs. (REF ) and (REF ) into (REF ):\frac{1}{2... | {
"cite_spans": []
} | 10.1088/1751-8113/41/50/505304 | 0807.1804 | A study of two-qubit density matrices with fermionic purifications | [
"Szilárd Szalay",
"Péter Lévay",
"Szilvia Nagy",
"János Pipek"
] | [
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b520a71dee49b88b93c2f25fcfb4f6193964f9c9 | abstract | 0 | 7 | Abstract | One of the most challenging issues in the characterization of magnetic
materials is to obtain quantitative analysis on the nanometer scale. Here we
describe how electron magnetic circular dichroism (EMCD) measurements using the
transmission electron microscope (TEM) can be used for that purpose, utilizing
reciprocal sp... | {
"cite_spans": []
} | 10.1103/PhysRevLett.102.037201 | 0807.1805 | Quantitative magnetic information from reciprocal space maps in
transmission electron microscopy | [
"Hans Lidbaum",
"Ján Rusz",
"Andreas Liebig",
"Björgvin Hjörvarsson",
"Peter M. Oppeneer",
"Ernesto Coronel",
"Olle Eriksson",
"Klaus Leifer"
] | [
"cond-mat.mtrl-sci"
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b5e385da70a11e42acf6ce84d6e93a27ace30556 | subsection | 1 | 7 | Body | =1Quantitative magnetic information from reciprocal space maps in transmission electron microscopyHans Lidbaum
Department of Engineering Sciences, Uppsala University, Box 534, S-751 21 Uppsala, Sweden
Ján Rusz
Department of Physics and Materials Science, Uppsala University, Box 530, S-751 21 Uppsala, Sweden
Institute o... | {
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... | 10.1103/PhysRevLett.102.037201 | 0807.1805 | Quantitative magnetic information from reciprocal space maps in
transmission electron microscopy | [
"Hans Lidbaum",
"Ján Rusz",
"Andreas Liebig",
"Björgvin Hjörvarsson",
"Peter M. Oppeneer",
"Ernesto Coronel",
"Olle Eriksson",
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43747e6ece43e18ea78dc03f4ee86f7756e01005 | subsection | 2 | 7 | Body | The recent
derivation of the EMCD sum rules for extraction of spin (m_S) and orbital (m_L) magnetic
moments represents an important step in that direction , . As EMCD relies on reciprocal
space vectors, proper \mathbf {k}-space selection of detector positions is essential. So far, most measurements are carried
out by s... | {
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"en... | 10.1103/PhysRevLett.102.037201 | 0807.1805 | Quantitative magnetic information from reciprocal space maps in
transmission electron microscopy | [
"Hans Lidbaum",
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"Andreas Liebig",
"Björgvin Hjörvarsson",
"Peter M. Oppeneer",
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"Olle Eriksson",
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fe1351c8c5b197a8912b422318aaaf45736ca9f0 | subsection | 3 | 7 | Body | By tilting the sample further in the perpendicular
direction by a small angle of \beta \sim 0.4^\circ the 2BC geometry is obtained. In this geometry the
transmitted and Bragg scattered beam \mathbf {G} = (200) in Fe are strongly excited, while all others
are weak. To use the sum rules , allowing a quantitative assessme... | {
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"... | 10.1103/PhysRevLett.102.037201 | 0807.1805 | Quantitative magnetic information from reciprocal space maps in
transmission electron microscopy | [
"Hans Lidbaum",
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"Andreas Liebig",
"Björgvin Hjörvarsson",
"Peter M. Oppeneer",
"Ernesto Coronel",
"Olle Eriksson",
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c650a2c528eee11af71a6ee1591672052f93d60e | subsection | 4 | 7 | Body | REF , in agreement with Refs. , . A reduced EMCD signal is also present around the weakly excited reflection (-\mathbf {G}).After applying cross-correlation on the transmitted beam, each spectrum in the experimental data-cube was pre-edge background subtracted (power-law model) and normalized in the post-edge region (a... | {
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"end":... | 10.1103/PhysRevLett.102.037201 | 0807.1805 | Quantitative magnetic information from reciprocal space maps in
transmission electron microscopy | [
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"Ján Rusz",
"Andreas Liebig",
"Björgvin Hjörvarsson",
"Peter M. Oppeneer",
"Ernesto Coronel",
"Olle Eriksson",
"Klaus Leifer"
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d9a46878917f24d811de1b328ac8f5d06f86a08d | subsection | 5 | 7 | Body | The black lines indicate the applied mirror axes and blue spots the positions of the transmitted and Bragg scattered \mathbf {G} = (200) and -\mathbf {G} = (\bar{2}00) beams. The insets in b) and f) show the diffraction patterns averaged over an energy interval from 695 eV to 740 eV. In d) and h) the experimental m_L/m... | {
