Discrete Laplace operator

9869: 7521: 7053: 6477:-dimensions. In other words, the discrete Laplacian filter of any size can be generated conveniently as the sampled Laplacian of Gaussian with spatial size befitting the needs of a particular application as controlled by its variance. Monomials which are non-linear operators can also be implemented using a similar reconstruction and approximation approach provided that the signal is sufficiently over-sampled. Thereby, such non-linear operators e.g. 32: 8465: 4135: 7516:{\displaystyle {\begin{aligned}{\frac {d\phi _{i}}{dt}}&=-k\sum _{j}A_{ij}\left(\phi _{i}-\phi _{j}\right)\\&=-k\left(\phi _{i}\sum _{j}A_{ij}-\sum _{j}A_{ij}\phi _{j}\right)\\&=-k\left(\phi _{i}\ \deg(v_{i})-\sum _{j}A_{ij}\phi _{j}\right)\\&=-k\sum _{j}\left(\delta _{ij}\ \deg(v_{i})-A_{ij}\right)\phi _{j}\\&=-k\sum _{j}\left(L_{ij}\right)\phi _{j}.\end{aligned}}} 5078: 7963: 1767: 3866: 5465: 4326: 3675: 9251: 9859:

converges to the same value at each of the vertices of the graph, which is the average of the initial values at all of the vertices. Since this is the solution to the heat diffusion equation, this makes perfect sense intuitively. We expect that neighboring elements in the graph will exchange energy

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for γ ∈ is compatible with discrete scale-space properties, where specifically the value γ = 1/3 gives the best approximation of rotational symmetry. Regarding three-dimensional signals, it is shown that the Laplacian operator can be approximated by the two-parameter family of difference operators

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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

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This GIF shows the progression of diffusion, as solved by the graph laplacian technique. A graph is constructed over a grid, where each pixel in the graph is connected to its 8 bordering pixels. Values in the image then diffuse smoothly to their neighbors over time via these connections. This

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and calculated as sum of differences over the nearest neighbours of the central pixel. Since derivative filters are often sensitive to noise in an image, the Laplace operator is often preceded by a smoothing filter (such as a Gaussian filter) in order to remove the noise before calculating the

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The complete Matlab source code that was used to generate this animation is provided below. It shows the process of specifying initial conditions, projecting these initial conditions onto the eigenvalues of the Laplacian Matrix, and simulating the exponential decay of these projected initial

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diffusing over time through a graph. The graph in this example is constructed on a 2D discrete grid, with points on the grid connected to their eight neighbors. Three initial points are specified to have a positive value, while the rest of the values in the grid are zero. Over time, the

4597: 8460:{\displaystyle {\begin{aligned}0={}&{\frac {d\left(\sum _{i}c_{i}(t)\mathbf {v} _{i}\right)}{dt}}+kL\left(\sum _{i}c_{i}(t)\mathbf {v} _{i}\right)\\{}={}&\sum _{i}\left\\{}={}&\sum _{i}\left\\\Rightarrow 0={}&{\frac {dc_{i}(t)}{dt}}+k\lambda _{i}c_{i}(t),\\\end{aligned}}} 5601:. In particular any discrete image, with reasonable presumptions on the discretization process, e.g. assuming band limited functions, or wavelets expandable functions, etc. can be reconstructed by means of well-behaving interpolation functions underlying the reconstruction formulation, 1560: 2785: 4130:{\displaystyle \nabla _{\gamma }^{2}=(1-\gamma )\nabla _{5}^{2}+\gamma \nabla _{\times }^{2}=(1-\gamma ){\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}+\gamma {\begin{bmatrix}1/2&0&1/2\\0&-2&0\\1/2&0&1/2\end{bmatrix}}} 4591: 11233: 1200: 6799: 11542:

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.

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domain but also an effective range in the frequency domain (alternatively Gaussian scale space) which can be controlled explicitly via the variance of the Gaussian in a principled manner. The resulting filtering can be implemented by separable filters and

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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.

9113: 9833: 2673:) finite-difference formula seen previously. It is stable for very smoothly varying fields, but for equations with rapidly varying solutions a more stable and isotropic form of the Laplacian operator is required, such as the 807: 2518: 3740:

D 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:

8884: 5073:{\displaystyle (\nabla _{+^{3}}^{2}f)_{0,0,0}={\frac {1}{4}}(f_{-1,-1,0}+f_{-1,+1,0}+f_{+1,-1,0}+f_{+1,+1,0}+f_{-1,0,-1}+f_{-1,0,+1}+f_{+1,0,-1}+f_{+1,0,+1}+f_{0,-1,-1}+f_{0,-1,+1}+f_{0,+1,-1}+f_{0,+1,+1}-12f_{0,0,0}),} 991: 1762:{\displaystyle C_{ij}={\begin{cases}{\frac {1}{2}}(\cot \alpha _{ij}+\cot \beta _{ij})&ij{\text{ is an edge, that is }}j\in N(i),\\-\sum \limits _{k\in N(i)}C_{ik}&i=j,\\0&{\text{otherwise}}\end{cases}}} 5697: 5911: 6131: 6616:). This may also be seen by applying the Fourier transform. Note that the discrete Laplacian on an infinite grid has purely absolutely continuous spectrum, and therefore, no eigenvalues or eigenfunctions. 3516: 11472: 7844: 2684: 498: 2331: 7920: 9655: 8558: 6352:. An advantage of using Gaussians as interpolation functions is that they yield linear operators, including Laplacians, that are free from rotational artifacts of the coordinate frame in which 1001:

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 a two-dimensional

6042: 7968: 7537: 7058: 9072: 1484: 7689: 6628:, 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 4337: 2423:

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

11138: 3440: 11319: 1055: 11035: 8988: 6634: 5460:{\displaystyle (\nabla _{\times ^{3}}^{2}f)_{0,0,0}={\frac {1}{4}}(f_{-1,-1,-1}+f_{-1,-1,+1}+f_{-1,+1,-1}+f_{-1,+1,+1}+f_{+1,-1,-1}+f_{+1,-1,+1}+f_{+1,+1,-1}+f_{+1,+1,+1}-8f_{0,0,0}).} 3748: 8594:

are non-negative, showing that the solution to the diffusion equation approaches an equilibrium, because it only exponentially decays or remains constant. This also shows that given

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particular image starts off with three strong point values which spill over to their neighbors slowly. The whole system eventually settles out to the same value at equilibrium.

331: 3185: 1529: 629: 6985: 3278: 3095: 9327: 9356: 8805: 7955: 7783: 6124: 5511: 4321:{\displaystyle \nabla _{\gamma _{1},\gamma _{2}}^{2}=(1-\gamma _{1}-\gamma _{2})\,\nabla _{7}^{2}+\gamma _{1}\,\nabla _{+^{3}}^{2}+\gamma _{2}\,\nabla _{\times ^{3}}^{2}),} 7532: 1273: 840: 591: 11263: 11070: 9105: 6527: 1303: 6446: 5733: 8619: 8588: 1992: 1950: 11384: 9428:

eigenvectors of ones and zeros, where each connected component corresponds to an eigenvector with ones at the elements in the connected component and zeros elsewhere.

7727: 5800: 9426: 3670:{\displaystyle a_{x_{1},x_{2},\dots ,x_{n}}=\left\{{\begin{array}{ll}-2n&{\text{if }}s=n,\\1&{\text{if }}s=n-1,\\0&{\text{otherwise,}}\end{array}}\right.} 869: 359: 259: 11364: 9685: 6881: 5579: 536: 10965: 8776: 8747: 8698: 8655: 6397: 1911: 1353: 568: 379: 6062: 5931: 1798: 2360:) on the boundary of the lattice grid, thus this is the case of no source at the boundary, that is, a no-flux boundary condition (aka, insulation, or homogeneous 9895: 9857: 9705: 9246:{\displaystyle \lim _{t\to \infty }e^{-k\lambda _{i}t}={\begin{cases}0,&{\text{if}}&\lambda _{i}>0\\1,&{\text{if}}&\lambda _{i}=0\end{cases}}} 7754: 6850: 1326: 9458: 6547: 6475: 6417: 6370: 6350: 6086: 5955: 5820: 5773: 5753: 5599: 2920: 1884: 1864: 1841: 1818: 1549: 1397: 1377: 1243: 1223: 1047: 1027: 299: 279: 6614: 6579: 3003: 2840: 8937:

is simply an orthogonal coordinate transformation of the initial condition to a set of coordinates which decay exponentially and independently of each other.

9725: 9478: 9400: 9376: 9278: 8935: 8911: 8718: 7045: 7025: 7005: 6945: 6925: 6901: 2831: 9733: 3086: 676: 148:. 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 212:, differing by sign and scale factor (sometimes one averages over the neighboring vertices, other times one just sums; this makes no difference for a 2447: 3860:

It can be shown that the following discrete approximation of the two-dimensional Laplacian operator as a convex combination of difference operators

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Reuter, M.; Biasotti, S.; Giorgi, D.; Patane, G.; Spagnuolo, M. (2009). "Discrete Laplace-Beltrami operators for shape analysis and segmentation".

8813: 884: 6322:{\displaystyle \nabla ^{2}f({\bar {r}}_{k})=\sum _{k'\in K}f_{k'}(\nabla ^{2}\mu ({\bar {r}}-{\bar {r}}_{k'}))|_{{\bar {r}}={\bar {r}}_{k}}} 5607: 5825: 96: 49: 68: 12189: 2780:{\displaystyle \mathbf {D} _{xy}^{2}={\begin{bmatrix}0.25&0.5&0.25\\0.5&-3&0.5\\0.25&0.5&0.25\end{bmatrix}}} 12156: 3445: 11396: 5822:. On a uniform grid, such as images, and for bandlimited functions, interpolation functions are shift invariant amounting to 387: 75: 7792: 2199: 12137: 12104: 11956: 11819: 9589: 7849: 11618: 8476: 2395: 82: 6485:

which are used in pattern recognition for their total least-square optimality in orientation estimation, can be realized.

5960: 4586:{\displaystyle (\nabla _{7}^{2}f)_{0,0,0}=f_{-1,0,0}+f_{+1,0,0}+f_{0,-1,0}+f_{0,+1,0}+f_{0,0,-1}+f_{0,0,+1}-6f_{0,0,0},} 11984: 9379: 1428: 11644: 8993: 2391: 64: 11883: 11228:{\displaystyle G(v,w;\lambda )=\left\langle \delta _{v}\left|{\frac {1}{H-\lambda }}\right|\delta _{w}\right\rangle } 7639: 115: 11896:

Patra, Michael; Karttunen, Mikko (2006). "Stencils with isotropic discretization error for differential operators".

1195:{\displaystyle (\Delta u)_{i}\equiv {\frac {1}{2A_{i}}}\sum _{j}(\cot \alpha _{ij}+\cot \beta _{ij})(u_{j}-u_{i}),} 6794:{\displaystyle {\frac {\partial ^{2}F}{\partial x^{2}}}=\lim _{\epsilon \rightarrow 0}{\frac {-}{\epsilon ^{2}}}.} 3379: 11859: 11272: 6088:-dimensions. Accordingly, the discrete Laplacian becomes a discrete version of the Laplacian of the continuous 3847:{\displaystyle \mathbf {D} _{xy}^{2}={\begin{bmatrix}1&1&1\\1&-8&1\\1&1&1\end{bmatrix}}.} 2622:{\displaystyle \mathbf {D} _{xy}^{2}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}} 10972: 6450: 53: 8948: 11798: 9573:{\displaystyle \lim _{t\to \infty }\phi (t)=\left\langle c(0),\mathbf {v^{1}} \right\rangle \mathbf {v^{1}} } 12132:. Proceedings of Symposia in Pure Mathematics. Vol. 77. American Mathematical Society. pp. 51–86. 10910: 6332:

which in turn is a convolution with the Laplacian of the interpolation function on the uniform (image) grid

2793: 12204: 11560: 7733: 6853: 6482: 6419:-dimensions, and are frequency aware by definition. A linear operator has not only a limited range in the 5475:

A discrete signal, comprising images, can be viewed as a discrete representation of a continuous function

2635: 2377: 209: 141: 3268:{\displaystyle {\frac {1}{26}}{\begin{bmatrix}3&6&3\\6&-88&6\\3&6&3\end{bmatrix}}} 2424: 641: 12194: 11502: 11107: 5516: 11546:

The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:

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until that energy is spread out evenly throughout all of the elements that are connected to each other.

3358:{\displaystyle {\frac {1}{26}}{\begin{bmatrix}2&3&2\\3&6&3\\2&3&2\end{bmatrix}}} 3175:{\displaystyle {\frac {1}{26}}{\begin{bmatrix}2&3&2\\3&6&3\\2&3&2\end{bmatrix}}} 304: 6904: 6454: 2361: 1489: 1416: 596: 89: 7732:

To find a solution to this differential equation, apply standard techniques for solving a first-order

546:. For a graph with a finite number of edges and vertices, this definition is identical to that of the 12043:"Discrete Green's functions and spectral graph theory for computationally efficient thermal modeling" 7623:{\displaystyle {\begin{aligned}{\frac {d\phi }{dt}}&=-k(D-A)\phi \\&=-kL\phi ,\end{aligned}}} 6950: 1400: 1006: 11083:

If the number of edges meeting at a vertex is uniformly bounded, and the potential is bounded, then

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exponential decay acts to distribute the values at these points evenly throughout the entire grid.

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The resulting numbering is unique (scale is specified by the "2"), and consists of integers; for E

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are connected (if they are not connected, no heat is transferred). Then, for thermal conductivity

2171:{\displaystyle \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}}},} 1248: 815: 573: 12199: 11608: 11241: 11095: 11043: 9257: 6500: 1278: 539: 42: 11125: 11077: 9077: 6422: 5705: 11744: 11707: 1916: 334: 11369: 8890:

This approach has been applied to quantitative heat transfer modelling on unstructured grids.

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The spectrum of the discrete Laplacian on an infinite grid is of key interest; since it is a

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derivative. The smoothing filter and Laplace filter are often combined into a single filter.

1972: 1408: 845: 344: 226: 12094: 11860:

Lindeberg, T., "Scale-space for discrete signals", PAMI(12), No. 3, March 1990, pp. 234–254.

