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| :''For the discrete equivalent of the [[Laplace transform]], see [[Z-transform]].''
| | == Gu Nanhai brow is wrinkled == |
| {{refimprove|date=December 2007}}
| |
| In [[mathematics]], the '''discrete Laplace operator''' is an analog of the continuous [[Laplace operator]], defined so that it has meaning on a [[graph (mathematics)|graph]] or a [[lattice (group)|discrete grid]]. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the [[Laplacian matrix]].
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| The discrete Laplace operator occurs in [[physics]] problems such as the [[Ising model]] and [[loop quantum gravity]], as well as in the study of discrete [[dynamical system]]s. It is also used in [[numerical analysis]] as a stand-in for the continuous Laplace operator. Common applications include [[image processing]], where it is known as the [[Laplace filter]], and in machine learning for [[cluster analysis|clustering]] and [[semi-supervised learning]] on neighborhood graphs.
| | Sea faint smile, Gang Yu once again to say something, the surrounding space, sudden severe fluctuations since a thrill of horror temperature, [http://www.nnyagdev.org/sitemap.xml http://www.nnyagdev.org/sitemap.xml] sailed from the surrounding diffuse open.<br><br>'Be careful!'<br><br>see this scene, Gu Nanhai brow is wrinkled, said.<br><br>course, no reminding, at the moment, Xiao go far people have filed a heart, looking more and more twisted [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ レディース 電波ソーラー腕時計] around watchful eyes of space, this space [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ 電波時計] is too strange fire demon, have to be cautious.<br><br>'roaring!'<br><br>With more distorted space to get later, a road crack suddenly emerged out of these cracks, like the waves of a Unit as 'milk' white 'color' flame, since which swept open, followed by, covered [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオの時計] with a road filled with 'milk' white 'color' flame figure, such as 'boom' like water from the storm surge out of the many cracks in the blink of an eye, the square outside the sky, who is to be the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio 腕時計 データバンク] flame |
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| | <li>[http://thinkchen.66rt.com/viewthread.php?tid=177953&extra= http://thinkchen.66rt.com/viewthread.php?tid=177953&extra=]</li> |
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| | <li>[http://bbs.ec01.cn/home.php?mod=space&uid=360578 http://bbs.ec01.cn/home.php?mod=space&uid=360578]</li> |
| | |
| | </ul> |
|
| |
|
| ==Definitions== | | == it seems you are very Yunshan fear ah == |
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| ===Graph Laplacians===
| | .<br><br>'Hey, you old dog Yunshan this hand it very ruthless [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html casio 腕時計 デジタル] ah. Hell, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] Gouji you still jump off the wall, the rabbit [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 電波ソーラー時計] is also anxious to bite you it was' forced 'After two days I was on cloud Arashiyama.'<br><br>finger on dark refers slight flick, 'medicine' that old figure is slowly wafting [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ] illusory, he looked at Xiao Yan, said: 'battlefield Arashiyama in the cloud, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps] cloud-lan has the terrain, the disciples were also numerous In the mountains, there are more people this trend, hey, it seems you are very Yunshan fear ah, otherwise do not use so much thought. '<br><br>Xiao Yan nodded, cloud-lan many people, but also know how to punch battle, but that punch battle for power, he is personally experienced the year, the natural thing is to know that tyranny.<br><br>'you rhyme with those between clouds, Yunshan probably also know, and so he moves, there is a fear also angered and disturbed' mess 'the intent of your heart,' 'medicine' old Shen ' |
| There are various definitions of the ''discrete Laplacian'' for [[Graph (mathematics)|graphs]], differing by sign and scale factor (sometimes one averages over the neighboring vertices, other times one just sums; this makes no difference for a [[regular graph]]). The traditional definition of the graph Laplacian, given below, corresponds to the '''negative''' continuous Laplacian on a domain with a free boundary.