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"... | 10.1103/PhysRevLett.102.037201 | 0807.1805 | Quantitative magnetic information from reciprocal space maps in
transmission electron microscopy | [
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"Björgvin Hjörvarsson",
"Peter M. Oppeneer",
"Ernesto Coronel",
"Olle Eriksson",
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6991dd09aad72e043221f0ba1db30cc30b1bccf0 | subsection | 6 | 7 | Body | We
obtain a consistent m_L/m_S ratio, depicted in Fig. REF b, of 0.09 \pm 0.01 in the 2BC and 0.08 \pm 0.01
in 3BC geometry using the double difference maps. The standard error s = 0.01 was
estimated using N=1225 individual m_L/m_S ratios within the selection window (standard deviation of an individual m_L/m_S ratio is... | {
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98eec72d9fa7b4699a84acff5566721ae4af049b | abstract | 0 | 9 | Abstract | We consider the problem of determining a pair of functions $(u,f)$ satisfying
the heat equation $u_t -\Delta u =\varphi(t)f (x,y)$, where $(x,y)\in
\Omega=(0,1)\times (0,1)$ and the function $\varphi$ is given. The problem is
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determined uniq... | {
"cite_spans": []
} | 0807.1806 | Determine the spacial term of a two-dimensional heat source | [
"Dang Duc Trong",
"Alain Pham Ngoc Dinh",
"Phan Thanh Nam"
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4c48e896ee9005f4a317016e688826bd99584305 | subsection | 1 | 9 | Body | Determine the spacial term of a two-dimensional heat source
Dang Duc Trong^a, Pham Ngoc Dinh Alain^b and Phan Thanh Nam^a^aMathematics Department, HoChiMinh City National University, Viet Nam^bMathematics Department, Mapmo UMR 6628, BP 67-59, 45067 Orleans cedex, France
We consider the problem of determining a pair... | {
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e6d5aecf6242b4d37dae74eb9062534a28fdfe0a | subsection | 2 | 9 | Body | Of course, the problem with approximate data is even more difficult because of the ill-posedness.Under a slight condition on \varphi , we shall use the variational method and some properties of analytic functions to show the uniqueness of the solution. In particular, this result makes a regularization theorem of trivia... | {
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"Dang Duc Trong",
"Alain Pham Ngoc Dinh",
"Phan Thanh Nam"
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5e78fd3fe9753a687e7aefac31fcc5f1f61665f5 | subsection | 3 | 9 | Body | This condition holds with respect to \theta =0, for example, if \varphi is continuous at t=0 and \varphi (0)\ne 0. To compare, we refer to the condition \varphi \in C^1[0,T] and \varphi (0)\ne 0 in \cite { Y93,Y94}.Under the condition (H), we will obtain the uniqueness of the problem (\ref {1}).Theorem 1 Assume that g\... | {
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"Dang Duc Trong",
"Alain Pham Ngoc Dinh",
"Phan Thanh Nam"
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2fab1d77d9f70fd3d4421bf85b686a08e4641e39 | subsection | 4 | 9 | Body | Getting the inner product in L^2(\Omega ) of the first equation of the system (1) and W(x,y)=\cosh (\alpha x)\cos (n\pi y), then using the integral by part we haveNext, we multiply the latter equality with e^{-(\alpha ^2 -n^2 \pi ^2)t} to getFinally, integrating (\ref {tam}) with respect to t from 0 to T we obtain the ... | {
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4b1c577e893405373394d98a3c25f00fcd52533f | subsection | 5 | 9 | Body | Since \mathop {\lim }\limits _{\lambda \rightarrow + \infty } \left( {\lambda ^{\theta + 1} e^{ - \lambda T_0 } } \right) = 0, it is sufficient to show thatwhereUsing the integral by part we getTherefore, it is enough to prove (\ref {D1}) for all \theta \in [0,1). Indeed, by direct calculus we obtain \mathop {\lim }\li... | {
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c1ccbdb3414e653ea4fa55c4f771f791f1b1fba4 | subsection | 6 | 9 | Body | Let w and \widetilde{w} be two even complex function such that w is an entire function and |w(z)|\le Ae^{|z|} for all z\in C, where A is independent on z. ThenFix z\in C, |z|\le \pi r and denote z_j=4r+j for each j=1,2,...,20r. We shall use the triangle inequalityW̉e first estimate |w(z) - L(B_r ;w)(z)|. Let \gamma =\l... | {
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} | 0807.1806 | Determine the spacial term of a two-dimensional heat source | [
"Dang Duc Trong",
"Alain Pham Ngoc Dinh",
"Phan Thanh Nam"