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Certain equations involving the discrete Laplacian only have solutions on the simply-laced

10950: 9663: 6859: 6375: 1889: 1412: 1331: 553: 364: 164: 8752: 8723: 8667: 8624: 6047: 5916: 2993:{\displaystyle {\begin{bmatrix}0&1&0\\1&-6&1\\0&1&0\end{bmatrix}}} 1774: 1407:. The above cotangent formula can be derived using many different methods among which are 8: 11773: 11129: 11121: 3073:{\displaystyle {\begin{bmatrix}0&0&0\\0&1&0\\0&0&0\end{bmatrix}}} 2910:{\displaystyle {\begin{bmatrix}0&0&0\\0&1&0\\0&0&0\end{bmatrix}}} 1420: 12160: 12058: 12017: 1308: 12072: 12026: 12001: 11921: 11613: 11493: 11483: 9880: 9842: 9690: 7739: 6835: 6805: 6532: 6460: 6402: 6355: 6335: 6071: 5940: 5805: 5758: 5738: 5584: 2674: 2670: 1980: 1869: 1849: 1826: 1803: 1534: 1382: 1362: 1228: 1208: 1032: 1012: 338: 284: 264: 172: 11876:

Lindeberg, T., Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, 1994

9828:{\displaystyle \lim _{t\to \infty }\phi _{j}(t)={\frac {1}{N}}\sum _{i=1}^{N}c_{i}(0)} 9434: 6584: 6552: 216:). The traditional definition of the graph Laplacian, given below, corresponds to the 12133: 12100: 12076: 11980: 11952: 11925: 11913: 11879: 11815: 11623: 6065: 2436:

For one-, two- and three-dimensional signals, the discrete Laplacian can be given as

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between vertices w and v. Thus, this sum is over the nearest neighbors of the vertex

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In other words, the equilibrium state of the system is determined completely by the

5581:. Derivation operation is therefore directly applicable to the continuous function, 12119: 12090: 12062: 12021: 12002:"Computational heat transfer with spectral graph theory: Quantitative verification" 11944: 11905: 11807: 11754: 11717: 11680: 9710: 9463: 9385: 9361: 9263: 8920: 8917:, its eigenvectors are all orthogonal. So the projection onto the eigenvectors of 8914: 8896: 8703: 7030: 7010: 6990: 6930: 6910: 6886: 6809: 6478: 1404: 547: 184: 176: 168: 149: 137: 2384: 11837:"A Family of Large-Stencil Discrete Laplacian Approximations in Three Dimensions" 11554:

In terms of the Laplacian, the positive solutions to the inhomogeneous equation:

11266: 802:{\displaystyle (\Delta _{\gamma }\phi )(v)=\sum _{w:\,d(w,v)=1}\gamma _{wv}\left} 145: 11489: 11113:

On regular lattices, the operator typically has both traveling-wave as well as

10279:% BELOW IS THE KEY CODE THAT COMPUTES THE SOLUTION TO THE DIFFERENTIAL EQUATION 6821: 6813: 6625: 2513:{\displaystyle {\vec {D}}_{x}^{2}={\begin{bmatrix}1&-2&1\end{bmatrix}}} 2345: 180: 175:

as a stand-in for the continuous Laplace operator. Common applications include

11758: 11721: 3583: 2380:), is rarely used for graph Laplacians, but is common in other applications. 1979:. For example, the Laplacian in two dimensions can be approximated using the 12173: 12123: 11917: 11672: 8879:{\displaystyle c_{i}(0)=\left\langle \phi (0),\mathbf {v} _{i}\right\rangle } 7695: 5934: 213: 11684: 10636:% Transform from eigenvector coordinate system to original coordinate system 10333:% Compute the laplacian matrix in terms of the degree and adjacency matrices 986:{\displaystyle (M\phi )(v)={\frac {1}{\deg v}}\sum _{w:\,d(w,v)=1}\phi (w).} 11088: 2403: 2387: 11811: 11972: 11496:. Specifically, the only positive solutions to the homogeneous equation: 11099: 6817: 2437: 160: 129: 20: 11875: 11806:. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA. p. 219. 11550:

Twice any label minus two is the sum of the labels on adjacent vertices.

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in the graph, then this vector of all ones can be split into the sum of

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can be considered to be a multiplicative operator acting diagonally on

2399: 1379:; that is, e.g. one third of the summed areas of triangles incident to 12127: 11909: 11386:

a complex number, the Green's function considered to be a function of

12099:, Elements of Mathematics, translated by Pressley, Andrew, Springer, 11117:

solutions, depending on whether the potential is periodic or random.

10940: 5692:{\displaystyle f({\bar {r}})=\sum _{k\in K}f_{k}\mu _{k}({\bar {r}})} 1964: 11948: 11599:

they range from 58 to 270, and have been observed as early as 1968.

5906:{\displaystyle \mu _{k}({\bar {r}})=\mu ({\bar {r}}-{\bar {r}}_{k})} 3857:

These kernels are deduced by using discrete differential quotients.

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is the product of the column vector and the Laplacian matrix, while

31: 1002: 12040: 10372:% Initial condition (place a few large positive values around and 156: 11538:"Twice any label is the sum of the labels on adjacent vertices," 1425:

To facilitate computation, the Laplacian is encoded in a matrix

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in both dimensions, so that the five-point stencil of a point (

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If the graph has weighted edges, that is, a weighting function

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The consequence of this is that for a given initial condition

2348:. There are no constraints here on the values of the function 11673:"Digital geometry processing with discrete exterior calculus" 11103: 2431: 10549:% Transform the initial condition into the coordinate system 3511:{\displaystyle \mathbf {D} _{x_{1},x_{2},\dots ,x_{n}}^{2},} 11670: 11467:{\displaystyle (H-\lambda )G(v,w;\lambda )=\delta _{w}(v).} 9239: 7839:{\textstyle L\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}} 3664: 1755: 493:{\displaystyle (\Delta \phi )(v)=\sum _{w:\,d(w,v)=1}\left} 11844:

International Journal for Numerical Methods in Engineering

11697: 11671:

Crane, K.; de Goes, F.; Desbrun, M.; Schröder, P. (2013).

2364:). The control of the state variable at the boundary, as 2326:{\displaystyle \{(x-h,y),(x,y),(x+h,y),(x,y-h),(x,y+h)\}.} 19:

For the discrete equivalent of the Laplace transform, see

10351:% Compute the eigenvalues/vectors of the laplacian matrix 11800:

Phase-Field Methods in Materials Science and Engineering

7915:{\textstyle \phi (t)=\sum _{i}c_{i}(t)\mathbf {v} _{i}.} 6812:. In image processing, it is considered to be a type of 6457:

representations for further computational efficiency in

1399:. It is important to note that the sign of the discrete 11734: 9650:{\displaystyle \mathbf {v^{1}} ={\frac {1}{\sqrt {N}}}} 1955:

A more general overview of mesh operators is given in.

220:

continuous Laplacian on a domain with a free boundary.

11492:(all edges multiplicity 1), and are an example of the 9883: 9845: 9713: 9693: 9666: 9466: 9437: 9408: 9388: 9364: 9335: 9289: 9266: 9080: 8996: 8951: 8923: 8899: 8784: 8755: 8726: 8706: 8670: 8627: 8600: 8569: 7934: 7852: 7795: 7762: 7742: 7708: 7033: 7013: 6993: 6953: 6933: 6913: 6889: 6862: 6838: 4034: 3962: 3780: 3297: 3204: 3114: 3012: 2929: 2849: 2716: 2558: 2483: 11735:

Forsyth, D. A.; Ponce, J. (2003). "Computer Vision".

11563: 11505: 11399: 11372: 11346: 11275: 11244: 11141: 11046: 10975: 10953: 10913: 10579:% Loop through times and decay each initial component 9919:% The number of pixels along a dimension of the image 9736: 9592: 9489: 9116: 8893:

In the case of undirected graphs, this works because

8816: 8553:{\displaystyle c_{i}(t)=c_{i}(0)e^{-k\lambda _{i}t}.} 8479: 7966: 7694:

Notice that this equation takes the same form as the

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This definition of the Laplacian is commonly used in

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Numerical Methods for Partial Differential Equations

9863: 6037:{\displaystyle {\bar {r}}=(x_{1},x_{2}...x_{n})^{T}} 1403:

is conventionally opposite the sign of the ordinary

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is given, then the definition can be generalized to