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| | <ul> |
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| | <li>[http://www.yg6688.com/plus/view.php?aid=10441 http://www.yg6688.com/plus/view.php?aid=10441]</li> |
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| | |
| | </ul> |
|
| |
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| Let <math>G = (V,E)</math> be a graph with vertices <math>\scriptstyle V</math> and edges <math>\scriptstyle E</math>. Let <math>\phi\colon V\to R</math> be a [[function (mathematics)|function]] of the vertices taking values in a [[ring (mathematics)|ring]]. Then, the discrete Laplacian <math>\Delta</math> acting on <math>\phi</math> is defined by
| | == if re-Bushi Xiang == |
|
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| :<math>(\Delta \phi)(v)=\sum_{w:\,d(w,v)=1}\left[\phi(w)-\phi(v)\right]\,</math> | | The two city in the horizon of two old reputation is not [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps] weak, rapid footsteps mounted the pedal back, while the other hand young black robes that name, it was like a rock-solid, motionless.<br><br>'today than, blame, if re-Bushi Xiang, can [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 バンド] not blame the next ruthless!'<br><br>Xiao Yan eyes looking at the two chill man, in the eyes of the intention to kill, reveals the slightest cold, today this thing is already going beyond his bottom line.<br><br>Xiao Yan out [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン] previously demonstrated the strength [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ レディース 電波ソーラー腕時計] of the shock and awe, two old man is afraid to be Alert, but the hearts crying, this sister, 'milk', 'milk' run amok for so many years, finally hit the muzzle up, This kid may [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] look young, but the strength is not extremely weak, and that the previous single-handedly, even family, is only very few of the few people able to cast out.<br><br>'the friend, the next is the home of the Romanian people, who previously collision your girl, Luo home |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.k9158.com/user/space.php?uid=23177&do=blog&id=47671 http://www.k9158.com/user/space.php?uid=23177&do=blog&id=47671]</li> |
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| | |
| | </ul> |
|
| |
|
| where <math>d(w,v)</math> is the [[Distance (graph theory)|graph distance]] between vertices w and v. Thus, this sum is over the nearest neighbors of the vertex ''v''. For a graph with a finite number of edges and vertices, this definition is identical to that of the [[Laplacian matrix]]. That is, <math> \phi</math> can be written as a column vector; and so <math>\Delta\phi</math> is the product of the column vector and the Laplacian matrix, while <math>(\Delta \phi)(v)</math> is just the ''v'''th entry of the product vector.
| | == Xiao Yan far behind == |
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| If the graph has weighted edges, that is, a weighting function <math>\gamma\colon E\to R</math> is given, then the definition can be generalized to
| | Proud of her, for a few different 'sex' feel admiration, the only exception to this person in front of<br>They contact time<br>not long, 'medicine' Nine mysterious tribe of Dan when refining, Xiao Yan serious and calm, and [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] in the face of ethnic [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html casio 腕時計 ゴールド] extermination soul, he unfolded, but it is like [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html 電波時計 casio] a blade like a sharp, decisive rage, all the way grappling, eventually escaped with their situation and that this is a dead end.<br><br>Americans love a hero, no matter how cold, and arrogant once this beauty, but it seems it is still inevitable that law.<br><br>idea for the children's arms, Xiao Yan'd [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html casio 腕時計 スタンダード] never bother him now, most of the mind, are pressing and refining of the vast energy of the body out of control, but [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps] he also must maintain a speed, these kinds of , for him, but a considerable challenge.<br><br>'separated it.'<br><br>Xiao Yan far behind, his face dark soul of evil |
| | | 相关的主题文章： |
| :<math>(\Delta_\gamma\phi)(v)=\sum_{w:\,d(w,v)=1}\gamma_{wv}\left[\phi(w)-\phi(v)\right]</math>
| | <ul> |
| | | |
| where <math>\gamma_{wv}</math> is the weight value on the edge <math>wv\in E</math>.