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4a60791a4d6ef75c9d43ca7d7eca737f74884830 | subsection | 7 | 9 | Body | Hence, for \varepsilon >0 small enough one hasThus, according to Lemma REF , there exists C(\varphi _0)>0 depending only on \varphi _0 such thatand consequently,It follows from Lemma REF and Lemma REF , for \varepsilon >0 small enough, thatMoreover, for \varepsilon >0 small enough,The desired result follows the two lat... | {
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} | 0807.1806 | Determine the spacial term of a two-dimensional heat source | [
"Dang Duc Trong",
"Alain Pham Ngoc Dinh",
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ce29b1c735199bd43f511d621b018e3a3d50b102 | subsection | 8 | 9 | Body | Hence it follows from \mathop {\lim }\limits _{\varepsilon \rightarrow 0^ + } \Gamma _{r_\varepsilon } (f_0 ) = f_0 in L^2(\Omega ) that \mathop {\lim }\limits _{\varepsilon \rightarrow 0^ + } f_\varepsilon = f_0 in L^2(\Omega ).Now assume in addition that f_0\in H^1(\Omega ). Then Lemma REF leads to \mathop {\lim }\li... | {
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} | 0807.1806 | Determine the spacial term of a two-dimensional heat source | [
"Dang Duc Trong",
"Alain Pham Ngoc Dinh",
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dd2a9a5575b8b78438d521c715bb620d53ba85a5 | abstract | 0 | 8 | Abstract | We provide a comprehensive view on the role of Abelian symmetry and
stochasticity in the universality class of directed sandpile models, in context
of the underlying spatial correlations of metastable patterns and scars. It is
argued that the relevance of Abelian symmetry may depend on whether the dynamic
rule is stoch... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.218001 | 0807.1807 | Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles | [
"Hang-Hyun Jo",
"Meesoon Ha"
] | [
"cond-mat.stat-mech"
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06dc453c325e6a71301273c64c717c9a9f77f89e | subsection | 1 | 8 | Body | Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles
Hang-Hyun Jo
School of Physics, Korea
Institute for Advanced Study, Seoul 130-722, KoreaMeesoon Ha
[Corresponding author: ]msha@kaist.ac.kr
Department of Physics, Korea
Advanced Institute of Science and Technology, Daejeon 305-701,
KoreaWe provide a ... | {
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... | 10.1103/PhysRevLett.101.218001 | 0807.1807 | Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles | [
"Hang-Hyun Jo",
"Meesoon Ha"
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b5d553bc36c4db801f1d9754187c58d5987ff621 | subsection | 2 | 8 | Body | Finally, we
reinterpret the earlier known results for the Abelian case by our
conjecture, and confirm those for non-Abelian case by large-scale
numerical simulations with various data analysis techniques
developed so far.Consider DSMs defined on a (1+1)-dimensional tilted square
lattice of size (L,T). The preferred dir... | {
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78d9b77ad09a06daab11f5abb645e199973e14bd | subsection | 3 | 8 | Body | REF ,
which shows the case of z_i(t)=3.Each avalanche can be characterized by the following quantities:
mass s (the number of toppled grains), duration t (the number
of affected layers), area a (the number of distinct toppled
sites), width w (the mean distance between left and right
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} | 10.1103/PhysRevLett.101.218001 | 0807.1807 | Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles | [
"Hang-Hyun Jo",
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3c8296ec675cb67d81bcc9ecc86e381ed2d7a3b0 | subsection | 4 | 8 | Body | In the AD, it is well-known that D_h=0
by definition and D_w=1/2 by mapping avalanche boundaries onto
the random walks . The avalanche flow of the AD can
be written as \frac{dN}{dt}\approx \eta . An uncorrelated noise
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4540614b77b7cdb0c19171dc18d116b2b957391f | subsection | 5 | 8 | Body | Another scenario for \alpha =1/2
can be found by mapping metastable patterns onto the space-time
configuration of 2A\rightarrow A coagulation-diffusion model defined in
d=1, where the particle density decays as
t^{-1/2} . One can say that the NS belongs
to the same universality class as the AS in the following sense:
F... | {
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f6a6c5a6b078371ab7c043d10c1acde8f40fce18 | subsection | 6 | 8 | Body | Moreover, we
like to note that the resultant D_h=0 indicates the MF behavior
for the non-Abelian case, whereas D_h=0 for any dimension in the
AD.We performed extensive numerical simulations for all DSMs to
confirm our conjecture about the avalanche exponents in terms of