9988:% Use 8 neighbors, and fill in the adjacency matrix 56:. Unsourced material may be challenged and removed. 11889: 11584: 11523: 11466: 11378: 11358: 11313: 11257: 11227: 11064: 11029: 10959: 10931: 9889: 9851: 9827: 9719: 9699: 9679: 9649: 9572: 9472: 9452: 9420: 9394: 9370: 9350: 9321: 9272: 9245: 9099: 9066: 8982: 8929: 8905: 8878: 8799: 8770: 8741: 8712: 8692: 8649: 8613: 8582: 8552: 8459: 7949: 7928:is a symmetric matrix, its unit-norm eigenvectors 7914: 7838: 7777: 7748: 7721: 7683: 7622: 7515: 7039: 7019: 6999: 6979: 6939: 6919: 6895: 6875: 6844: 6793: 6608: 6581:(as the averaging operator has spectral values in 6573: 6541: 6521: 6469: 6440: 6411: 6391: 6364: 6344: 6321: 6118: 6080: 6056: 6036: 5949: 5925: 5905: 5814: 5794: 5767: 5747: 5727: 5691: 5593: 5573: 5547: 5505: 5459: 5072: 4585: 4320: 4129: 3846: 3669: 3510: 3434: 3357: 3267: 3174: 3072: 2992: 2909: 2822: 2779: 2661: 2621: 2512: 2325: 2170: 1944: 1905: 1878: 1858: 1835: 1812: 1792: 1761: 1543: 1523: 1478: 1391: 1371: 1347: 1320: 1297: 1267: 1237: 1217: 1194: 1041: 1021: 985: 863: 834: 801: 662: 623: 585: 562: 530: 492: 373: 353: 325: 293: 273: 253: 9067:{\textstyle c_{i}(t)=c_{i}(0)e^{-k\lambda _{i}t}} 5802:are interpolation functions specific to the grid 1479:{\displaystyle L\in \mathbb {R} ^{|V|\times |V|}} 874:Closely related to the discrete Laplacian is the 12171: 11834: 11790: 10902: 9738: 9491: 9118: 8953: 6676: 1886:-th entry along the diagonal is the vertex area 12047:International Journal of Heat and Mass Transfer 11102:; this is a consequence of the duality between 7924:Plugging into the original expression (because 7684:{\displaystyle {\frac {d\phi }{dt}}+kL\phi =0.} 1952:is the sought discretization of the Laplacian. 12041:Cole, K. D.; Riensche, A.; Rao, P. K. (2022). 11895: 9839:In other words, at steady state, the value of 6852:describes a temperature distribution across a 2344:discrete Laplacian on the graph, which is the 11796: 11679:. SIGGRAPH '13. Vol. 7. pp. 1–126. 6497:, it has a real spectrum. For the convention 2390:have very special properties, e.g., they are 12000:Yavari, R.; Cole, K. D.; Rao, P. K. (2020). 11797:Provatas, Nikolas; Elder, Ken (2010-10-13). 9877:This section shows an example of a function 6064:on uniform grids, are appropriately dilated 5471:Implementation via continuous reconstruction 3435:{\displaystyle a_{x_{1},x_{2},\dots ,x_{n}}} 2394:of one-dimensional discrete Laplacians, see 2317: 2203: 1205:where the sum is over all adjacent vertices 11314:{\displaystyle \delta _{w}(v)=\delta _{wv}} 10669:% Display the results and write to GIF file 9358:of all ones is in the kernel. If there are 11993: 11828: 10943:function defined on the graph. Note that 8720:in terms of the overall initial condition 2432:Implementation via operator discretization 2376:) given on the boundary of the grid (aka, 570:can be written as a column vector; and so 12066: 12025: 12006:International Journal of Thermal Sciences 11938: 11932: 11748: 11711: 11642: 11030:{\displaystyle (P\phi )(v)=P(v)\phi (v).} 8983:{\textstyle \lim _{t\to \infty }\phi (t)} 6827: 4289: 4253: 4224: 1440: 941: 722: 426: 116:Learn how and when to remove this message 16:Analog of the continuous Laplace operator 12154: 12096:Groupes et algebres de Lie: Chapters 4–6 12089: 12083: 12068:10.1016/j.ijheatmasstransfer.2021.122112 12034: 11728: 11098:of this Hamiltonian can be studied with 9867: 7756:as a linear combination of eigenvectors 2409: 2383:Multidimensional discrete Laplacians on 1305:are the two angles opposite of the edge 155:The discrete Laplace operator occurs in 12129:Analysis on Graphs and Its Applications 11871: 11869: 11867: 11855: 11853: 11835:O'Reilly, H.; Beck, Jeffrey M. (2006). 11771: 11691: 8940: 2425:Laplace operator#Coordinate expressions 12172: 12118: 11971: 11664: 10932:{\displaystyle P\colon V\rightarrow R} 10615:% Exponential decay for each component 3704:) of the element in the kernel in the 2823:{\displaystyle \mathbf {D} _{xyz}^{2}} 167:, as well as in the study of discrete 140:, defined so that it has meaning on a 11585:{\displaystyle \Delta \phi =\phi -2.} 11477: 9727:in the graph, it can be rewritten as 2662:{\displaystyle \mathbf {D} _{xy}^{2}} 1958: 204:There are various definitions of the 11864: 11850: 11619:Kronecker sum of discrete Laplacians 7846:) with time-dependent coefficients, 7702:is replacing the Laplacian operator 2396:Kronecker sum of discrete Laplacians 663:{\displaystyle \gamma \colon E\to R} 54:adding citations to reliable sources 25: 11524:{\displaystyle \Delta \phi =\phi ,} 5548:{\displaystyle {\bar {r}}\in R^{n}} 2418: 1687: 337:of the vertices taking values in a 199: 13: 12027:10.1016/j.ijthermalsci.2020.106383 11564: 11506: 11053: 9748: 9501: 9128: 8963: 7710: 6656: 6642: 6504: 6216: 6136: 5092: 4605: 4345: 4291: 4255: 4226: 4152: 3925: 3904: 3871: 1996: 1505: 1062: 996: 684: 603: 577: 394: 348: 326:{\displaystyle \phi \colon V\to R} 14: 12216: 12148: 11643:Leventhal, Daniel (Autumn 2011). 9864:Example of the operator on a grid 8563:As shown before, the eigenvalues 6907:, the heat transferred from node 1524:{\displaystyle Lu=(\Delta u)_{i}} 624:{\displaystyle (\Delta \phi )(v)} 12190:Numerical differential equations 9599: 9595: 9564: 9560: 9547: 9543: 9338: 8861: 8787: 8778:onto the unit-norm eigenvectors 8347: 8300: 8217: 8177: 8094: 8023: 7937: 7899: 7826: 7801: 7765: 7729:; hence, the "graph Laplacian". 6980:{\textstyle \phi _{i}-\phi _{j}} 5735:are discrete representations of 3754: 3451: 2799: 2690: 2677:, which includes the diagonals: 2641: 2532: 1409:piecewise linear finite elements 842:is the weight value on the edge 635:th entry of the product vector. 30: 11965: 5933:being an appropriately dilated 1655: is an edge, that is 341:. Then, the discrete Laplacian 136:is an analog of the continuous 41:needs additional citations for 11765: 11636: 11458: 11452: 11436: 11418: 11412: 11400: 11292: 11286: 11163: 11145: 11076:, an analog of the continuous 11021: 11015: 11009: 11003: 10994: 10988: 10985: 10976: 10923: 9822: 9816: 9769: 9763: 9745: 9644: 9620: 9535: 9529: 9515: 9509: 9498: 9447: 9441: 9322:{\textstyle \sum _{j}L_{ij}=0} 9125: 9035: 9029: 9013: 9007: 8977: 8971: 8960: 8853: 8847: 8833: 8827: 8765: 8759: 8736: 8730: 8687: 8681: 8644: 8638: 8518: 8512: 8496: 8490: 8447: 8441: 8401: 8395: 8366: 8332: 8326: 8284: 8278: 8209: 8203: 8161: 8155: 8089: 8083: 8018: 8012: 7894: 7888: 7862: 7856: 7585: 7573: 7415: 7402: 7305: 7292: 6772: 6769: 6757: 6748: 6742: 6736: 6730: 6727: 6721: 6712: 6700: 6694: 6683: 6603: 6588: 6568: 6556: 6451:decimation (signal processing) 6432: 6305: 6289: 6278: 6273: 6270: 6253: 6237: 6228: 6212: 6170: 6158: 6148: 6113: 6107: 6098: 6025: 5979: 5970: 5900: 5888: 5872: 5863: 5854: 5848: 5839: 5686: 5680: 5671: 5629: 5623: 5614: 5526: 5513:, where the coordinate vector 5500: 5494: 5485: 5451: 5151: 5117: 5088: 5064: 4664: 4630: 4601: 4363: 4341: 4312: 4221: 4189: 3954: 3942: 3900: 3888: 2458: 2406:can be explicitly calculated. 2314: 2296: 2290: 2272: 2266: 2248: 2242: 2230: 2224: 2206: 2149: 2137: 2125: 2107: 2098: 2080: 2071: 2053: 2044: 2026: 2014: 2002: 1787: 1781: 1706: 1700: 1673: 1667: 1642: 1598: 1512: 1502: 1470: 1462: 1454: 1446: 1186: 1160: 1157: 1113: 1069: 1059: 977: 971: 957: 945: 906: 900: 897: 888: 791: 785: 776: 770: 738: 726: 705: 699: 696: 680: 654: 618: 612: 609: 600: 525: 513: 482: 476: 467: 461: 442: 430: 409: 403: 400: 391: 317: 248: 236: 194: 183:, and in machine learning for 1: 12124:"Discrete geometric analysis" 11772:Matthys, Don (Feb 14, 2001). 11629: 11074:discrete Schrödinger operator 10903:Discrete Schrödinger operator 9351:{\textstyle \mathbf {v} ^{1}} 8800:{\textstyle \mathbf {v} _{i}} 7950:{\textstyle \mathbf {v} _{i}} 7778:{\textstyle \mathbf {v} _{i}} 6883:is the temperature at vertex 6119:{\displaystyle f({\bar {r}})} 5555:and the value domain is real 5506:{\displaystyle f({\bar {r}})} 10375:% make everything else zero) 9074:that remain are those where 7734:matrix differential equation 6624:If the graph is an infinite 6483:Generalized Structure Tensor 3714:is the number of directions 2440:with the following kernels: 2378:Dirichlet boundary condition 1800:denotes the neighborhood of 1268:{\displaystyle \alpha _{ij}} 835:{\displaystyle \gamma _{wv}} 586:{\displaystyle \Delta \phi } 7: 11979:. Oxford University Press. 11602: 11258:{\displaystyle \delta _{w}} 11065:{\displaystyle H=\Delta +P} 10702:'Diffusion t = %3f' 10312:% Compute the degree matrix 9100:{\textstyle \lambda _{i}=0} 8657:, the solution at any time 7526:In matrix-vector notation, 6619: 6549:, the spectrum lies within 6522:{\displaystyle \Delta =I-M} 6488: 1298:{\displaystyle \beta _{ij}} 179:, where it is known as the 65:"Discrete Laplace operator" 10: 12221: 11481: 11390:is the unique solution to 8621:and the initial condition 6455:pyramid (image processing) 6441:{\displaystyle {\bar {r}}} 6044:. Other approximations of 5728:{\displaystyle f_{k}\in R} 2362:Neumann boundary condition 1417:discrete exterior calculus 18: 12157:"Spectral gap of a graph" 11977:Networks: An Introduction 11759:10.1016/j.cag.2009.03.005 11722:10.1016/j.cag.2009.03.005 11677:ACM SIGGRAPH 2013 Courses 8913:is symmetric, and by the 8614:{\textstyle \lambda _{i}} 8583:{\textstyle \lambda _{i}} 1945:{\displaystyle L=M^{-1}C} 1401:Laplace-Beltrami operator 1007:Laplace-Beltrami operator 261:be a graph with vertices 134:discrete Laplace operator 11737:Computers & Graphics 11700:Computers & Graphics 11652:University of Washington 11379:{\displaystyle \lambda } 11265:is understood to be the 9904: 7722:{\textstyle \nabla ^{2}} 5795:{\displaystyle \mu _{k}} 3692:is the position (either 2398:, in which case all its 1984:finite-difference method 1969:finite-difference method 191:on neighborhood graphs. 189:semi-supervised learning 12155:Ollivier, Yann (2004). 