| | <li>[http://www.wa3key.com/cgi-bin/ibook/ibook.cgi http://www.wa3key.com/cgi-bin/ibook/ibook.cgi]</li> |
| | | |
| Closely related to the discrete Laplacian is the '''averaging operator''':
| | <li>[http://www.age.ne.jp/x/koryu/bbs/light.cgi http://www.age.ne.jp/x/koryu/bbs/light.cgi]</li> |
| | | |
| :<math>(M\phi)(v)=\frac{1}{\deg v}\sum_{w:\,d(w,v)=1}\phi(w).</math>
| | <li>[http://cgi.www7a.biglobe.ne.jp/~respective-style/joyful/joyful.cgi http://cgi.www7a.biglobe.ne.jp/~respective-style/joyful/joyful.cgi]</li> |
| | | |
| ===Mesh Laplacians===
| | </ul> |
| | |
| In addition to considering the connectivity of nodes and edges in a graph, mesh laplace operators take into account the geometry of a surface (e.g. the angles at the nodes). For triangle meshes, for example, different discretizations exist, some of them are an extension of the graph operator, while other approaches are based on the [[finite element method]] (see below) and allow for higher order approximations. An overview on some mesh operators and a comparison is given in.<ref name="reuter06">{{cite journal
| |
| |author = Reuter, M. and Biasotti, S. and Giorgi, D. and Patane, G. and Spagnuolo, M.",
| |
| | year = 2009
| |
| | title = Discrete Laplace-Beltrami operators for shape analysis and segmentation
| |
| | journal = Computers & Graphics
| |
| | publisher = Elsevier
| |
| | volume = 33
| |
| | issue = 3
| |
| | pages = 381–390
| |
| | doi=10.1016/j.cag.2009.03.005
| |
| }}</ref>
| |
| | |
| === Finite Differences ===
| |
| | |
| Approximations of the [[Laplacian]], obtained by the [[finite difference method]] or by the [[finite element method]] can also be called '''Discrete Laplacians'''. For example, the Laplacian in two dimensions can be approximated using the [[five-point stencil]] [[finite difference method]], resulting in
| |
| | |
| :<math> \Delta f(x,y) \approx \frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}, \,</math>
| |
| | |
| where the grid size is ''h'' in both dimensions, so that the five point stencil of a point (''x'', ''y'') in the grid is
| |
| | |
| :<math>\{(x-h, y), (x, y), (x+h, y), (x, y-h), (x, y+h)\}. \,</math> | |
| | |
| If the grid size ''h=1'', the result is the '''negative''' discrete Laplacian on the graph, which is the [[square lattice|square lattice grid]]. There are no constraints here on the values of the function ''f(x,y)'' on the boundary of the lattice grid, thus this is the case of the homogeneous [[Neumann boundary condition]], i.e., free boundary. Other types of boundary conditions, e.g., the homogeneous [[Dirichlet boundary condition]], where ''f(x,y)=0'' on the boundary of the grid, are rarely used for graph Laplacians, but are common in other applications.
| |
| | |
| Multidimensional discrete Laplacians on [[Cuboid#Rectangular_cuboid|rectangular cuboid]] [[regular grid]]s have very special properties, e.g., they are [[Kronecker_product#Kronecker_sum_and_exponentiation|Kronecker sums]] of one-dimensional discrete Laplacians, see [[Kronecker sum of discrete Laplacians]], in which case all its [[eigenvalue]]s and [[eigenvectors]] can be explicitly calculated.
| |
| | |
| ===Finite Element Method===
| |
| | |
| In this approach, the domain is discretized into smaller elements, often triangles or tetrahedra, but other elements such as rectangles or cuboids are possible. The solution space is then approximated using so called form-functions of a pre-defined degree. The differential equation containing the Laplace operator is then transformed into a variational formulation, and a system of equations is constructed (linear or eigenvalue problems). The resulting matrices are usually very sparse and can be solved with iterative methods.
| |
| | |
| ===Image Processing===
| |
| Discrete Laplace operator is often used in image processing e.g. in edge detection and motion estimation applications. The discrete Laplacian is defined as the sum of the second derivatives [[Laplace operator#Coordinate expressions]] and calculated as sum of differences over the nearest neighbours of the central pixel.