the scar exponent, \alpha _{\rm sc}=\alpha , up t... | {
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"end": 2534,... | 10.1103/PhysRevLett.101.218001 | 0807.1807 | Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles | [
"Hang-Hyun Jo",
"Meesoon Ha"
] | [
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8add19cc9449808ef02837ae2b01ac24b0f3628e | subsection | 7 | 8 | Body | Moreover, our
results provide essential information on analyzing the
self-organized criticality in real systems as well as answering
how ubiquitous long-range spatial correlations in nature can be
developed and affect real avalanche dynamics.This work was supported by the BK21 project and Acceleration
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e3142699983c4f2f966d9bc29a1a87add8d30489 | abstract | 0 | 49 | Abstract | Two holomorphic Hopf differentials for surfaces of non-null parallel mean
curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence
between these surfaces and pairs of constant mean curvature surfaces of S^2xR
and H^2xR is established. Using that, surfaces with vanishing Hopf
differentials (in parti... | {
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48f1b8b4778b78d74d58095d588b5b527af76a10 | subsection | 1 | 49 | Introduction | Surfaces with constant mean curvature (CMC-surfaces) in three manifolds is a classic topic in differential geometry and it has been extensively studied when the ambient manifold has constant curvature. In 2004, Abresh and Rosenberg studied CMC-surfaces in \mathbb {S}^2\times \mathbb {R} and \mathbb {H}^2\times \mathbb ... | {
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"Francisco Torralbo",
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07ab49ff3e53acaac96b0bc976aa30dabce3c8ae | subsection | 2 | 49 | Introduction | In this case, although there are umbilical hypersurfaces of the ambient space, only the totally geodesic ones (up to congruences \mathbb {S}^2\times \mathbb {R} and \mathbb {H}^2\times \mathbb {R}) have constant mean curvature (see Proposition REF ) and so CMC-surfaces of \mathbb {S}^2\times \mathbb {R} and \mathbb {H}... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
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39ef670925f1894d0dd30d7f33c42c69ada9e8bd | subsection | 3 | 49 | Introduction | Theorem REF is the most important contribution of the paper, it classifies the surfaces with parallel mean curvature vector with null extrinsic normal curvature. In the classification it appears the CMC-surfaces of \mathbb {S}^2\times \mathbb {R} and \mathbb {H}^2\times \mathbb {R}, the Lagrangian PMC-surfaces and a ne... | {
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"Francisco Torralbo",
"Francisco Urbano"
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bb83d5b2438102e6996fb9f1f3ea26ea5d253684 | subsection | 4 | 49 | Preliminaries and examples | We denote by M^2(\epsilon ), \epsilon =1,-1, the two-dimensional sphere \mathbb {S}^2=\lbrace x\in \mathbb {R}^3\,|\,x_1^2+x_2^2+x_3^2=1\rbrace endowed with the canonical metric of constant curvature 1 when \epsilon =1 and the hyperbolic plane \mathbb {H}^2=\lbrace x\in \mathbb {R}^3\,|\,x_1^2+x_2^2-x_3^2=-1, \, x_3 > ... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
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"math.DG"
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5958d86922f6f30f4522d49155346e83578df49e | subsection | 5 | 49 | Preliminaries and examples | If \lbrace e_1,e_2,e_3,e_4\rbrace is an oriented orthonormal local frame on \Phi ^*T(M^2(\epsilon )\times M^2(\epsilon )) such that \lbrace e_1,e_2\rbrace is an oriented frame on T\Sigma , then we define the normal curvature K^{\perp } of the immersion \Phi byK^{\perp }=R^{\perp }(e_1,e_2,e_3,e_4),where R^{\perp } is t... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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4640d546bc0e018ec6933254af95154f48180c92 | subsection | 6 | 49 | Preliminaries and examples | Then \Psi is totally geodesic and it is locally congruent to the totally geodesic immersion:\epsilon &= 1 & \epsilon &= -1 \\
\mathbb {S}^2\times \mathbb {R}&\rightarrow \mathbb {S}^2\times \mathbb {S}^2 & \mathbb {H}^2\times \mathbb {R}&\rightarrow \mathbb {H}^2\times \mathbb {H}^2 \\