11685:10.1145/2504435.2504442 11609:Spectral shape analysis 11269:function on the graph: 9707:, i.e. for each vertex 9421:{\textstyle \lambda =0} 6905:Newton's law of cooling 6816:, more specifically an 2340:= 1, the result is the 2181:where the grid size is 1049:can be approximated as 864:{\displaystyle wv\in E} 354:{\displaystyle \Delta } 254:{\displaystyle G=(V,E)} 11586: 11525: 11468: 11380: 11360: 11359:{\displaystyle w\in V} 11315: 11259: 11229: 11066: 11031: 10961: 10933: 9985:% The adjacency matrix 9891: 9874: 9853: 9829: 9805: 9721: 9701: 9681: 9680:{\textstyle \phi _{j}} 9651: 9574: 9474: 9454: 9422: 9396: 9372: 9352: 9323: 9274: 9247: 9101: 9068: 8984: 8931: 8907: 8880: 8801: 8772: 8743: 8714: 8694: 8651: 8615: 8584: 8554: 8461: 7951: 7916: 7840: 7779: 7750: 7723: 7685: 7624: 7517: 7041: 7021: 7001: 6981: 6941: 6921: 6897: 6877: 6876:{\textstyle \phi _{i}} 6846: 6828:Discrete heat equation 6795: 6610: 6575: 6543: 6523: 6471: 6442: 6413: 6393: 6366: 6346: 6323: 6120: 6082: 6058: 6038: 5951: 5927: 5907: 5816: 5796: 5769: 5749: 5729: 5693: 5595: 5575: 5574:{\displaystyle f\in R} 5549: 5507: 5461: 5074: 4587: 4322: 4131: 3848: 3671: 3512: 3436: 3359: 3269: 3176: 3074: 2994: 2911: 2824: 2781: 2663: 2623: 2514: 2327: 2172: 1963:Approximations of the 1946: 1907: 1880: 1860: 1837: 1814: 1794: 1763: 1545: 1525: 1480: 1393: 1373: 1349: 1322: 1299: 1269: 1239: 1219: 1196: 1043: 1023: 987: 865: 836: 803: 664: 625: 587: 564: 532: 531:{\displaystyle d(w,v)} 494: 375: 355: 327: 295: 275: 255: 11941:Vision with Direction 11812:10.1002/9783527631520 11587: 11526: 11482:Further information: 11469: 11381: 11361: 11321:; that is, it equals 11316: 11260: 11230: 11115:Anderson localization 11067: 11032: 10962: 10960:{\displaystyle \phi } 10934: 10552:% of the eigenvectors 9892: 9871: 9854: 9830: 9785: 9722: 9702: 9682: 9652: 9575: 9475: 9455: 9423: 9397: 9373: 9353: 9324: 9283:Since by definition, 9275: 9248: 9102: 9069: 8985: 8932: 8908: 8881: 8802: 8773: 8771:{\textstyle \phi (0)} 8744: 8742:{\textstyle \phi (0)} 8715: 8695: 8693:{\textstyle c_{i}(0)} 8652: 8650:{\textstyle c_{i}(0)} 8616: 8585: 8555: 8462: 7952: 7917: 7841: 7780: 7751: 7724: 7686: 7625: 7518: 7042: 7022: 7002: 6982: 6942: 6922: 6898: 6878: 6847: 6796: 6611: 6576: 6544: 6524: 6495:self-adjoint operator 6472: 6443: 6414: 6394: 6392:{\displaystyle f_{k}} 6367: 6347: 6324: 6121: 6083: 6059: 6039: 5952: 5928: 5908: 5817: 5797: 5770: 5750: 5730: 5694: 5596: 5576: 5550: 5508: 5462: 5075: 4588: 4323: 4132: 3849: 3672: 3513: 3437: 3360: 3270: 3177: 3075: 2995: 2912: 2825: 2782: 2664: 2624: 2515: 2410:Finite-element method 2328: 2173: 1975:, can also be called 1973:finite-element method 1947: 1908: 1906:{\displaystyle A_{i}} 1881: 1861: 1838: 1815: 1795: 1764: 1546: 1526: 1481: 1394: 1374: 1350: 1348:{\displaystyle A_{i}} 1323: 1300: 1270: 1240: 1220: 1197: 1044: 1024: 1009:of a scalar function 988: 866: 837: 804: 665: 626: 588: 565: 563:{\displaystyle \phi } 533: 495: 376: 374:{\displaystyle \phi } 356: 328: 296: 276: 256: 171:. It is also used in 159:problems such as the 11778:Marquette University 11561: 11503: 11397: 11370: 11344: 11273: 11242: 11139: 11126:Schrödinger operator 11078:Schrödinger operator 11044: 10973: 10951: 10911: 9881: 9843: 9734: 9711: 9691: 9664: 9590: 9487: 9464: 9435: 9406: 9386: 9380:connected components 9362: 9333: 9287: 9264: 9114: 9078: 8994: 8949: 8941:Equilibrium behavior 8921: 8897: 8814: 8782: 8753: 8724: 8704: 8668: 8625: 8598: 8567: 8477: 7964: 7932: 7850: 7793: 7760: 7740: 7706: 7698:, where the matrix − 7640: 7533: 7054: 7031: 7011: 6991: 6951: 6931: 6911: 6887: 6860: 6836: 6635: 6585: 6553: 6533: 6501: 6461: 6423: 6403: 6376: 6356: 6336: 6132: 6092: 6072: 6057:{\displaystyle \mu } 6048: 5961: 5941: 5926:{\displaystyle \mu } 5917: 5826: 5806: 5779: 5759: 5739: 5706: 5608: 5585: 5559: 5517: 5479: 5085: 4598: 4338: 4148: 3867: 3749: 3523: 3446: 3380: 3279: 3186: 3096: 3004: 2921: 2841: 2794: 2685: 2669:corresponds to the ( 2636: 2527: 2448: 2200: 1993: 1917: 1890: 1870: 1850: 1827: 1804: 1793:{\displaystyle N(i)} 1775: 1561: 1535: 1490: 1429: 1383: 1363: 1332: 1309: 1279: 1249: 1229: 1209: 1056: 1033: 1013: 885: 846: 816: 677: 642: 597: 574: 554: 507: 388: 365: 345: 305: 285: 265: 227: 165:loop quantum gravity 50:improve this article 12205:Geometry processing 12059:2022IJHMT.18322112C 12018:2020IJTS..15306383C 11130:resolvent formalism 11096:spectral properties 10882:'DelayTime' 10870:'WriteMode' 10825:'DelayTime' 10813:'Loopcount' 6947:is proportional to 6626:square lattice grid 6372:is represented via 5112: 4625: 4358: 4311: 4275: 4239: 4185: 3938: 3917: 3884: 3771: 3708:-th direction, and 3504: 2832:seven-point stencil 2819: 2707: 2658: 2549: 2474: 2346:square lattice grid 1977:discrete Laplacians 1005:triangle mesh, the 12195:Finite differences 11939:Bigun, J. (2006). 11645:"Image processing" 11614:Electrical network 11582: 11521: 11494:ADE classification 11484:ADE classification 11478:ADE classification 11464: 11376: 11356: 11311: 11255: 11225: 11062: 11027: 10957: 10929: 9890:{\textstyle \phi } 9887: 9875: 9852:{\textstyle \phi } 9849: 9825: 9752: 9717: 9700:{\textstyle \phi } 9697: 9677: 9647: 9570: 9505: 9470: 9450: 9418: 9392: 9368: 9348: 9319: 9299: 9270: 9243: 9238: 9132: 9097: 9064: 8980: 8967: 8927: 8903: 8876: 8797: 8768: 8739: 8710: 8690: 8647: 8611: 8580: 8550: 8470:whose solution is 8457: 8455: 8256: 8133: 8072: 8001: 7947: 7912: 7877: 7836: 7775: 7749:{\textstyle \phi } 7746: 7736:. That is, write 7719: 7681: 7620: 7618: 7513: 7511: 7474: 7374: 7320: 7223: 7197: 7110: 7037: 7017: 6997: 6977: 6937: 6917: 6893: 6873: 6845:{\textstyle \phi } 6842: 6806:numerical analysis 6791: 6690: 6606: 6571: 6539: 6519: 6467: 6438: 6409: 6389: 6362: 6342: 6319: 6196: 6116: 6078: 6066:Gaussian functions 6054: 6034: 5947: 5923: 5903: 5812: 5792: 5765: 5745: 5725: 5689: 5650: 5591: 5571: 5545: 5503: 5457: 5091: 5070: 4604: 4583: 4344: 4318: 4290: 4254: 4225: 4151: 4127: 4121: 4017: 3924: 3903: 3870: 3844: 3835: 3752: 3667: 3662: 3508: 3449: 3432: 3376:: For the element 3355: 3349: 3265: 3259: 3172: 3166: 3070: 3064: 2990: 2984: 2907: 2901: 2820: 2797: 2777: 2771: 2688: 2675:nine-point stencil 2671:Five-point stencil 2659: 2639: 2619: 2613: 2530: 2510: 2504: 2451: 2385:rectangular cuboid 2323: 2168: 1981:five-point stencil 1967:, obtained by the 1959:Finite differences 1942: 1903: 1876: 1856: 1833: 1810: 1790: 1759: 1754: 1710: 1541: 1521: 1476: 1389: 1369: 1345: 1321:{\displaystyle ij} 1318: 1295: 1265: 1235: 1215: 1192: 1112: 1039: 1019: 983: 967: 876:averaging operator 861: 832: 799: 748: 660: 621: 583: 560: 528: 490: 452: 371: 351: 323: 291: 271: 251: 206:discrete Laplacian 173:numerical analysis 12139:978-0-8218-9384-5 12106:978-3-540-69171-6 12091:Bourbaki, Nicolas 11958:978-3-540-27322-6 11910:10.1002/num.20129 11821:978-3-527-63152-0 11624:Discrete calculus 11204: 10858:'out.gif' 10801:'out.gif' 9783: 9737: 9660:For each element 9618: 9617: 9490: 9460:for a graph with 9453:{\textstyle c(0)} 9290: 9216: 9181: 9117: 8990:, the only terms 8952: 8749:, simply project 8413: 8296: 8247: 8173: 8124: 8063: 8047: 7992: 7957:are orthogonal): 7868: 7661: 7558: 7465: 7395: 7365: 7311: 7285: 7214: 7188: 7101: 7086: 6786: 6675: 6670: 6542:{\displaystyle Z} 6470:{\displaystyle n} 6435: 6412:{\displaystyle n} 6365:{\displaystyle f} 6345:{\displaystyle K} 6308: 6292: 6256: 6240: 6176: 6161: 6110: 6081:{\displaystyle n} 5973: 5957:-dimensions i.e. 5950:{\displaystyle n} 5891: 5875: 5851: 5815:{\displaystyle K} 5768:{\displaystyle K} 5748:{\displaystyle f} 5683: 5635: 5626: 5594:{\displaystyle f} 5529: 5497: 5149: 4662: 3658: 3626: 3600: 3290: 3197: 3182:; second plane = 3107: 2917:; second plane = 2461: 2336:If the grid size 2193:) in the grid is 2163: 1879:{\displaystyle i} 1859:{\displaystyle M} 1836:{\displaystyle M} 1813:{\displaystyle i} 1750: 1686: 1656: 1596: 1544:{\displaystyle C} 1392:{\displaystyle i} 1372:{\displaystyle i} 1238:{\displaystyle i} 1218:{\displaystyle j} 1103: 1101: 1042:{\displaystyle i} 1022:{\displaystyle u} 930: 928: 711: 415: 294:{\displaystyle E} 274:{\displaystyle V} 169:dynamical systems 126: 125: 118: 100: 12212: 12164: 12159:. Archived from 12143: 12110: 12109: 12087: 12081: 12080: 12070: 12038: 12032: 12031: 12029: 11997: 11991: 11990: 11969: 11963: 11962: 11936: 11930: 11929: 11893: 11887: 11873: 11862: 11857: 11848: 11847: 11841: 11832: 11826: 11825: 11805: 11794: 11788: 11787: 11785: 11784: 11769: 11763: 11762: 11752: 11732: 11726: 11725: 11715: 11706:(3): 381–390df. 11695: 11689: 11688: 11668: 11662: 11661: 11659: 11658: 11649: 11640: 11591: 11589: 11588: 11583: 11530: 11528: 11527: 11522: 11473: 11471: 11470: 11465: 11451: 11450: 11385: 11383: 11382: 11377: 11365: 11363: 11362: 11357: 11320: 11318: 11317: 11312: 11310: 11309: 11285: 11284: 11264: 11262: 11261: 11256: 11254: 11253: 11234: 11232: 11231: 11226: 11224: 11220: 11219: 11218: 11209: 11205: 11203: 11189: 11183: 11182: 11128:is given in the 11124:of the discrete 11122:Green's function 11108:Boolean algebras 11071: 11069: 11068: 11063: 11036: 11034: 11033: 11028: 10966: 10964: 10963: 10958: 10938: 10936: 10935: 10930: 10898: 10895: 10892: 10889: 10886: 10883: 10880: 10877: 10876:'append' 10874: 10871: 10868: 10865: 10862: 10859: 10856: 10853: 10850: 10847: 10844: 10841: 10838: 10835: 10832: 10829: 10826: 10823: 10820: 10817: 10814: 10811: 10808: 10805: 10802: 10799: 10796: 10793: 10790: 10787: 10784: 10781: 10778: 10775: 10772: 10769: 10766: 10763: 10760: 10757: 10754: 10751: 10748: 10745: 10742: 10739: 10736: 10733: 10730: 10727: 10724: 10721: 10718: 10715: 10712: 10709: 10706: 10703: 10700: 10697: 10694: 10691: 10688: 10685: 10682: 10679: 10676: 10673: 10670: 10667: 10664: 10661: 10658: 10655: 10652: 10649: 10646: 10643: 10640: 10637: 10634: 10631: 10628: 10625: 10622: 10619: 10616: 10613: 10610: 10607: 10604: 10601: 10598: 10595: 10592: 10589: 10586: 10583: 10580: 10577: 10574: 10571: 10568: 10565: 10562: 10559: 10556: 10553: 10550: 10547: 10544: 10541: 10538: 10535: 10532: 10529: 10526: 10523: 10520: 10517: 10514: 10511: 10508: 10505: 10502: 10499: 10496: 10493: 10490: 10487: 10484: 10481: 10478: 10475: 10472: 10469: 10466: 10463: 10460: 10457: 10454: 10451: 10448: 10445: 10442: 10439: 10436: 10433: 10430: 10427: 10424: 10421: 10418: 10415: 10412: 10409: 10406: 10403: 10400: 10397: 10394: 10391: 10388: 10385: 10382: 10379: 10376: 10373: 10370: 10367: 10364: 10361: 10358: 10355: 10352: 10349: 10346: 10343: 10340: 10337: 10334: 10331: 10328: 10325: 10322: 10319: 10316: 10313: 10310: 10307: 10304: 10301: 10298: 10295: 10292: 10289: 10286: 10283: 10280: 10277: 10274: 10271: 10268: 10265: 10262: 10259: 10256: 10253: 10250: 10247: 10244: 10241: 10238: 10235: 10232: 10229: 10226: 10223: 10220: 10217: 10214: 10211: 10208: 10205: 10202: 10199: 10196: 10193: 10190: 10187: 10184: 10181: 10178: 10175: 10172: 10169: 10166: 10163: 10160: 10157: 10154: 10151: 10148: 10145: 10142: 10139: 10136: 10133: 10130: 10127: 10124: 10121: 10118: 10115: 10112: 10109: 10106: 10103: 10100: 10097: 10094: 10091: 10088: 10085: 10082: 10079: 10076: 10073: 10070: 10067: 10064: 10061: 10058: 10055: 10052: 10049: 10046: 10043: 10040: 10037: 10034: 10031: 10028: 10025: 10022: 10019: 10016: 10013: 10010: 10007: 10004: 10001: 9998: 9995: 9992: 9989: 9986: 9983: 9980: 9977: 9974: 9971: 9968: 9965: 9962: 9959: 9956: 9953: 9950: 9947: 9944: 9941: 9938: 9935: 9932: 9929: 9926: 9923: 9920: 9917: 9914: 9911: 9908: 9896: 9894: 9893: 9888: 9858: 9856: 9855: 9850: 9834: 9832: 9831: 9826: 9815: 9814: 9804: 9799: 9784: 9776: 9762: 9761: 9751: 9726: 9724: 9723: 9718: 9706: 9704: 9703: 9698: 9686: 9684: 9683: 9678: 9676: 9675: 9656: 9654: 9653: 9648: 9619: 9613: 9609: 9604: 9603: 9602: 9579: 9577: 9576: 9571: 9569: 9568: 9567: 9557: 9553: 9552: 9551: 9550: 9504: 9479: 9477: 9476: 9471: 9459: 9457: 9456: 9451: 9427: 9425: 9424: 9419: 9401: 9399: 9398: 9393: 9377: 9375: 9374: 9369: 9357: 9355: 9354: 9349: 9347: 9346: 9341: 9328: 9326: 9325: 9320: 9312: 9311: 9298: 9279: 9277: 9276: 9271: 9252: 9250: 9249: 9244: 9242: 9241: 9229: 9228: 9217: 9214: 9194: 9193: 9182: 9179: 9158: 9157: 9153: 9152: 9131: 9106: 9104: 9103: 9098: 9090: 9089: 9073: 9071: 9070: 9065: 9063: 9062: 9058: 9057: 9028: 9027: 9006: 9005: 8989: 8987: 8986: 8981: 8966: 8936: 8934: 8933: 8928: 8915:spectral theorem 8912: 8910: 8909: 8904: 8885: 8883: 8882: 8877: 8875: 8871: 8870: 8869: 