| |
| | |
| ====Implementation in Image Processing====
| |
| For one, two and three dimensional signals, the discrete Laplacian can be given as [[convolution]] with the following kernels:
| |
| :''1D-filter:'' <math>\vec{D}^2_x=\begin{bmatrix}1 & -2 & 1\end{bmatrix}</math>
| |
| :''2D-filter:'' <math>\mathbf{D}^2_{xy}=\begin{bmatrix}0 & 1 & 0\\1 & -4 & 1\\0 & 1 & 0\end{bmatrix}</math>
| |
| or, including the diagonals:
| |
| :''2D-filter:'' <math>\mathbf{D}^2_{xy}=\begin{bmatrix}0.5 & 1 & 0.5\\1 & -6 & 1\\0.5 & 1 & 0.5\end{bmatrix}</math>
| |
| :''3D-filter:'' <math>\mathbf{D}^2_{xyz}</math> is given by: first plane = <math>\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix}</math> ; second plane = <math>\begin{bmatrix}0 & 1 & 0\\1 & -6 & 1\\0 & 1 & 0\end{bmatrix}</math> ; third plane = <math>\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix}</math>
| |
| <br />
| |
| :''{{var|n}}D-filter'': For the element <math>a_{x_1, x_2, \dots , x_n}</math> of the kernel <math>\mathbf{D}^2_{x_1, x_2, \dots , x_n},</math>
| |
| | |
| ::<math>a_{x_1, x_2, \dots , x_n} = \left\{\begin{array}{ll}
| |
| -2n, & \text{if } s = n \\
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| 1, & \text{if } s = n - 1 \\
| |
| 0, & \text{otherwise}
| |
| \end{array}\right.</math>
| |
| | |
| :where {{math|{{var|x}}{{sub|{{var|i}}}}}} is the position (either {{math|-1}}, {{math|0}} or {{math|1}}) of the element in the kernel in the {{math|{{var|i}}:th}} direction, and {{math|{{var|s}}}} is the number of directions {{math|{{var|i}}}} for which {{math|{{var|x}}{{sub|{{var|i}}}} {{=}} 0}}.
| |
| | |
| Note that the nD version, which is based on the graph generalization of the Laplacian, assumes all neighbors to be at an equal distance, and hence, leads to the following 2D-filter with diagonals included, rather than the version above:
| |
| :''2D-filter:'' <math>\mathbf{D}^2_{xy}=\begin{bmatrix}1 & 1 & 1\\1 & -8 & 1\\1 & 1 & 1\end{bmatrix}.</math>
| |
| | |
| These kernels are deduced by using discrete differential quotients.
| |
| | |
| ==Spectrum==
| |
| The spectrum of the discrete Laplacian is of key interest; since it is a [[self-adjoint operator]], it has a real spectrum. For the convention <math>\Delta = I - M</math>, the spectrum lies within <math>[0,2]</math> (as the averaging operator has spectral values in <math>[-1,1]</math>). The smallest non-zero eigenvalue is denoted <math>\lambda_1</math> and is called the [[spectral gap]]. There is also the notion of the [[spectral radius]], commonly taken as the largest eigenvalue.
| |
| | |
| The eigenvectors don't depend on the convention above (for [[regular graph]]s), and are the same as for the averaging operator (as they differ by adding a multiple of the identity), though the eigenvalues differ according to the convention.
| |
| | |
| For operators that approximate the underlying continuous Laplacian the eigenvalues are a sequence of positive real numbers. The first eigenvalue is zero, if the domain has a boundary and the Neumann boundary condition is used, or if the shape contains no boundary (e.g. the sphere).
| |
| | |
| ==Theorems==
| |
| If the graph is an infinite [[square lattice|square lattice grid]], then this definition of the Laplacian can be shown to correspond to the continuous Laplacian in the limit of an infinitely fine grid. Thus, for example, on a one-dimensional grid we have
| |
| | |
| :<math>\frac{\partial^2F}{\partial x^2} =
| |
| \lim_{\epsilon \rightarrow 0}
| |
| \frac{[F(x+\epsilon)-F(x)]+[F(x-\epsilon)-F(x)]}{\epsilon^2}.
| |
| </math> | |
| | |
| This definition of the Laplacian is commonly used in [[numerical analysis]] and in [[image processing]]. In image processing, it is considered to be a type of [[digital filter]], more specifically an [[edge filter]], called the [[Laplace filter]].