(p,t) &\mapsto (p,(\cos t,\sin t,... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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c0f86cb866990db1fb2bf726566c334e4a69026a | subsection | 7 | 49 | Preliminaries and examples | Now the second equation says that \eta _2=A_2 and so \langle \Psi _2,A_2\rangle =\langle \Psi ,\eta \rangle =0 with |A_2|=|\eta _2|=1. This proves that \Psi _2(N) is a geodesic of \mathbb {S}^2 or \mathbb {H}^2 and the proof finishes.As a consequence of this result we obtain thatCMC-surfaces of M^2(\epsilon )\times \ma... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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fff8aaa7cc95198b53c88c30cc63ad95a3f1dcb8 | subsection | 8 | 49 | Preliminaries and examples | It is interesting to remark that the induced metric on I\times I^{\prime } by \Phi is flat.Taking into account the curves of constant curvature of \mathbb {S}^2 and \mathbb {H}^2 we have that
the above examples are, up to congruences, open subsets of the following family of complete and embedded PMC-surfaces:Example 1 ... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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1a5cc80d99cbb9b849a93f057b26d8b694177c38 | subsection | 9 | 49 | Hopf differentials. | In order to have a deep understanding of the geometry of M^2(\epsilon )\times M^2(\epsilon ) and of
its surfaces we need to introduce the two Kähler structures that M^2(\epsilon )\times M^2(\epsilon ) has. We can define two complex structures on M^2(\epsilon )\times M^2(\epsilon ) byJ_1=(J,J),\quad J_2=(J,-J),whose Käh... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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00a7d2cc7291d1ee01ba689ca5a28c66f92fb5bf | subsection | 10 | 49 | Hopf differentials. | It is interesting to remark that C_j^2 is well defined even when the surface is not orientable.Now it is easy to check that the Jacobians of \phi and \psi are given by\hbox{Jac}\,(\phi )=\frac{C_1+C_2}{2},\quad \hbox{Jac}\,(\psi )=\frac{C_1-C_2}{2},and that the extrinsic curvature \bar{K}=\bar{R}(e_1,e_2,e_2,e_1), wher... | {
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2ac3f46fe264d7159dc804ca261e2c5d0717a1c6 | subsection | 11 | 49 | Hopf differentials. | Denoting\xi =\frac{1}{\sqrt{2}|H|}(H-i\tilde{H}),we have that |\xi |^2=1, \langle \xi ,\xi \rangle =0, \nabla ^{\perp }\xi =0 and \lbrace \xi ,\bar{\xi }\rbrace is a reference of the complexified normal bundle. | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
] | 2,008 | en | Mathematics | [
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29ded319652eb91b18e866d32aea9f2cd8acb094 | subsection | 12 | 49 | Hopf differentials. | Using known arguments in theory of surfaces in Kähler surfaces (see for instance ) and taking into account the chosen orientations it is easy to prove thatJ_1\Phi _z=iC_1\Phi _z+\gamma _1\xi ,\quad \quad J_1\xi =-2e^{-2u}\bar{\gamma }_1\Phi _z-iC_1\xi ,\\
J_2\Phi _z=iC_2\Phi _z+\gamma _2\bar{\xi },\quad \quad J_2\xi =-... | {
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"raw": "J.H. Eschenburg, I.V. Guadalupe and R.A. Tribuzy. The fundamental equations of minimal surfaces in \\mathbb {C}\\mathbb {P}^2. Math. Ann. 270 (1985) 571–598.",
"source_ref_id": "bb0ef67b14ac24... | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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85e784dc06399b5e5cfcfddbf3010ff87085d03d | subsection | 13 | 49 | Hopf differentials. | \, j=1,2.But the third equation in (REF ) can be easily deduced from the equations in (REF ) and from that equation and using again (REF ) we obtain Gauss and Ricci equations. So, really, the integrability conditions of the above Frenet system are (REF ).Proposition 2
Let \Phi :\Sigma \rightarrow M^2(\epsilon )\times ... | {
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"raw": "U. Abresch and H. Rosenberg. A Hopf differential for constant mean curvature surfaces in \\mathbb {S}^2\\times \\mathbb {R} and \\mathbb {H}^2\\times \\mathbb {R}. Acta Math. 193 (2004) 141–174.",
... | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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0d017d2516c9fee7669e18809ac2fb85c182d34c | subsection | 14 | 49 | Hopf differentials. | It is clear that, in this case, \eta =(0,(0,0,1)) (respectively \eta =(0,(1,0,0))) when \epsilon =1 (respectively \epsilon =-1) is a unit normal field to the totally geodesic immersion M^2(\epsilon )\times \mathbb {R}\hookrightarrow M^2(\epsilon )\times M^2(\epsilon ) given in Proposition REF . So \tilde{H}=|H|\eta . I... | {
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} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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177b2ed7d01f055d950302fa7693ec33473215ff | subsection | 15 | 49 | Hopf differentials. | Second, using (REF ) and the integrability conditions, we obtain the following relation between |\Theta _j|^2 and |\nabla C_j|^2
\begin{split}
&|\nabla C_j|^2+4\epsilon e^{-4u}|\Theta _j|^2 = \\
&=(1-C_j^2+4\epsilon |H|^2)\left(\frac{\epsilon (1-C_j^2)}{4}+|H|^2+\epsilon C_j^2-K\right), \quad j =1,2.