8864: 8826: 8825: 8806: 8804: 8803: 8798: 8796: 8795: 8790: 8777: 8775: 8774: 8769: 8748: 8746: 8745: 8740: 8719: 8717: 8716: 8711: 8699: 8697: 8696: 8691: 8680: 8679: 8656: 8654: 8653: 8648: 8637: 8636: 8620: 8618: 8617: 8612: 8610: 8609: 8589: 8587: 8586: 8581: 8579: 8578: 8559: 8557: 8556: 8551: 8546: 8545: 8541: 8540: 8511: 8510: 8489: 8488: 8466: 8464: 8463: 8458: 8456: 8440: 8439: 8430: 8429: 8414: 8412: 8404: 8394: 8393: 8380: 8376: 8361: 8357: 8356: 8355: 8350: 8344: 8343: 8325: 8324: 8309: 8308: 8303: 8297: 8295: 8287: 8277: 8276: 8263: 8255: 8242: 8237: 8231: 8227: 8226: 8225: 8220: 8202: 8201: 8186: 8185: 8180: 8174: 8172: 8164: 8154: 8153: 8140: 8132: 8119: 8114: 8108: 8104: 8103: 8102: 8097: 8082: 8081: 8071: 8048: 8046: 8038: 8037: 8033: 8032: 8031: 8026: 8011: 8010: 8000: 7982: 7978: 7956: 7954: 7953: 7948: 7946: 7945: 7940: 7921: 7919: 7918: 7913: 7908: 7907: 7902: 7887: 7886: 7876: 7845: 7843: 7842: 7837: 7835: 7834: 7829: 7823: 7822: 7810: 7809: 7804: 7784: 7782: 7781: 7776: 7774: 7773: 7768: 7755: 7753: 7752: 7747: 7728: 7726: 7725: 7720: 7718: 7717: 7690: 7688: 7687: 7682: 7662: 7660: 7652: 7644: 7629: 7627: 7626: 7621: 7619: 7594: 7559: 7557: 7549: 7541: 7522: 7520: 7519: 7514: 7512: 7505: 7504: 7495: 7491: 7490: 7473: 7452: 7448: 7447: 7438: 7434: 7433: 7432: 7414: 7413: 7393: 7392: 7391: 7373: 7352: 7348: 7344: 7343: 7342: 7333: 7332: 7319: 7304: 7303: 7283: 7282: 7281: 7255: 7251: 7247: 7246: 7245: 7236: 7235: 7222: 7210: 7209: 7196: 7187: 7186: 7160: 7156: 7152: 7151: 7150: 7138: 7137: 7123: 7122: 7109: 7087: 7085: 7077: 7076: 7075: 7062: 7046: 7044: 7043: 7038: 7026: 7024: 7023: 7018: 7006: 7004: 7003: 6998: 6986: 6984: 6983: 6978: 6976: 6975: 6963: 6962: 6946: 6944: 6943: 6938: 6926: 6924: 6923: 6918: 6902: 6900: 6899: 6894: 6882: 6880: 6879: 6874: 6872: 6871: 6851: 6849: 6848: 6843: 6810:image processing 6800: 6798: 6797: 6792: 6787: 6785: 6784: 6775: 6692: 6689: 6671: 6669: 6668: 6667: 6654: 6650: 6649: 6639: 6615: 6613: 6612: 6609:{\displaystyle } 6607: 6580: 6578: 6577: 6574:{\displaystyle } 6572: 6548: 6546: 6545: 6540: 6528: 6526: 6525: 6520: 6479:Structure Tensor 6476: 6474: 6473: 6468: 6447: 6445: 6444: 6439: 6437: 6436: 6428: 6418: 6416: 6415: 6410: 6398: 6396: 6395: 6390: 6388: 6387: 6371: 6369: 6368: 6363: 6351: 6349: 6348: 6343: 6328: 6326: 6325: 6320: 6318: 6317: 6316: 6315: 6310: 6309: 6301: 6294: 6293: 6285: 6281: 6269: 6268: 6267: 6258: 6257: 6249: 6242: 6241: 6233: 6224: 6223: 6211: 6210: 6209: 6195: 6188: 6169: 6168: 6163: 6162: 6154: 6144: 6143: 6125: 6123: 6122: 6117: 6112: 6111: 6103: 6087: 6085: 6084: 6079: 6063: 6061: 6060: 6055: 6043: 6041: 6040: 6035: 6033: 6032: 6023: 6022: 6004: 6003: 5991: 5990: 5975: 5974: 5966: 5956: 5954: 5953: 5948: 5932: 5930: 5929: 5924: 5912: 5910: 5909: 5904: 5899: 5898: 5893: 5892: 5884: 5877: 5876: 5868: 5853: 5852: 5844: 5838: 5837: 5821: 5819: 5818: 5813: 5801: 5799: 5798: 5793: 5791: 5790: 5774: 5772: 5771: 5766: 5754: 5752: 5751: 5746: 5734: 5732: 5731: 5726: 5718: 5717: 5698: 5696: 5695: 5690: 5685: 5684: 5676: 5670: 5669: 5660: 5659: 5649: 5628: 5627: 5619: 5600: 5598: 5597: 5592: 5580: 5578: 5577: 5572: 5554: 5552: 5551: 5546: 5544: 5543: 5531: 5530: 5522: 5512: 5510: 5509: 5504: 5499: 5498: 5490: 5466: 5464: 5463: 5458: 5450: 5449: 5422: 5421: 5388: 5387: 5354: 5353: 5320: 5319: 5286: 5285: 5252: 5251: 5218: 5217: 5184: 5183: 5150: 5142: 5137: 5136: 5111: 5106: 5105: 5104: 5079: 5077: 5076: 5071: 5063: 5062: 5035: 5034: 5004: 5003: 4973: 4972: 4942: 4941: 4911: 4910: 4880: 4879: 4849: 4848: 4818: 4817: 4787: 4786: 4756: 4755: 4725: 4724: 4694: 4693: 4663: 4655: 4650: 4649: 4624: 4619: 4618: 4617: 4592: 4590: 4589: 4584: 4579: 4578: 4551: 4550: 4523: 4522: 4495: 4494: 4467: 4466: 4439: 4438: 4411: 4410: 4383: 4382: 4357: 4352: 4327: 4325: 4324: 4319: 4310: 4305: 4304: 4303: 4288: 4287: 4274: 4269: 4268: 4267: 4252: 4251: 4238: 4233: 4220: 4219: 4207: 4206: 4184: 4179: 4178: 4177: 4165: 4164: 4136: 4134: 4133: 4128: 4126: 4125: 4115: 4097: 4062: 4044: 4022: 4021: 3937: 3932: 3916: 3911: 3883: 3878: 3853: 3851: 3850: 3845: 3840: 3839: 3770: 3765: 3757: 3731: 3719: 3713: 3703: 3699: 3695: 3691: 3676: 3674: 3673: 3668: 3666: 3663: 3659: 3656: 3627: 3624: 3601: 3598: 3574: 3573: 3572: 3571: 3553: 3552: 3540: 3539: 3517: 3515: 3514: 3509: 3503: 3498: 3497: 3496: 3478: 3477: 3465: 3464: 3454: 3441: 3439: 3438: 3433: 3431: 3430: 3429: 3428: 3410: 3409: 3397: 3396: 3364: 3362: 3361: 3356: 3354: 3353: 3291: 3283: 3275:; third plane = 3274: 3272: 3271: 3266: 3264: 3263: 3198: 3190: 3181: 3179: 3178: 3173: 3171: 3170: 3108: 3100: 3087:27-point stencil 3079: 3077: 3076: 3071: 3069: 3068: 3000:; third plane = 2999: 2997: 2996: 2991: 2989: 2988: 2916: 2914: 2913: 2908: 2906: 2905: 2829: 2827: 2826: 2821: 2818: 2813: 2802: 2786: 2784: 2783: 2778: 2776: 2775: 2706: 2701: 2693: 2668: 2666: 2665: 2660: 2657: 2652: 2644: 2628: 2626: 2625: 2620: 2618: 2617: 2548: 2543: 2535: 2519: 2517: 2516: 2511: 2509: 2508: 2473: 2468: 2463: 2462: 2454: 2419:Image processing 2332: 2330: 2329: 2324: 2177: 2175: 2174: 2169: 2164: 2162: 2161: 2152: 2021: 1951: 1949: 1948: 1943: 1938: 1937: 1912: 1910: 1909: 1904: 1902: 1901: 1885: 1883: 1882: 1877: 1865: 1863: 1862: 1857: 1843:be the diagonal 1842: 1840: 1839: 1834: 1819: 1817: 1816: 1811: 1799: 1797: 1796: 1791: 1768: 1766: 1765: 1760: 1758: 1757: 1751: 1748: 1723: 1722: 1709: 1657: 1654: 1641: 1640: 1619: 1618: 1597: 1589: 1576: 1575: 1553:cotangent matrix 1551:be the (sparse) 1550: 1548: 1547: 1542: 1530: 1528: 1527: 1522: 1520: 1519: 1485: 1483: 1482: 1477: 1475: 1474: 1473: 1465: 1457: 1449: 1443: 1405:Laplace operator 1398: 1396: 1395: 1390: 1378: 1376: 1375: 1370: 1354: 1352: 1351: 1346: 1344: 1343: 1327: 1325: 1324: 1319: 1304: 1302: 1301: 1296: 1294: 1293: 1274: 1272: 1271: 1266: 1264: 1263: 1244: 1242: 1241: 1236: 1224: 1222: 1221: 1216: 1201: 1199: 1198: 1193: 1185: 1184: 1172: 1171: 1156: 1155: 1134: 1133: 1111: 1102: 1100: 1099: 1098: 1082: 1077: 1076: 1048: 1046: 1045: 1040: 1028: 1026: 1025: 1020: 992: 990: 989: 984: 966: 929: 927: 913: 870: 868: 867: 862: 841: 839: 838: 833: 831: 830: 808: 806: 805: 800: 798: 794: 761: 760: 747: 692: 691: 669: 667: 666: 661: 630: 628: 627: 622: 592: 590: 589: 584: 569: 567: 566: 561: 548:Laplacian matrix 537: 535: 534: 529: 499: 497: 496: 491: 489: 485: 451: 380: 378: 377: 372: 360: 358: 357: 352: 332: 330: 329: 324: 300: 298: 297: 292: 280: 278: 277: 272: 260: 258: 257: 252: 200:Graph Laplacians 177:image processing 150:Laplacian matrix 138:Laplace operator 121: 114: 110: 107: 101: 99: 58: 34: 26: 12220: 12219: 12215: 12214: 12213: 12211: 12210: 12209: 12180:Operator theory 12170: 12169: 12151: 12146: 12140: 12114: 12113: 12107: 12088: 12084: 12039: 12035: 11998: 11994: 11987: 11970: 11966: 11959: 11949:10.1007/b138918 11937: 11933: 11894: 11890: 11874: 11865: 11858: 11851: 11839: 11833: 11829: 11822: 11803: 11795: 11791: 11782: 11780: 11770: 11766: 11733: 11729: 11696: 11692: 11669: 11665: 11656: 11654: 11647: 11641: 11637: 11632: 11605: 11598: 11562: 11559: 11558: 11504: 11501: 11500: 11490:Dynkin diagrams 11486: 11480: 11446: 11442: 11398: 11395: 11394: 11371: 11368: 11367: 11345: 11342: 11341: 11302: 11298: 11280: 11276: 11274: 11271: 11270: 11267:Kronecker delta 11249: 11245: 11243: 11240: 11239: 11214: 11210: 11193: 11188: 11184: 11178: 11174: 11173: 11169: 11140: 11137: 11136: 11100:Stone's theorem 11087:is bounded and 11045: 11042: 11041: 10974: 10971: 10970: 10952: 10949: 10948: 10912: 10909: 10908: 10905: 10900: 10899: 10896: 10893: 10890: 10887: 10884: 10881: 10878: 10875: 10872: 10869: 10866: 10863: 10860: 10857: 10854: 10851: 10848: 10845: 10842: 10839: 10836: 10833: 10830: 10827: 10824: 10821: 10818: 10815: 10812: 10809: 10806: 10803: 10800: 10797: 10794: 10791: 10788: 10785: 10782: 10779: 10776: 10773: 10770: 10767: 10764: 10761: 10758: 10755: 10752: 10749: 10746: 10743: 10740: 10737: 10734: 10731: 10728: 10725: 10722: 10719: 10716: 10713: 10710: 10707: 10704: 10701: 10698: 10695: 10692: 10689: 10686: 10683: 10680: 10677: 10674: 10671: 10668: 10665: 10662: 10659: 10656: 10653: 10650: 10647: 10644: 10641: 10638: 10635: 10632: 10629: 10626: 10623: 10620: 10617: 10614: 10611: 10608: 10605: 10602: 10599: 10596: 10593: 10590: 10587: 10584: 10581: 10578: 10575: 10572: 10569: 10566: 10563: 10560: 10557: 10554: 10551: 10548: 10545: 10542: 10539: 10536: 10533: 10530: 10527: 10524: 10521: 10518: 10515: 10512: 10509: 10506: 10503: 10500: 10497: 10494: 10491: 10488: 10485: 10482: 10479: 10476: 10473: 10470: 10467: 10464: 10461: 10458: 10455: 10452: 10449: 10446: 10443: 10440: 10437: 10434: 10431: 10428: 10425: 10422: 10419: 10416: 10413: 10410: 10407: 10404: 10401: 10398: 10395: 10392: 10389: 10386: 10383: 10380: 10377: 10374: 10371: 10368: 10365: 10362: 10359: 10356: 10353: 10350: 10347: 10344: 10341: 10338: 10335: 10332: 10329: 10326: 10323: 10320: 10317: 10314: 10311: 10308: 10305: 10302: 10299: 10296: 10293: 10290: 10287: 10284: 10281: 10278: 10275: 10272: 10269: 10266: 10263: 10260: 10257: 10254: 10251: 10248: 10245: 10242: 10239: 10236: 10233: 10230: 10227: 10224: 10221: 10218: 10215: 10212: 10209: 10206: 10203: 10200: 10197: 10194: 10191: 10188: 10185: 10182: 10179: 10176: 10173: 10170: 10167: 10164: 10161: 10158: 10155: 10152: 10149: 10146: 10143: 10140: 10137: 10134: 10131: 10128: 10125: 10122: 10119: 10116: 10113: 10110: 10107: 10104: 10101: 10098: 10095: 10092: 10089: 10086: 10083: 10080: 10077: 10074: 10071: 10068: 10065: 10062: 10059: 10056: 10053: 10050: 10047: 10044: 10041: 10038: 10035: 10032: 10029: 10026: 10023: 10020: 10017: 10014: 10011: 10008: 10005: 10002: 9999: 9996: 9993: 9990: 9987: 9984: 9981: 9978: 9975: 9972: 9969: 9966: 9963: 9960: 9957: 9954: 9951: 9948: 9945: 9942: 9939: 9936: 9933: 9930: 9927: 9924: 9921: 9918: 9915: 9912: 9909: 9906: 9882: 9879: 9878: 9866: 9844: 9841: 9840: 9810: 9806: 9800: 9789: 9775: 9757: 9753: 9741: 9735: 9732: 9731: 9712: 9709: 9708: 9692: 9689: 9688: 9671: 9667: 9665: 9662: 9661: 9608: 9598: 9594: 9593: 9591: 9588: 9587: 9563: 9559: 9558: 9546: 9542: 9541: 9525: 9521: 9494: 9488: 9485: 9484: 9465: 9462: 9461: 9436: 9433: 9432: 9407: 9404: 9403: 9387: 9384: 9383: 9363: 9360: 9359: 9342: 9337: 9336: 9334: 9331: 9330: 9304: 9300: 9294: 9288: 9285: 9284: 9265: 9262: 9261: 9237: 9236: 9224: 9220: 9218: 9213: 9211: 9202: 9201: 9189: 9185: 9183: 9178: 9176: 9163: 9162: 9148: 9144: 9137: 9133: 9121: 9115: 9112: 9111: 9085: 9081: 9079: 9076: 9075: 9053: 9049: 9042: 9038: 9023: 9019: 9001: 8997: 8995: 8992: 8991: 8956: 8950: 8947: 8946: 8943: 8922: 8919: 8918: 8898: 8895: 8894: 8865: 8860: 8859: 8843: 8839: 8821: 8817: 8815: 8812: 8811: 8791: 8786: 8785: 8783: 8780: 8779: 8754: 8751: 8750: 8725: 8722: 8721: 8705: 8702: 8701: 8675: 8671: 8669: 8666: 8665: 8632: 8628: 8626: 8623: 8622: 8605: 8601: 8599: 8596: 8595: 8574: 8570: 8568: 8565: 8564: 8536: 8532: 8525: 8521: 8506: 8502: 8484: 8480: 8478: 8475: 8474: 8454: 8453: 8435: 8431: 8425: 8421: 8405: 8389: 8385: 8381: 8379: 8377: 8375: 8363: 8362: 8351: 8346: 8345: 8339: 8335: 8320: 8316: 8304: 8299: 8298: 8288: 8272: 8268: 8264: 8262: 8261: 8257: 8251: 8243: 8241: 