| |
| | |
| ==Discrete Schrödinger operator==
| |
| Let <math>P:V\rightarrow R</math> be a [[potential]] function defined on the graph. Note that ''P'' can be considered to be a multiplicative operator acting diagonally on <math>\phi</math>
| |
| | |
| :<math>(P\phi)(v)=P(v)\phi(v).\,</math>
| |
| | |
| Then <math>H=\Delta+P</math> is the '''discrete Schrödinger operator''', an analog of the continuous [[Schrödinger equation|Schrödinger operator]].
| |
| | |
| If the number of edges meeting at a vertex is uniformly bounded, and the potential is bounded, then ''H'' is bounded and [[self-adjoint]].
| |
| | |
| The [[spectrum of an operator|spectral properties]] of this Hamiltonian can be studied with [[Stone space|Stone's theorem]]; this is a consequence of the duality between [[poset]]s and [[Boolean algebra (structure)|Boolean algebra]]s.
| |
| | |
| On regular lattices, the operator typically has both traveling-wave as well as [[Anderson localization]] solutions, depending on whether the potential is periodic or random.
| |
| | |
| ==Discrete Green's function==
| |
| The [[Green's function]] of the discrete [[Schrödinger operator]] is given in the [[resolvent formalism]] by
| |
| :<math>G(v,w;\lambda)=\left\langle\delta_v\left| \frac{1}{H-\lambda}\right| \delta_w\right\rangle </math>
| |
| where <math>\delta_w</math> is understood to be the [[Kronecker delta]] function on the graph: <math>\delta_w(v)=\delta_{wv}</math>; that is, it equals ''1'' if ''v''=''w'' and ''0'' otherwise.
| |
| | |
| For fixed <math>w\in V</math> and <math>\lambda</math> a complex number, the Green's function considered to be a function of ''v'' is the unique solution to
| |
| | |
| :<math>(H-\lambda)G(v,w;\lambda)=\delta_w(v).\,</math>
| |
| | |
| == ADE classification ==
| |
| {{further|ADE classification}}
| |
| Certain equations involving the discrete Laplacian only have solutions on the simply-laced [[Dynkin diagram]]s (all edges multiplicity 1), and are an example of the [[ADE classification]]. Specifically, the only positive solutions to the homogeneous equation:
| |
| :<math>\Delta \phi = \phi,</math>
| |
| in words,
| |
| :"Twice any label is the sum of the labels on adjacent vertices,"
| |
| are on the extended (affine) ADE Dynkin diagrams, of which there are 2 infinite families (A and D) and 3 exceptions (E). The resulting numbering is unique up to scale, and if the smallest value is set at 1, the other numbers are integers, ranging up to 6.
| |
| | |
| The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:
| |
| :Twice any label minus two is the sum of the labels on adjacent vertices.
| |
| In terms of the Laplacian, the positive solutions to the inhomogeneous equation:
| |
| :<math>\Delta \phi = \phi - 2.</math>
| |
| The resulting numbering is unique (scale is specified by the "2"), and consists of integers; for E<sub>8</sub> they range from 58 to 270, and have been observed as early as {{Harv|Bourbaki|1968}}.
| |
| | |
| == See also ==
| |
| * [[Spectral shape analysis]]
| |
| * [[Electrical network]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| {{refbegin}}
| |
| * {{citation
| |
| | authorlink = Nicolas Bourbaki
| |
| | first = Nicolas | last = Bourbaki | year = 1968 | title = Groupes et algebres de Lie | chapter = Chapters 4–6 | publisher = Hermann | location = Paris }}
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| *[http://www.yann-ollivier.org/specgraph/specgraph.html Definition and application to spectral gap]
| |
| *[https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkYXZlZGNsdWJ8Z3g6Mzk2ZTUxYzA4MzI5MzBiNw Layered networks, the discrete Laplacian, and a continued fraction identity], Owen D. Biesel, David V. Ingerman, James A. Morrow, and William T. Shore
| |
| | |
| [[Category:Operator theory]]
| |
| [[Category:Graph theory]]
| |
| [[Category:Numerical differential equations]]
| |
| [[Category:Finite differences]]
| |
| [[Category:Feature detection]]
| |
Gu Nanhai brow is wrinkled
Sea faint smile, Gang Yu once again to say something, the surrounding space, sudden severe fluctuations since a thrill of horror temperature, http://www.nnyagdev.org/sitemap.xml sailed from the surrounding diffuse open.