\end{split}
Al... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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25c4fdd1e1191b9425f65b8bc0f4a1af0dd9f6a8 | subsection | 16 | 49 | Hopf differentials. | Therefore, in this case, p_0 is a non-degenerate critical point.If \Theta _j(p_0)\ne 0, in a neighborhood of p_0 we can normalize \Theta _j=\lambda \in \mathbb {C}^*. In particular \lambda =\Theta _j(p_0)=2\sqrt{2}|H|f_j(p_0). In this case we get
that the determinant of the Hessian of C_j at p_0 is4\left(\frac{|\lambda... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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6585cdab209cc488a432c1cf46454837575e4d9e | subsection | 17 | 49 | Main Results | The integrability equations given in the previous section allow to relate, at least in the simply connected case, PMC-immersions in M^2(\epsilon )\times M^2(\epsilon ) with pairs of CMC-immersions in M^2(\epsilon ) \times \mathbb {R} with the same induced metric and the same length of the mean curvature. We concrete th... | {
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"start": ... | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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20e5ee5d6e3362cfc4459c43aa471402a1ec3be3 | subsection | 18 | 49 | Main Results | Then the Frenet equations of \Psi : \Sigma \longrightarrow M^2(\epsilon ) \times \mathbb {R}\subset \mathbb {R}^3\times \mathbb {R} (or \mathbb {R}^3_1) are given by:\begin{aligned}\Psi _{zz} &= 2u_z \Psi _z + p N + \epsilon \eta _z^2 \hat{\Psi } \\
\Psi _{z\bar{z}} &= \frac{e^{2u}}{2}H N + \epsilon \left( |\eta _z|^2 ... | {
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"Francisco Torralbo",
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"math.DG"
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1932382e89a190001340441158e9454807b2170b | subsection | 19 | 49 | Main Results | We consider the data\bigl (u, H_j = |H|, \nu _j = C_j, \eta _j, p_j = \sqrt{2}f_j\bigr ), \quad j= 1,2.From (REF ), it is followed that these data satisfy (REF ), and so there exist two CMC-isometric immersions \Phi _j: (\Sigma , g) \rightarrow M^2(\epsilon ) \times \mathbb {R} with |H_j| = |H|, j = 1, 2.Moreover, it i... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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96334a373f982ac5285f73af97db6f53564aaae6 | subsection | 20 | 49 | Main Results | Then, given an isothermal parameter z, and possible up to a congruence, we can take the data of \Phi _j as |H_1| = |H_2|, p_1 = p_2, \nu _1 = \nu _2 and \eta _1 = \eta _2. Therefore the associated PMC-isometric immersion \Phi = (\phi , \psi ):(\Sigma , g) \rightarrow M^2(\epsilon )\times M^2(\epsilon ) has f_1 = f_2, \... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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e5f0a34c1d8847ba0b62a69c7491c690df405eea | subsection | 21 | 49 | Main Results | In the following result we classify (even locally) those PMC-surfaces of M^2(\epsilon )\times M^2(\epsilon ) which are Lagrangian with respect to some of the complex structures.Theorem 2
Let \Phi :\Sigma \rightarrow M^2(\epsilon )\times M^2(\epsilon ) be a PMC-immersion of a surface \Sigma . If \Phi is Lagrangian with... | {
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"Francisco Torralbo",
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"math.DG"
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4aff7898d5bab247c11fbc4e3454057d047d8455 | subsection | 22 | 49 | Main Results | We have proved that U = \emptyset . Therefore C_2 is constant. But (C_2)_{z}=0 implies that (1-C_2^2)f_2=\frac{|H|}{\sqrt{2}}\gamma _2^2. From here and (REF ) one obtains that C_2^2=\epsilon K=0. So in this case our immersion \Phi is also Lagrangian with respect to J_2.Secondly if \Theta _2\ne 0, then it has isolated z... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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af97c099a1ad3bebf3d94e9f9238cf2b5a69c523 | subsection | 23 | 49 | Main Results | Beside the above family, an interesting family of examples appears in the classification which we describe in the next result.Proposition 4