8236: 8233: 8232: 8221: 8216: 8215: 8197: 8193: 8181: 8176: 8175: 8165: 8149: 8145: 8141: 8139: 8138: 8134: 8128: 8120: 8118: 8113: 8110: 8109: 8098: 8093: 8092: 8077: 8073: 8067: 8062: 8058: 8039: 8027: 8022: 8021: 8006: 8002: 7996: 7991: 7987: 7983: 7981: 7979: 7977: 7967: 7965: 7962: 7961: 7941: 7936: 7935: 7933: 7930: 7929: 7903: 7898: 7897: 7882: 7878: 7872: 7851: 7848: 7847: 7830: 7825: 7824: 7818: 7814: 7805: 7800: 7799: 7794: 7791: 7790: 7769: 7764: 7763: 7761: 7758: 7757: 7741: 7738: 7737: 7713: 7709: 7707: 7704: 7703: 7653: 7645: 7643: 7641: 7638: 7637: 7617: 7616: 7592: 7591: 7560: 7550: 7542: 7540: 7536: 7534: 7531: 7530: 7510: 7509: 7500: 7496: 7483: 7479: 7475: 7469: 7450: 7449: 7443: 7439: 7425: 7421: 7409: 7405: 7384: 7380: 7379: 7375: 7369: 7350: 7349: 7338: 7334: 7325: 7321: 7315: 7299: 7295: 7277: 7273: 7272: 7268: 7253: 7252: 7241: 7237: 7228: 7224: 7218: 7202: 7198: 7192: 7182: 7178: 7177: 7173: 7158: 7157: 7146: 7142: 7133: 7129: 7128: 7124: 7115: 7111: 7105: 7088: 7078: 7071: 7067: 7063: 7061: 7057: 7055: 7052: 7051: 7032: 7029: 7028: 7012: 7009: 7008: 6992: 6989: 6988: 6971: 6967: 6958: 6954: 6952: 6949: 6948: 6932: 6929: 6928: 6912: 6909: 6908: 6903:. According to 6888: 6885: 6884: 6867: 6863: 6861: 6858: 6857: 6837: 6834: 6833: 6830: 6780: 6776: 6693: 6691: 6679: 6663: 6659: 6655: 6645: 6641: 6640: 6638: 6636: 6633: 6632: 6622: 6586: 6583: 6582: 6554: 6551: 6550: 6534: 6531: 6530: 6502: 6499: 6498: 6491: 6462: 6459: 6458: 6427: 6426: 6424: 6421: 6420: 6404: 6401: 6400: 6383: 6379: 6377: 6374: 6373: 6357: 6354: 6353: 6337: 6334: 6333: 6311: 6300: 6299: 6298: 6284: 6283: 6282: 6277: 6276: 6260: 6259: 6248: 6247: 6246: 6232: 6231: 6219: 6215: 6202: 6201: 6197: 6181: 6180: 6164: 6153: 6152: 6151: 6139: 6135: 6133: 6130: 6129: 6102: 6101: 6093: 6090: 6089: 6073: 6070: 6069: 6049: 6046: 6045: 6028: 6024: 6018: 6014: 5999: 5995: 5986: 5982: 5965: 5964: 5962: 5959: 5958: 5942: 5939: 5938: 5918: 5915: 5914: 5894: 5883: 5882: 5881: 5867: 5866: 5843: 5842: 5833: 5829: 5827: 5824: 5823: 5807: 5804: 5803: 5786: 5782: 5780: 5777: 5776: 5760: 5757: 5756: 5740: 5737: 5736: 5713: 5709: 5707: 5704: 5703: 5675: 5674: 5665: 5661: 5655: 5651: 5639: 5618: 5617: 5609: 5606: 5605: 5586: 5583: 5582: 5560: 5557: 5556: 5539: 5535: 5521: 5520: 5518: 5515: 5514: 5489: 5488: 5480: 5477: 5476: 5473: 5433: 5429: 5396: 5392: 5362: 5358: 5328: 5324: 5294: 5290: 5260: 5256: 5226: 5222: 5192: 5188: 5158: 5154: 5141: 5120: 5116: 5107: 5100: 5096: 5095: 5086: 5083: 5082: 5046: 5042: 5012: 5008: 4981: 4977: 4950: 4946: 4919: 4915: 4888: 4884: 4857: 4853: 4826: 4822: 4795: 4791: 4764: 4760: 4733: 4729: 4702: 4698: 4671: 4667: 4654: 4633: 4629: 4620: 4613: 4609: 4608: 4599: 4596: 4595: 4562: 4558: 4531: 4527: 4503: 4499: 4475: 4471: 4447: 4443: 4419: 4415: 4391: 4387: 4366: 4362: 4353: 4348: 4339: 4336: 4335: 4306: 4299: 4295: 4294: 4283: 4279: 4270: 4263: 4259: 4258: 4247: 4243: 4234: 4229: 4215: 4211: 4202: 4198: 4180: 4173: 4169: 4160: 4156: 4155: 4149: 4146: 4145: 4120: 4119: 4111: 4106: 4101: 4093: 4087: 4086: 4081: 4073: 4067: 4066: 4058: 4053: 4048: 4040: 4030: 4029: 4016: 4015: 4010: 4005: 3999: 3998: 3993: 3985: 3979: 3978: 3973: 3968: 3958: 3957: 3933: 3928: 3912: 3907: 3879: 3874: 3868: 3865: 3864: 3834: 3833: 3828: 3823: 3817: 3816: 3811: 3803: 3797: 3796: 3791: 3786: 3776: 3775: 3766: 3758: 3753: 3750: 3747: 3746: 3729: 3728: 3724: 3721: 3718: 3715: 3712: 3709: 3707: 3701: 3697: 3693: 3690: 3689: 3685: 3682: 3661: 3660: 3655: 3653: 3647: 3646: 3623: 3621: 3615: 3614: 3597: 3595: 3582: 3578: 3567: 3563: 3548: 3544: 3535: 3531: 3530: 3526: 3524: 3521: 3520: 3499: 3492: 3488: 3473: 3469: 3460: 3456: 3455: 3450: 3447: 3444: 3443: 3424: 3420: 3405: 3401: 3392: 3388: 3387: 3383: 3381: 3378: 3377: 3373: 3348: 3347: 3342: 3337: 3331: 3330: 3325: 3320: 3314: 3313: 3308: 3303: 3293: 3292: 3282: 3280: 3277: 3276: 3258: 3257: 3252: 3247: 3241: 3240: 3235: 3227: 3221: 3220: 3215: 3210: 3200: 3199: 3189: 3187: 3184: 3183: 3165: 3164: 3159: 3154: 3148: 3147: 3142: 3137: 3131: 3130: 3125: 3120: 3110: 3109: 3099: 3097: 3094: 3093: 3063: 3062: 3057: 3052: 3046: 3045: 3040: 3035: 3029: 3028: 3023: 3018: 3008: 3007: 3005: 3002: 3001: 2983: 2982: 2977: 2972: 2966: 2965: 2960: 2952: 2946: 2945: 2940: 2935: 2925: 2924: 2922: 2919: 2918: 2900: 2899: 2894: 2889: 2883: 2882: 2877: 2872: 2866: 2865: 2860: 2855: 2845: 2844: 2842: 2839: 2838: 2814: 2803: 2798: 2795: 2792: 2791: 2770: 2769: 2764: 2759: 2753: 2752: 2747: 2739: 2733: 2732: 2727: 2722: 2712: 2711: 2702: 2694: 2689: 2686: 2683: 2682: 2653: 2645: 2640: 2637: 2634: 2633: 2612: 2611: 2606: 2601: 2595: 2594: 2589: 2581: 2575: 2574: 2569: 2564: 2554: 2553: 2544: 2536: 2531: 2528: 2525: 2524: 2503: 2502: 2497: 2489: 2479: 2478: 2469: 2464: 2453: 2452: 2449: 2446: 2445: 2434: 2421: 2412: 2201: 2198: 2197: 2157: 2153: 2022: 2020: 1994: 1991: 1990: 1986:, resulting in 1961: 1930: 1926: 1918: 1915: 1914: 1897: 1893: 1891: 1888: 1887: 1871: 1868: 1867: 1851: 1848: 1847: 1828: 1825: 1824: 1805: 1802: 1801: 1776: 1773: 1772: 1753: 1752: 1747: 1745: 1739: 1738: 1724: 1715: 1711: 1690: 1680: 1679: 1653: 1645: 1633: 1629: 1611: 1607: 1588: 1581: 1580: 1568: 1564: 1562: 1559: 1558: 1536: 1533: 1532: 1515: 1511: 1491: 1488: 1487: 1469: 1461: 1453: 1445: 1444: 1439: 1438: 1430: 1427: 1426: 1419:(PDF download: 1384: 1381: 1380: 1364: 1361: 1360: 1339: 1335: 1333: 1330: 1329: 1310: 1307: 1306: 1286: 1282: 1280: 1277: 1276: 1256: 1252: 1250: 1247: 1246: 1230: 1227: 1226: 1210: 1207: 1206: 1180: 1176: 1167: 1163: 1148: 1144: 1126: 1122: 1107: 1094: 1090: 1086: 1081: 1072: 1068: 1057: 1054: 1053: 1034: 1031: 1030: 1014: 1011: 1010: 999: 997:Mesh Laplacians 934: 917: 912: 886: 883: 882: 847: 844: 843: 823: 819: 817: 814: 813: 766: 762: 753: 749: 715: 687: 683: 678: 675: 674: 643: 640: 639: 598: 595: 594: 575: 572: 571: 555: 552: 551: 508: 505: 504: 457: 453: 419: 389: 386: 385: 366: 363: 362: 346: 343: 342: 306: 303: 302: 286: 283: 282: 266: 263: 262: 228: 225: 224: 202: 197: 122: 111: 105: 102: 59: 57: 47: 35: 24: 17: 12: 11: 5: 12218: 12208: 12207: 12202: 12200:Edge detection 12197: 12192: 12187: 12182: 12168: 12167: 12165: 12163:on 2007-05-23. 12150: 12149:External links 12147: 12145: 12144: 12138: 12115: 12112: 12111: 12105: 12082: 12033: 11992: 11986:978-0199206650 11985: 11964: 11957: 11931: 11904:(4): 936–953. 11888: 11863: 11849: 11827: 11820: 11789: 11764: 11750:10.1.1.157.757 11743:(3): 381–390. 11727: 11713:10.1.1.157.757 11690: 11663: 11634: 11633: 11631: 11628: 11627: 11626: 11621: 11616: 11611: 11604: 11601: 11596: 11593: 11592: 11581: 11578: 11575: 11572: 11569: 11566: 11552: 11551: 11540: 11539: 11532: 11531: 11520: 11517: 11514: 11511: 11508: 11479: 11476: 11475: 11474: 11463: 11460: 11457: 11454: 11449: 11445: 11441: 11438: 11435: 11432: 11429: 11426: 11423: 11420: 11417: 11414: 11411: 11408: 11405: 11402: 11375: 11355: 11352: 11349: 11308: 11305: 11301: 11297: 11294: 11291: 11288: 11283: 11279: 11252: 11248: 11236: 11235: 11223: 11217: 11213: 11208: 11202: 11199: 11196: 11192: 11187: 11181: 11177: 11172: 11168: 11165: 11162: 11159: 11156: 11153: 11150: 11147: 11144: 11061: 11058: 11055: 11052: 11049: 11038: 11037: 11026: 11023: 11020: 11017: 11014: 11011: 11008: 11005: 11002: 10999: 10996: 10993: 10990: 10987: 10984: 10981: 10978: 10956: 10928: 10925: 10922: 10919: 10916: 10904: 10901: 9905: 9886: 9865: 9862: 9848: 9837: 9836: 9824: 9821: 9818: 9813: 9809: 9803: 9798: 9795: 9792: 9788: 9782: 9779: 9774: 9771: 9768: 9765: 9760: 9756: 9750: 9747: 9744: 9740: 9720:{\textstyle j} 9716: 9696: 9674: 9670: 9658: 9657: 9646: 9643: 9640: 9637: 9634: 9631: 9628: 9625: 9622: 9616: 9612: 9607: 9601: 9597: 9581: 9580: 9566: 9562: 9556: 9549: 9545: 9540: 9537: 9534: 9531: 9528: 9524: 9520: 9517: 9514: 9511: 9508: 9503: 9500: 9497: 9493: 9473:{\textstyle N} 9469: 9449: 9446: 9443: 9440: 9417: 9414: 9411: 9395:{\textstyle k} 9391: 9371:{\textstyle k} 9367: 9345: 9340: 9318: 9315: 9310: 9307: 9303: 9297: 9293: 9273:{\textstyle L} 9269: 9254: 9253: 9240: 9235: 9232: 9227: 9223: 9219: 9212: 9210: 9207: 9204: 9203: 9200: 9197: 9192: 9188: 9184: 9177: 9175: 9172: 9169: 9168: 9166: 9161: 9156: 9151: 9147: 9143: 9140: 9136: 9130: 9127: 9124: 9120: 9096: 9093: 9088: 9084: 9061: 9056: 9052: 9048: 9045: 9041: 9037: 9034: 9031: 9026: 9022: 9018: 9015: 9012: 9009: 9004: 9000: 8979: 8976: 8973: 8970: 8965: 8962: 8959: 8955: 8945:To understand 8942: 8939: 8930:{\textstyle L} 8926: 8906:{\textstyle L} 8902: 8888: 8887: 8874: 8868: 8863: 8858: 8855: 8852: 8849: 8846: 8842: 8838: 8835: 8832: 8829: 8824: 8820: 8794: 8789: 8767: 8764: 8761: 8758: 8738: 8735: 8732: 8729: 8713:{\textstyle i} 8709: 8689: 8686: 8683: 8678: 8674: 8661:can be found. 8646: 8643: 8640: 8635: 8631: 8608: 8604: 8577: 8573: 8561: 8560: 8549: 8544: 8539: 8535: 8531: 8528: 8524: 8520: 8517: 8514: 8509: 8505: 8501: 8498: 8495: 8492: 8487: 8483: 8468: 8467: 8452: 8449: 8446: 8443: 8438: 8434: 8428: 8424: 8420: 8417: 8411: 8408: 8403: 8400: 8397: 8392: 8388: 8384: 8378: 8374: 8371: 8368: 8365: 8364: 8360: 8354: 8349: 8342: 8338: 8334: 8331: 8328: 8323: 8319: 8315: 8312: 8307: 8302: 8294: 8291: 8286: 8283: 8280: 8275: 8271: 8267: 8260: 8254: 8250: 8246: 8244: 8240: 8235: 8234: 8230: 8224: 8219: 8214: 8211: 8208: 8205: 8200: 8196: 8192: 8189: 8184: 8179: 8171: 8168: 8163: 8160: 8157: 8152: 8148: 8144: 8137: 8131: 8127: 8123: 8121: 8117: 8112: 8111: 8107: 8101: 8096: 8091: 8088: 8085: 8080: 8076: 8070: 8066: 8061: 8057: 8054: 8051: 8045: 8042: 8036: 8030: 8025: 8020: 8017: 8014: 8009: 8005: 7999: 7995: 7990: 7986: 7980: 7976: 7973: 7970: 7969: 7944: 7939: 7911: 7906: 7901: 7896: 7893: 7890: 7885: 7881: 7875: 7871: 7867: 7864: 7861: 7858: 7855: 7833: 7828: 7821: 7817: 7813: 7808: 7803: 7798: 7772: 7767: 7745: 7716: 7712: 7692: 7691: 7680: 7677: 7674: 7671: 7668: 7665: 7659: 7656: 7651: 7648: 7631: 7630: 7615: 7612: 7609: 7606: 7603: 7600: 7597: 7595: 7593: 7590: 7587: 7584: 7581: 7578: 7575: 7572: 7569: 7566: 7563: 7561: 7556: 7553: 7548: 7545: 7539: 7538: 7524: 7523: 7508: 7503: 7499: 7494: 7489: 7486: 7482: 7478: 7472: 7468: 7464: 7461: 7458: 7455: 7453: 7451: 7446: 7442: 7437: 7431: 7428: 7424: 7420: 7417: 7412: 7408: 7404: 7401: 7398: 7390: 7387: 7383: 7378: 7372: 7368: 7364: 7361: 7358: 7355: 7353: 7351: 7347: 7341: 7337: 7331: 7328: 7324: 7318: 7314: 7310: 7307: 7302: 7298: 7294: 7291: 7288: 7280: 7276: 7271: 7267: 7264: 7261: 7258: 7256: 7254: 7250: 7244: 7240: 7234: 7231: 7227: 7221: 7217: 7213: 7208: 7205: 7201: 7195: 7191: 7185: 7181: 7176: 7172: 7169: 7166: 7163: 7161: 7159: 7155: 7149: 7145: 7141: 7136: 7132: 7127: 7121: 7118: 7114: 7108: 7104: 7100: 7097: 7094: 7091: 7089: 7084: 7081: 7074: 7070: 7066: 7060: 7059: 7040:{\textstyle k} 7036: 7020:{\textstyle j} 7016: 7000:{\textstyle i} 6996: 6974: 6970: 6966: 6961: 6957: 6940:{\textstyle j} 6936: 6920:{\textstyle i} 6916: 6896:{\textstyle i} 6892: 6870: 6866: 6841: 6829: 6826: 6822:Laplace filter 6814:digital filter 6802: 6801: 6790: 6783: 6779: 6774: 6771: 6768: 6765: 6762: 6759: 6756: 6753: 6750: 6747: 6744: 6741: 6738: 6735: 6732: 6729: 6726: 6723: 6720: 6717: 6714: 6711: 6708: 6705: 6702: 