'Be careful!'
see this scene, Gu Nanhai brow is wrinkled, said.
course, no reminding, at the moment, Xiao go far people have filed a heart, looking more and more twisted カシオ レディース 電波ソーラー腕時計 around watchful eyes of space, this space カシオ 電波時計 is too strange fire demon, have to be cautious.
'roaring!'
With more distorted space to get later, a road crack suddenly emerged out of these cracks, like the waves of a Unit as 'milk' white 'color' flame, since which swept open, followed by, covered カシオの時計 with a road filled with 'milk' white 'color' flame figure, such as 'boom' like water from the storm surge out of the many cracks in the blink of an eye, the square outside the sky, who is to be the casio 腕時計 データバンク flame
相关的主题文章：
it seems you are very Yunshan fear ah
.
'Hey, you old dog Yunshan this hand it very ruthless casio 腕時計 デジタル ah. Hell, カシオ ソーラー 腕時計 Gouji you still jump off the wall, the rabbit カシオ 電波ソーラー時計 is also anxious to bite you it was' forced 'After two days I was on cloud Arashiyama.'
finger on dark refers slight flick, 'medicine' that old figure is slowly wafting 時計 カシオ illusory, he looked at Xiao Yan, said: 'battlefield Arashiyama in the cloud, カシオ 腕時計 gps cloud-lan has the terrain, the disciples were also numerous In the mountains, there are more people this trend, hey, it seems you are very Yunshan fear ah, otherwise do not use so much thought. '
Xiao Yan nodded, cloud-lan many people, but also know how to punch battle, but that punch battle for power, he is personally experienced the year, the natural thing is to know that tyranny.
'you rhyme with those between clouds, Yunshan probably also know, and so he moves, there is a fear also angered and disturbed' mess 'the intent of your heart,' 'medicine' old Shen '
相关的主题文章：
if re-Bushi Xiang
The two city in the horizon of two old reputation is not カシオ 腕時計 gps weak, rapid footsteps mounted the pedal back, while the other hand young black robes that name, it was like a rock-solid, motionless.
'today than, blame, if re-Bushi Xiang, can カシオ 腕時計 バンド not blame the next ruthless!'
Xiao Yan eyes looking at the two chill man, in the eyes of the intention to kill, reveals the slightest cold, today this thing is already going beyond his bottom line.
Xiao Yan out カシオ 腕時計 チタン previously demonstrated the strength カシオ レディース 電波ソーラー腕時計 of the shock and awe, two old man is afraid to be Alert, but the hearts crying, this sister, 'milk', 'milk' run amok for so many years, finally hit the muzzle up, This kid may カシオ 腕時計 スタンダード look young, but the strength is not extremely weak, and that the previous single-handedly, even family, is only very few of the few people able to cast out.
'the friend, the next is the home of the Romanian people, who previously collision your girl, Luo home
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Xiao Yan far behind
Proud of her, for a few different 'sex' feel admiration, the only exception to this person in front of
They contact time
not long, 'medicine' Nine mysterious tribe of Dan when refining, Xiao Yan serious and calm, and カシオ腕時計 g-shock in the face of ethnic casio 腕時計 ゴールド extermination soul, he unfolded, but it is like 電波時計 casio a blade like a sharp, decisive rage, all the way grappling, eventually escaped with their situation and that this is a dead end.
Americans love a hero, no matter how cold, and arrogant once this beauty, but it seems it is still inevitable that law.
idea for the children's arms, Xiao Yan'd casio 腕時計 スタンダード never bother him now, most of the mind, are pressing and refining of the vast energy of the body out of control, but カシオ 腕時計 gps he also must maintain a speed, these kinds of , for him, but a considerable challenge.
'separated it.'
Xiao Yan far behind, his face dark soul of evil
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