Let a,b,c be real numbers with b>0 and h:I\subset \mathbb {R}\rightarrow \mathbb {R} a non-constant solution of the O.D.E.(h^{\prime })^2(x)=(a-h^2(x))\left((a-h^2(x))-\epsilon b(... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
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"math.DG"
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dc2e293567fa1ef0575213f2a5471c930d8ca635 | subsection | 24 | 49 | Main Results | All the previous examples are invariant under the 1-parametric group of isometries \lbrace I(\theta ) \times \mathrm {Id},\, \theta \in \mathbb {R}\rbrace of M^2(\epsilon ) \times M^2(\epsilon ), where I(\theta ):M^2(\epsilon )\rightarrow M^2(\epsilon ) is the isometry given by:
[Table: NO_CAPTION]First it is easy to c... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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d5efc8b29acee8c5ed56cc096c1201cc30471aa6 | subsection | 25 | 49 | Main Results | Using the above formulae we getH = \frac{1}{2\epsilon (a-h^2)} \left( -\frac{b h^{\prime }(h-c)}{|\phi _x|^2}\phi _x, \frac{b h^{\prime }(h-c)}{|\psi _x|^2}\psi _x - \frac{\epsilon b(a-h^2)}{|\psi _x|^2}J\psi _x \right).From this equation the length of H is |H|^2 = b/4 and after a long straightforward computation we ob... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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dab18044bb0912ba89bc1705416407505a674c50 | subsection | 26 | 49 | Main Results | On the other hand, it is possible to obtain all the solutions of equation (REF ) in terms of Jacobi elliptic functions (see ) and a deep analysis of them shows that the conditions appearing in (REF ) are also sufficient in order to the solutions of equation (REF ) satisfied \epsilon (a-h^2)>0. SoThe solutions h of the ... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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a2a261cf921b52ab8569b6291ebf965581417bd4 | subsection | 27 | 49 | Main Results | When \lambda = 0, that is a = -1, \Phi _0 is the product of a geodesic and a horocycle, i.e. \hat{P}_0 in example REF .Theorem 3
Let \Phi :\Sigma \rightarrow M^2(\epsilon )\times M^2(\epsilon ) be a PMC-immersion of a surface \Sigma . Then the extrinsic normal curvature vanishes, \bar{K}^{\perp }=0, if and only if \Ph... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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26720cafee27833a82dbf582b55aa6907795ddc6 | subsection | 28 | 49 | Main Results | Outside its zeroes we can normalize it as \gamma _2-\gamma _1=2\sqrt{2}|H|. As C_1=C_2 we have that |\gamma _1|^2=|\gamma _2|^2 and so \Re \,(\gamma _1)=-\sqrt{2}|H|. Hence\gamma _1=-\sqrt{2}|H|+ig,\quad \gamma _2=-\bar{\gamma }_1,for certain function g:\Sigma \rightarrow \mathbb {R}. Now, using the integrability equat... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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24e30c242758bafe39c4a7dd436e31ccabbe62f5 | subsection | 29 | 49 | Main Results | \end{split}We are going to integrate the Frenet equations. First of all, from (REF ) and () we obtain that J_1\Phi _z-J_2\Phi _z=2\Re \,(\gamma _1\xi ) and J_1\Phi _z+J_2\Phi _z=2iC_1\Phi _z+2i\Im \,(\gamma _1\xi ). So, taking into account the definitions of J_j, we get that (0,J\psi _z)=\Re \,(\gamma _1\xi ) and (J\ph... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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618ed099070129fce19f960c638b4f260f686419 | subsection | 30 | 49 | Main Results | So from the above equations we finally obtain that\frac{u^{\prime }(x)^2}{C_1(x)^2}+\epsilon e^{2u(x)}=a\in \mathbb {R},\quad \tilde{G}(x)=-\tilde{G}_0\in \mathbb {R}^3\,(\mathbb {R}^3_1),\quad \forall (x,y)\in \Sigma ,and so finally F satisfies the following O.D.E.F^{\prime \prime }(y)+aF(y)-\tilde{G}{_0}=0.The soluti... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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9b8cf6d9176b30a44312a99d724353f0129e3207 | subsection | 31 | 49 | Main Results | Now, up to an isometry in \mathbb {R}^3 or \mathbb {R}^3_1 we can choose H_1 = (1/\sqrt{a},0,0), H_2 = (0,1/\sqrt{a},0) and \hat{G} = h(x)(0, 0, 1/\sqrt{a}) when a > 0, H_1 = (0,0,1/\sqrt{-a}), H_2 = (0,1/\sqrt{-a},0) and \hat{G} = h(x)(1/\sqrt{-a}, 0, 0) when a < 0, and H_1 = (0,1,0), \tilde{G}_0 = (1,0,1) and \hat{G}... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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16afc7805a71960ae0e81046f1fe97c40acaca61 | subsection | 32 | 49 | Main Results | Therefore, the above equations become in\phi (x,y) &= \frac{1}{\sqrt{a}}\Bigl (e^{u(x)} \cos (\sqrt{a}y), e^{u(x)} \sin (\sqrt{a}y), h(x)\Bigr ), & a>0 \\