6699: 6696: 6688: 6685: 6682: 6678: 6674: 6666: 6662: 6658: 6653: 6648: 6644: 6621: 6618: 6605: 6602: 6599: 6596: 6593: 6590: 6570: 6567: 6564: 6561: 6558: 6538: 6518: 6515: 6512: 6509: 6506: 6490: 6487: 6466: 6434: 6431: 6408: 6386: 6382: 6361: 6341: 6330: 6329: 6314: 6307: 6304: 6297: 6291: 6288: 6280: 6275: 6272: 6266: 6263: 6255: 6252: 6245: 6239: 6236: 6230: 6227: 6222: 6218: 6214: 6208: 6205: 6200: 6194: 6191: 6187: 6184: 6179: 6175: 6172: 6167: 6160: 6157: 6150: 6147: 6142: 6138: 6115: 6109: 6106: 6100: 6097: 6077: 6053: 6031: 6027: 6021: 6017: 6013: 6010: 6007: 6002: 5998: 5994: 5989: 5985: 5981: 5978: 5972: 5969: 5946: 5922: 5902: 5897: 5890: 5887: 5880: 5874: 5871: 5865: 5862: 5859: 5856: 5850: 5847: 5841: 5836: 5832: 5811: 5789: 5785: 5764: 5744: 5724: 5721: 5716: 5712: 5700: 5699: 5688: 5682: 5679: 5673: 5668: 5664: 5658: 5654: 5648: 5645: 5642: 5638: 5634: 5631: 5625: 5622: 5616: 5613: 5590: 5570: 5567: 5564: 5542: 5538: 5534: 5528: 5525: 5502: 5496: 5493: 5487: 5484: 5472: 5469: 5468: 5467: 5456: 5453: 5448: 5445: 5442: 5439: 5436: 5432: 5428: 5425: 5420: 5417: 5414: 5411: 5408: 5405: 5402: 5399: 5395: 5391: 5386: 5383: 5380: 5377: 5374: 5371: 5368: 5365: 5361: 5357: 5352: 5349: 5346: 5343: 5340: 5337: 5334: 5331: 5327: 5323: 5318: 5315: 5312: 5309: 5306: 5303: 5300: 5297: 5293: 5289: 5284: 5281: 5278: 5275: 5272: 5269: 5266: 5263: 5259: 5255: 5250: 5247: 5244: 5241: 5238: 5235: 5232: 5229: 5225: 5221: 5216: 5213: 5210: 5207: 5204: 5201: 5198: 5195: 5191: 5187: 5182: 5179: 5176: 5173: 5170: 5167: 5164: 5161: 5157: 5153: 5148: 5145: 5140: 5135: 5132: 5129: 5126: 5123: 5119: 5115: 5110: 5103: 5099: 5094: 5090: 5080: 5069: 5066: 5061: 5058: 5055: 5052: 5049: 5045: 5041: 5038: 5033: 5030: 5027: 5024: 5021: 5018: 5015: 5011: 5007: 5002: 4999: 4996: 4993: 4990: 4987: 4984: 4980: 4976: 4971: 4968: 4965: 4962: 4959: 4956: 4953: 4949: 4945: 4940: 4937: 4934: 4931: 4928: 4925: 4922: 4918: 4914: 4909: 4906: 4903: 4900: 4897: 4894: 4891: 4887: 4883: 4878: 4875: 4872: 4869: 4866: 4863: 4860: 4856: 4852: 4847: 4844: 4841: 4838: 4835: 4832: 4829: 4825: 4821: 4816: 4813: 4810: 4807: 4804: 4801: 4798: 4794: 4790: 4785: 4782: 4779: 4776: 4773: 4770: 4767: 4763: 4759: 4754: 4751: 4748: 4745: 4742: 4739: 4736: 4732: 4728: 4723: 4720: 4717: 4714: 4711: 4708: 4705: 4701: 4697: 4692: 4689: 4686: 4683: 4680: 4677: 4674: 4670: 4666: 4661: 4658: 4653: 4648: 4645: 4642: 4639: 4636: 4632: 4628: 4623: 4616: 4612: 4607: 4603: 4593: 4582: 4577: 4574: 4571: 4568: 4565: 4561: 4557: 4554: 4549: 4546: 4543: 4540: 4537: 4534: 4530: 4526: 4521: 4518: 4515: 4512: 4509: 4506: 4502: 4498: 4493: 4490: 4487: 4484: 4481: 4478: 4474: 4470: 4465: 4462: 4459: 4456: 4453: 4450: 4446: 4442: 4437: 4434: 4431: 4428: 4425: 4422: 4418: 4414: 4409: 4406: 4403: 4400: 4397: 4394: 4390: 4386: 4381: 4378: 4375: 4372: 4369: 4365: 4361: 4356: 4351: 4347: 4343: 4329: 4328: 4317: 4314: 4309: 4302: 4298: 4293: 4286: 4282: 4278: 4273: 4266: 4262: 4257: 4250: 4246: 4242: 4237: 4232: 4228: 4223: 4218: 4214: 4210: 4205: 4201: 4197: 4194: 4191: 4188: 4183: 4176: 4172: 4168: 4163: 4159: 4154: 4138: 4137: 4124: 4118: 4114: 4110: 4107: 4105: 4102: 4100: 4096: 4092: 4089: 4088: 4085: 4082: 4080: 4077: 4074: 4072: 4069: 4068: 4065: 4061: 4057: 4054: 4052: 4049: 4047: 4043: 4039: 4036: 4035: 4033: 4028: 4025: 4020: 4014: 4011: 4009: 4006: 4004: 4001: 4000: 3997: 3994: 3992: 3989: 3986: 3984: 3981: 3980: 3977: 3974: 3972: 3969: 3967: 3964: 3963: 3961: 3956: 3953: 3950: 3947: 3944: 3941: 3936: 3931: 3927: 3923: 3920: 3915: 3910: 3906: 3902: 3899: 3896: 3893: 3890: 3887: 3882: 3877: 3873: 3855: 3854: 3843: 3838: 3832: 3829: 3827: 3824: 3822: 3819: 3818: 3815: 3812: 3810: 3807: 3804: 3802: 3799: 3798: 3795: 3792: 3790: 3787: 3785: 3782: 3781: 3779: 3774: 3769: 3764: 3761: 3756: 3736:Note that the 3734: 3733: 3726: 3725: 3722: 3716: 3710: 3705: 3687: 3686: 3683: 3679: 3678: 3677: 3665: 3654: 3652: 3649: 3648: 3645: 3642: 3639: 3636: 3633: 3630: 3622: 3620: 3617: 3616: 3613: 3610: 3607: 3604: 3596: 3594: 3591: 3588: 3585: 3584: 3581: 3577: 3570: 3566: 3562: 3559: 3556: 3551: 3547: 3543: 3538: 3534: 3529: 3507: 3502: 3495: 3491: 3487: 3484: 3481: 3476: 3472: 3468: 3463: 3459: 3453: 3442:of the kernel 3427: 3423: 3419: 3416: 3413: 3408: 3404: 3400: 3395: 3391: 3386: 3371: 3368: 3367: 3366: 3352: 3346: 3343: 3341: 3338: 3336: 3333: 3332: 3329: 3326: 3324: 3321: 3319: 3316: 3315: 3312: 3309: 3307: 3304: 3302: 3299: 3298: 3296: 3289: 3286: 3262: 3256: 3253: 3251: 3248: 3246: 3243: 3242: 3239: 3236: 3234: 3231: 3228: 3226: 3223: 3222: 3219: 3216: 3214: 3211: 3209: 3206: 3205: 3203: 3196: 3193: 3169: 3163: 3160: 3158: 3155: 3153: 3150: 3149: 3146: 3143: 3141: 3138: 3136: 3133: 3132: 3129: 3126: 3124: 3121: 3119: 3116: 3115: 3113: 3106: 3103: 3092:first plane = 3083: 3082: 3081: 3067: 3061: 3058: 3056: 3053: 3051: 3048: 3047: 3044: 3041: 3039: 3036: 3034: 3031: 3030: 3027: 3024: 3022: 3019: 3017: 3014: 3013: 3011: 2987: 2981: 2978: 2976: 2973: 2971: 2968: 2967: 2964: 2961: 2959: 2956: 2953: 2951: 2948: 2947: 2944: 2941: 2939: 2936: 2934: 2931: 2930: 2928: 2904: 2898: 2895: 2893: 2890: 2888: 2885: 2884: 2881: 2878: 2876: 2873: 2871: 2868: 2867: 2864: 2861: 2859: 2856: 2854: 2851: 2850: 2848: 2837:first plane = 2817: 2812: 2809: 2806: 2801: 2788: 2774: 2768: 2765: 2763: 2760: 2758: 2755: 2754: 2751: 2748: 2746: 2743: 2740: 2738: 2735: 2734: 2731: 2728: 2726: 2723: 2721: 2718: 2717: 2715: 2710: 2705: 2700: 2697: 2692: 2656: 2651: 2648: 2643: 2631: 2630: 2616: 2610: 2607: 2605: 2602: 2600: 2597: 2596: 2593: 2590: 2588: 2585: 2582: 2580: 2577: 2576: 2573: 2570: 2568: 2565: 2563: 2560: 2559: 2557: 2552: 2547: 2542: 2539: 2534: 2521: 2507: 2501: 2498: 2496: 2493: 2490: 2488: 2485: 2484: 2482: 2477: 2472: 2467: 2460: 2457: 2433: 2430: 2420: 2417: 2411: 2408: 2392:Kronecker sums 2334: 2333: 2322: 2319: 2316: 2313: 2310: 2307: 2304: 2301: 2298: 2295: 2292: 2289: 2286: 2283: 2280: 2277: 2274: 2271: 2268: 2265: 2262: 2259: 2256: 2253: 2250: 2247: 2244: 2241: 2238: 2235: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2211: 2208: 2205: 2179: 2178: 2167: 2160: 2156: 2151: 2148: 2145: 2142: 2139: 2136: 2133: 2130: 2127: 2124: 2121: 2118: 2115: 2112: 2109: 2106: 2103: 2100: 2097: 2094: 2091: 2088: 2085: 2082: 2079: 2076: 2073: 2070: 2067: 2064: 2061: 2058: 2055: 2052: 2049: 2046: 2043: 2040: 2037: 2034: 2031: 2028: 2025: 2019: 2016: 2013: 2010: 2007: 2004: 2001: 1998: 1960: 1957: 1941: 1936: 1933: 1929: 1925: 1922: 1900: 1896: 1875: 1855: 1832: 1809: 1789: 1786: 1783: 1780: 1756: 1746: 1744: 1741: 1740: 1737: 1734: 1731: 1728: 1725: 1721: 1718: 1714: 1708: 1705: 1702: 1699: 1696: 1693: 1689: 1685: 1682: 1681: 1678: 1675: 1672: 1669: 1666: 1663: 1660: 1652: 1649: 1646: 1644: 1639: 1636: 1632: 1628: 1625: 1622: 1617: 1614: 1610: 1606: 1603: 1600: 1595: 1592: 1587: 1586: 1584: 1579: 1574: 1571: 1567: 1540: 1518: 1514: 1510: 1507: 1504: 1501: 1498: 1495: 1472: 1468: 1464: 1460: 1456: 1452: 1448: 1442: 1437: 1434: 1413:finite volumes 1388: 1368: 1342: 1338: 1317: 1314: 1292: 1289: 1285: 1262: 1259: 1255: 1234: 1214: 1203: 1202: 1191: 1188: 1183: 1179: 1175: 1170: 1166: 1162: 1159: 1154: 1151: 1147: 1143: 1140: 1137: 1132: 1129: 1125: 1121: 1118: 1115: 1110: 1106: 1097: 1093: 1089: 1085: 1080: 1075: 1071: 1067: 1064: 1061: 1038: 1018: 998: 995: 994: 993: 982: 979: 976: 973: 970: 965: 962: 959: 956: 953: 950: 947: 944: 940: 937: 933: 926: 923: 920: 916: 911: 908: 905: 902: 899: 896: 893: 890: 860: 857: 854: 851: 829: 826: 822: 810: 809: 797: 793: 790: 787: 784: 781: 778: 775: 772: 769: 765: 759: 756: 752: 746: 743: 740: 737: 734: 731: 728: 725: 721: 718: 714: 710: 707: 704: 701: 698: 695: 690: 686: 682: 659: 656: 653: 650: 647: 620: 617: 614: 611: 608: 605: 602: 582: 579: 559: 540:graph distance 527: 524: 521: 518: 515: 512: 501: 500: 488: 484: 481: 478: 475: 472: 469: 466: 463: 460: 456: 450: 447: 444: 441: 438: 435: 432: 429: 425: 422: 418: 414: 411: 408: 405: 402: 399: 396: 393: 381:is defined by 370: 350: 322: 319: 316: 313: 310: 290: 270: 250: 247: 244: 241: 238: 235: 232: 201: 198: 196: 193: 181:Laplace filter 124: 123: 38: 36: 29: 15: 9: 6: 4: 3: 2: 12217: 12206: 12203: 12201: 12198: 12196: 12193: 12191: 12188: 12186: 12183: 12181: 12178: 12177: 12175: 12166: 12162: 12158: 12153: 12152: 12141: 12135: 12131: 12130: 12125: 12121: 12117: 12116: 12108: 12102: 12098: 12097: 12092: 12086: 12078: 12074: 12069: 12064: 12060: 12056: 12052: 12048: 12044: 12037: 12028: 12023: 12019: 12015: 12011: 12007: 12003: 11996: 11988: 11982: 11978: 11974: 11968: 11960: 11954: 11950: 11946: 11942: 11935: 11927: 11923: 11919: 11915: 11911: 11907: 11903: 11899: 11892: 11885: 11884:0-7923-9418-6 11881: 11877: 11872: 11870: 11868: 11861: 11856: 11854: 11845: 11838: 11831: 11823: 11817: 11813: 11809: 11802: 11801: 11793: 11779: 11775: 11768: 11760: 11756: 11751: 11746: 11742: 11738: 11731: 11723: 11719: 11714: 11709: 11705: 11701: 11694: 11686: 11682: 11678: 11674: 11667: 11653: 11646: 11639: 11635: 11625: 11622: 11620: 11617: 11615: 11612: 11610: 11607: 11606: 11600: 11579: 11576: 11573: 11570: 11567: 11557: 11556: 11555: 11549: 11548: 11547: 11544: 11537: 11536: 11535: 11518: 11515: 11512: 11509: 11499: 11498: 11497: 11495: 11491: 11485: 11461: 11455: 11447: 11443: 11439: 11433: 11430: 11427: 11424: 11421: 11415: 11409: 11406: 11403: 11393: 11392: 11391: 11389: 11373: 11353: 11350: 11347: 11338: 11336: 11332: 11328: 11324: 11306: 11303: 11299: 11295: 11289: 11281: 11277: 11268: 11250: 11246: 11221: 11215: 11211: 11206: 11200: 11197: 11194: 11190: 11185: 11179: 11175: 11170: 11166: 11160: 11157: 11154: 11151: 11148: 11142: 11135: 11134: 11133: 11131: 11127: 11123: 11118: 11116: 11111: 11109: 11105: 11101: 11097: 11092: 11090: 11086: 11081: 11079: 11075: 11059: 11056: 11050: 11047: 11024: 11018: 11012: 11006: 11000: 10997: 10991: 10982: 10979: 10969: 10968: 10967: 10954: 10946: 10942: 10926: 10920: 10917: 10914: 10864:'gif' 10807:'gif' 9903: 9899: 9884: 9870: 9861: 9846: 9819: 9811: 9807: 9801: 9796: 9793: 9790: 9786: 9780: 9777: 9772: 9766: 9758: 9754: 9742: 9730: 9729: 9728: 9714: 9694: 9672: 9668: 9641: 9638: 9635: 9632: 9629: 9626: 9623: 9614: 9610: 9605: 9586: 9585: 9584: 9554: 9538: 9532: 9526: 9522: 9518: 9512: 9506: 9495: 9483: 9482: 9481: 9467: 9444: 9438: 9429: 9415: 9412: 9409: 9389: 9381: 9365: 9343: 9329:, the vector 9316: 9313: 9308: 9305: 9301: 9295: 9291: 9281: 9267: 9259: 9233: 9230: 9225: 9221: 9208: 9205: 9198: 9195: 9190: 9186: 9173: 9170: 9164: 9159: 9154: 9149: 9145: 9141: 9138: 9134: 9122: 9110: 9109: 9108: 9094: 9091: 9086: 9082: 9059: 9054: 9050: 9046: 9043: 9039: 9032: 9024: 9020: 9016: 9010: 9002: 8998: 8974: 8968: 8957: 8938: 8924: 8916: 8900: 8891: 8872: 8866: 8856: 8850: 8844: 8840: 8836: 8830: 8822: 8818: 8810: 8809: 8808: 8792: 8762: 8756: 8733: 8727: 8707: 8684: 8676: 8672: 8662: 8660: 8641: 8633: 8629: 8606: 8602: 8593: 8575: 8571: 8547: 8542: 8537: 8533: 8529: 8526: 8522: 8515: 8507: 8503: 8499: 8493: 8485: 8481: 8473: 8472: 8471: 8450: 8444: 8436: 8432: 8426: 8422: 8418: 8415: 8409: 8406: 8398: 8390: 8386: 8382: 8372: 8369: 8358: 8352: 8340: 8336: 8329: 8321: 8317: 8313: 8310: 8305: 8292: 