\phi (x,y) &= \frac{1}{\sqrt{-a}}\Bigl (h(x), e^{u(x)} \sinh (\sqrt{-a}y), e^{u(x)} \cosh (\sqrt{-a}y)\Bigr ), & a<0 \\
\phi (x,y) &= \left( \frac{e^{u(x)}}{2}y^2 +\... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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2cc9fa63e16f4b95f54ed04bac300ca80e54557c | subsection | 33 | 49 | Main Results | From (REF ) and as h^2+\epsilon e^{2u}=a, we have that h=-\frac{\epsilon u^{\prime }}{C_1} and so (REF ) implies that h=-\frac{\epsilon \Im (\mu )}{2|H|^2}-\frac{ g}{\sqrt{2}|H|}. From (REF ) again we get that h^{\prime }= e^{2u}C_1 and then using one more time (REF ) we get(h^{\prime })^2=C_1^2e^{4u}=e^{2u}(e^{2u}-4|H... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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7d5412b1b9dfe8e41a8e9aaec01d7c709db8f6db | subsection | 34 | 49 | Main Results | \Theta _1=\Theta _2=0, if and only if one of the three following possibilities happens:\Phi (\Sigma ) lies in M^2(\epsilon )\times \mathbb {R} as a CMC-surface with vanishing Abresh-Rosenberg differential,
\epsilon =-1, 4|H|^2=1 and locally \Phi is the product of two hypercycles \alpha and \beta of \mathbb {H}^2 with ... | {
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"raw": "M.L. Leite. An elementary proof of the Abresh-Rosenberg theorem on constant mean curvature immersed surfaces in \\mathbb {S}^2\\times \\mathbb {R} and \\mathbb {H}^2\\times \\mathbb {R}. Quart.J. M... | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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fb0018bcb63cb4d40eb3758485b7a42e2e176406 | subsection | 35 | 49 | Main Results | If \epsilon =-1, on the open set O=\lbrace p\in \Sigma \,|\,C_1^2(p)\ne C_2^2(p)\rbrace , we have thatC_1^2+C_2^2=2(1-8|H|^2).But on O, C_1\nabla C_1=-C_2\nabla C_2, and then using (REF ), (REF ) and the integrability equations (REF ) we obtain thatC_1^2(1-C_1^2)(1-C_1^2-4|H|^2)^2=C_2^2(1-C_2^2)(1-C_2^2-4|H|^2)^2.As on... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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4cc62e89506925c18ad5709c0c5184bee73eed0e | subsection | 36 | 49 | Main Results | Then, up to congruences, \Phi is a CMC-sphere in M^2(\epsilon )\times \mathbb {R}.The examples described in Theorem REF .3) and the examples obtained by Leite in can be characterized in the following way.Corollary 2 Let \Phi :\Sigma \rightarrow M^2(\epsilon )\times M^2(\epsilon ) be a PMC-immersion of an orientable sur... | {
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"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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bc7ba0a73719d44890228c65d2b2f79e67f9fc59 | subsection | 37 | 49 | Examples of CMC-surfaces in | Following Theorem REF , the examples of PMC-surfaces of M^2(\epsilon )\times M^2(\epsilon ) described in Proposition REF have associated pairs of CMC-surfaces of M^2(\epsilon )\times \mathbb {R}. As these PMC-surfaces do not factorize through CMC-surfaces of M^2(\epsilon )\times \mathbb {R}, the pairs of CMC-surfaces a... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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5839dc030bb44a9e34b4593db92cd80f1c85ce64 | subsection | 38 | 49 | Examples of CMC-surfaces in | Then \Psi = (\psi , \eta ): I \times \mathbb {R}\rightarrow M^2(\epsilon )\times \mathbb {R} where \eta (x,y) = \sqrt{b}\left( y+\int _{x_0}^x \bigl (h(t) - c\bigr )\mathrm {d}t \right) and \psi :I\times \mathbb {R}\rightarrow M^2(\epsilon ) is given byIf E = a - \epsilon b > 0
\psi (x,y) = \frac{1}{\sqrt{E}}\left( \s... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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