8289: 8281: 8273: 8269: 8265: 8258: 8252: 8248: 8245: 8238: 8228: 8222: 8212: 8206: 8198: 8194: 8190: 8187: 8182: 8169: 8166: 8158: 8150: 8146: 8142: 8135: 8129: 8125: 8122: 8115: 8105: 8099: 8086: 8078: 8074: 8068: 8064: 8059: 8055: 8052: 8049: 8043: 8040: 8034: 8028: 8015: 8007: 8003: 7997: 7993: 7988: 7984: 7974: 7971: 7960: 7959: 7958: 7942: 7927: 7922: 7909: 7904: 7891: 7883: 7879: 7873: 7869: 7865: 7859: 7853: 7831: 7819: 7815: 7811: 7806: 7796: 7788: 7770: 7743: 7735: 7730: 7714: 7701: 7697: 7696:heat equation 7678: 7675: 7672: 7669: 7666: 7663: 7657: 7654: 7649: 7646: 7636: 7635: 7634: 7613: 7610: 7607: 7604: 7601: 7598: 7596: 7588: 7582: 7579: 7576: 7570: 7567: 7564: 7562: 7554: 7551: 7546: 7543: 7529: 7528: 7527: 7506: 7501: 7497: 7492: 7487: 7484: 7480: 7476: 7470: 7466: 7462: 7459: 7456: 7454: 7444: 7440: 7435: 7429: 7426: 7422: 7418: 7410: 7406: 7399: 7396: 7388: 7385: 7381: 7376: 7370: 7366: 7362: 7359: 7356: 7354: 7345: 7339: 7335: 7329: 7326: 7322: 7316: 7312: 7308: 7300: 7296: 7289: 7286: 7278: 7274: 7269: 7265: 7262: 7259: 7257: 7248: 7242: 7238: 7232: 7229: 7225: 7219: 7215: 7211: 7206: 7203: 7199: 7193: 7189: 7183: 7179: 7174: 7170: 7167: 7164: 7162: 7153: 7147: 7143: 7139: 7134: 7130: 7125: 7119: 7116: 7112: 7106: 7102: 7098: 7095: 7092: 7090: 7082: 7079: 7072: 7068: 7064: 7050: 7049: 7048: 7034: 7014: 6994: 6972: 6968: 6964: 6959: 6955: 6934: 6914: 6906: 6890: 6868: 6864: 6855: 6839: 6825: 6823: 6820:, called the 6819: 6815: 6811: 6807: 6788: 6781: 6777: 6766: 6763: 6760: 6754: 6751: 6745: 6739: 6733: 6724: 6718: 6715: 6709: 6706: 6703: 6697: 6686: 6680: 6672: 6664: 6660: 6651: 6646: 6631: 6630: 6629: 6627: 6617: 6600: 6597: 6594: 6591: 6565: 6562: 6559: 6536: 6516: 6513: 6510: 6507: 6496: 6486: 6484: 6480: 6464: 6456: 6452: 6429: 6406: 6384: 6380: 6359: 6339: 6312: 6302: 6295: 6286: 6264: 6261: 6250: 6243: 6234: 6225: 6220: 6206: 6203: 6198: 6192: 6189: 6185: 6182: 6177: 6173: 6165: 6155: 6145: 6140: 6128: 6127: 6126: 6104: 6095: 6075: 6067: 6051: 6029: 6019: 6015: 6011: 6008: 6005: 6000: 5996: 5992: 5987: 5983: 5976: 5967: 5944: 5936: 5935:sinc function 5920: 5895: 5885: 5878: 5869: 5860: 5857: 5845: 5834: 5830: 5809: 5787: 5783: 5762: 5742: 5722: 5719: 5714: 5710: 5677: 5666: 5662: 5656: 5652: 5646: 5643: 5640: 5636: 5632: 5620: 5611: 5604: 5603: 5602: 5588: 5568: 5565: 5562: 5540: 5536: 5532: 5523: 5491: 5482: 5454: 5446: 5443: 5440: 5437: 5434: 5430: 5426: 5423: 5418: 5415: 5412: 5409: 5406: 5403: 5400: 5397: 5393: 5389: 5384: 5381: 5378: 5375: 5372: 5369: 5366: 5363: 5359: 5355: 5350: 5347: 5344: 5341: 5338: 5335: 5332: 5329: 5325: 5321: 5316: 5313: 5310: 5307: 5304: 5301: 5298: 5295: 5291: 5287: 5282: 5279: 5276: 5273: 5270: 5267: 5264: 5261: 5257: 5253: 5248: 5245: 5242: 5239: 5236: 5233: 5230: 5227: 5223: 5219: 5214: 5211: 5208: 5205: 5202: 5199: 5196: 5193: 5189: 5185: 5180: 5177: 5174: 5171: 5168: 5165: 5162: 5159: 5155: 5146: 5143: 5138: 5133: 5130: 5127: 5124: 5121: 5113: 5108: 5101: 5097: 5081: 5067: 5059: 5056: 5053: 5050: 5047: 5043: 5039: 5036: 5031: 5028: 5025: 5022: 5019: 5016: 5013: 5009: 5005: 5000: 4997: 4994: 4991: 4988: 4985: 4982: 4978: 4974: 4969: 4966: 4963: 4960: 4957: 4954: 4951: 4947: 4943: 4938: 4935: 4932: 4929: 4926: 4923: 4920: 4916: 4912: 4907: 4904: 4901: 4898: 4895: 4892: 4889: 4885: 4881: 4876: 4873: 4870: 4867: 4864: 4861: 4858: 4854: 4850: 4845: 4842: 4839: 4836: 4833: 4830: 4827: 4823: 4819: 4814: 4811: 4808: 4805: 4802: 4799: 4796: 4792: 4788: 4783: 4780: 4777: 4774: 4771: 4768: 4765: 4761: 4757: 4752: 4749: 4746: 4743: 4740: 4737: 4734: 4730: 4726: 4721: 4718: 4715: 4712: 4709: 4706: 4703: 4699: 4695: 4690: 4687: 4684: 4681: 4678: 4675: 4672: 4668: 4659: 4656: 4651: 4646: 4643: 4640: 4637: 4634: 4626: 4621: 4614: 4610: 4594: 4580: 4575: 4572: 4569: 4566: 4563: 4559: 4555: 4552: 4547: 4544: 4541: 4538: 4535: 4532: 4528: 4524: 4519: 4516: 4513: 4510: 4507: 4504: 4500: 4496: 4491: 4488: 4485: 4482: 4479: 4476: 4472: 4468: 4463: 4460: 4457: 4454: 4451: 4448: 4444: 4440: 4435: 4432: 4429: 4426: 4423: 4420: 4416: 4412: 4407: 4404: 4401: 4398: 4395: 4392: 4388: 4384: 4379: 4376: 4373: 4370: 4367: 4359: 4354: 4349: 4334: 4333: 4332: 4315: 4307: 4300: 4296: 4284: 4280: 4276: 4271: 4264: 4260: 4248: 4244: 4240: 4235: 4230: 4216: 4212: 4208: 4203: 4199: 4195: 4192: 4186: 4181: 4174: 4170: 4166: 4161: 4157: 4144: 4143: 4142: 4122: 4116: 4112: 4108: 4103: 4098: 4094: 4090: 4083: 4078: 4075: 4070: 4063: 4059: 4055: 4050: 4045: 4041: 4037: 4031: 4026: 4023: 4018: 4012: 4007: 4002: 3995: 3990: 3987: 3982: 3975: 3970: 3965: 3959: 3951: 3948: 3945: 3939: 3934: 3929: 3921: 3918: 3913: 3908: 3897: 3894: 3891: 3885: 3880: 3875: 3863: 3862: 3861: 3858: 3841: 3836: 3830: 3825: 3820: 3813: 3808: 3805: 3800: 3793: 3788: 3783: 3777: 3772: 3767: 3762: 3759: 3744: 3743: 3742: 3739: 3680: 3650: 3643: 3640: 3637: 3634: 3631: 3628: 3618: 3611: 3608: 3605: 3602: 3592: 3589: 3586: 3579: 3575: 3568: 3564: 3560: 3557: 3554: 3549: 3545: 3541: 3536: 3532: 3527: 3519: 3518: 3505: 3500: 3493: 3489: 3485: 3482: 3479: 3474: 3470: 3466: 3461: 3457: 3425: 3421: 3417: 3414: 3411: 3406: 3402: 3398: 3393: 3389: 3384: 3375: 3369: 3350: 3344: 3339: 3334: 3327: 3322: 3317: 3310: 3305: 3300: 3294: 3287: 3284: 3260: 3254: 3249: 3244: 3237: 3232: 3229: 3224: 3217: 3212: 3207: 3201: 3194: 3191: 3167: 3161: 3156: 3151: 3144: 3139: 3134: 3127: 3122: 3117: 3111: 3104: 3101: 3091: 3090: 3088: 3084: 3065: 3059: 3054: 3049: 3042: 3037: 3032: 3025: 3020: 3015: 3009: 2985: 2979: 2974: 2969: 2962: 2957: 2954: 2949: 2942: 2937: 2932: 2926: 2902: 2896: 2891: 2886: 2879: 2874: 2869: 2862: 2857: 2852: 2846: 2836: 2835: 2834:is given by: 2833: 2815: 2810: 2807: 2804: 2789: 2772: 2766: 2761: 2756: 2749: 2744: 2741: 2736: 2729: 2724: 2719: 2713: 2708: 2703: 2698: 2695: 2680: 2679: 2678: 2676: 2672: 2654: 2649: 2646: 2614: 2608: 2603: 2598: 2591: 2586: 2583: 2578: 2571: 2566: 2561: 2555: 2550: 2545: 2540: 2537: 2522: 2505: 2499: 2494: 2491: 2486: 2480: 2475: 2470: 2465: 2455: 2443: 2442: 2441: 2439: 2429: 2426: 2416: 2407: 2405: 2401: 2397: 2393: 2389: 2388:regular grids 2386: 2381: 2379: 2375: 2371: 2367: 2363: 2359: 2355: 2351: 2347: 2343: 2339: 2320: 2311: 2308: 2305: 2302: 2299: 2293: 2287: 2284: 2281: 2278: 2275: 2269: 2263: 2260: 2257: 2254: 2251: 2245: 2239: 2236: 2233: 2227: 2221: 2218: 2215: 2212: 2209: 2196: 2195: 2194: 2192: 2188: 2184: 2165: 2158: 2154: 2146: 2143: 2140: 2134: 2131: 2128: 2122: 2119: 2116: 2113: 2110: 2104: 2101: 2095: 2092: 2089: 2086: 2083: 2077: 2074: 2068: 2065: 2062: 2059: 2056: 2050: 2047: 2041: 2038: 2035: 2032: 2029: 2023: 2017: 2011: 2008: 2005: 1999: 1989: 1988: 1987: 1985: 1982: 1978: 1974: 1970: 1966: 1956: 1953: 1939: 1934: 1931: 1927: 1923: 1920: 1898: 1894: 1873: 1853: 1846: 1830: 1821: 1807: 1784: 1778: 1769: 1742: 1735: 1732: 1729: 1726: 1719: 1716: 1712: 1703: 1697: 1694: 1691: 1683: 1676: 1670: 1664: 1661: 1658: 1650: 1647: 1637: 1634: 1630: 1626: 1623: 1620: 1615: 1612: 1608: 1604: 1601: 1593: 1590: 1582: 1577: 1572: 1569: 1565: 1556: 1555:with entries 1554: 1538: 1516: 1508: 1499: 1496: 1493: 1466: 1458: 1450: 1435: 1432: 1423: 1421: 1418: 1414: 1410: 1406: 1402: 1386: 1366: 1358: 1340: 1336: 1315: 1312: 1290: 1287: 1283: 1260: 1257: 1253: 1232: 1212: 1189: 1181: 1177: 1173: 1168: 1164: 1152: 1149: 1145: 1141: 1138: 1135: 1130: 1127: 1123: 1119: 1116: 1108: 1104: 1095: 1091: 1087: 1083: 1078: 1073: 1065: 1052: 1051: 1050: 1036: 1016: 1008: 1004: 980: 974: 968: 963: 960: 954: 951: 948: 942: 938: 935: 931: 924: 921: 918: 914: 909: 903: 894: 891: 881: 880: 879: 877: 872: 858: 855: 852: 849: 827: 824: 820: 795: 788: 782: 779: 773: 767: 763: 757: 754: 750: 744: 741: 735: 732: 729: 723: 719: 716: 712: 708: 702: 693: 688: 673: 672: 671: 657: 651: 648: 645: 636: 634: 615: 606: 580: 557: 549: 545: 541: 522: 519: 516: 510: 486: 479: 473: 470: 464: 458: 454: 448: 445: 439: 436: 433: 427: 423: 420: 416: 412: 406: 397: 384: 383: 382: 368: 340: 336: 320: 314: 311: 308: 288: 268: 245: 242: 239: 233: 230: 221: 219: 215: 214:regular graph 211: 207: 192: 190: 186: 182: 178: 174: 170: 166: 162: 158: 153: 151: 147: 146:discrete grid 143: 139: 135: 131: 120: 117: 109: 106:December 2007 98: 95: 91: 88: 84: 81: 77: 74: 70: 67: – 66: 62: 61:Find sources: 55: 51: 45: 44: 39:This article 37: 33: 28: 27: 22: 12185:Graph theory 12161:the original 12128: 12095: 12085: 12050: 12046: 12036: 12009: 12005: 11995: 11976: 11973:Newman, Mark 11967: 11943:. Springer. 11940: 11934: 11901: 11897: 11891: 11843: 11830: 11799: 11792: 11781:. Retrieved 11777: 11774:"LoG Filter" 11767: 11740: 11736: 11730: 11703: 11699: 11693: 11676: 11666: 11655:. Retrieved 11651: 11638: 11594: 11553: 11545: 11541: 11533: 11487: 11387: 11339: 11334: 11330: 11326: 11322: 11237: 11119: 11112: 11093: 11089:self-adjoint 11084: 11082: 11073: 11039: 10944: 10906: 9902:conditions. 9900: 9876: 9838: 9659: 9582: 9430: 9402:independent 9282: 9255: 8944: 8892: 8889: 8663: 8658: 8591: 8562: 8469: 7925: 7923: 7786: 7731: 7699: 7693: 7633:which gives 7632: 7525: 6831: 6803: 6623: 6492: 6331: 5701: 5474: 4330: 4139: 3859: 3856: 3737: 3735: 3370: 2632: 2435: 2422: 2413: 2404:eigenvectors 2382: 2373: 2369: 2365: 2357: 2353: 2349: 2341: 2337: 2335: 2190: 2186: 2182: 2180: 1976: 1962: 1954: 1844: 1822: 1770: 1557: 1552: 1424: 1356: 1204: 1029:at a vertex 1000: 875: 873: 811: 637: 632: 631:is just the 543: 502: 222: 217: 205: 203: 154: 133: 127: 112: 103: 93: 86: 79: 72: 60: 48:Please help 43:verification 40: 11337:otherwise. 9946:% The image 6818:edge filter 5937:defined in 3745:2D filter: 2790:3D filter: 2681:2D filter: 2523:2D filter: 2444:1D filter: 2438:convolution 2400:eigenvalues 1845:mass matrix 1357:vertex area 550:. That is, 195:Definitions 161:Ising model 130:mathematics 21:Z-transform 12174:Categories 12120:Sunada, T. 12053:: 122112. 12012:: 106383. 11783:2019-12-01 11657:2019-12-01 11630:References 11534:in words, 11340:For fixed 10192:&& 10180:&& 10168:&& 3720:for which 3657:otherwise, 3085:and using 1971:or by the 1486:such that 361:acting on 281:and edges 185:clustering 76:newspapers 12093:(2002) , 12077:244652819 11926:123145969 11918:0749-159X 11745:CiteSeerX 11708:CiteSeerX 11577:− 11574:ϕ 11568:ϕ 11565:Δ 11516:ϕ 11510:ϕ 11507:Δ 11444:δ 11434:λ 11410:λ 11407:− 11374:λ 11351:∈ 11300:δ 11278:δ 11247:δ 11212:δ 11201:λ 11198:− 11176:δ 11161:λ 11054:Δ 11013:ϕ 10983:ϕ 10955:ϕ 10941:potential 10924:→ 10918:: 9885:ϕ 9847:ϕ 9787:∑ 9755:ϕ 9749:∞ 9746:→ 9695:ϕ 9669:ϕ 9636:… 9507:ϕ 9502:∞ 9499:→ 9480:vertices 9410:λ 9378:disjoint 9292:∑ 9222:λ 9187:λ 9146:λ 9139:− 9129:∞ 9126:→ 9083:λ 9051:λ 9044:− 8969:ϕ 8964:∞ 8961:→ 8845:ϕ 8757:ϕ 8728:ϕ 8700:for each 8603:λ 8572:λ 8534:λ 8527:− 8423:λ 8367:⇒ 8337:λ 8249:∑ 8126:∑ 8065:∑ 7994:∑ 7870:∑ 7854:ϕ 7816:λ 7789:(so that 7744:ϕ 7711:∇ 7673:ϕ 7650:ϕ 7611:ϕ 7602:− 7589:ϕ 7580:− 7568:− 7547:ϕ 7498:ϕ 7467:∑ 7460:− 7441:ϕ 7419:− 7400:⁡ 7382:δ 7367:∑ 7360:− 7336:ϕ 7313:∑ 7309:− 7290:⁡ 7275:ϕ 7263:− 7239:ϕ 7216:∑ 7212:− 7190:∑ 7180:ϕ 7168:− 7144:ϕ 7140:− 7131:ϕ 7103:∑ 7096:− 7069:ϕ 6987:if nodes 6969:ϕ 6965:− 6956:ϕ 6865:ϕ 6840:ϕ 6778:ϵ 6767:ϵ 6764:− 6752:− 6734:− 6716:− 6710:ϵ 6684:→ 6681:ϵ 6657:∂ 6643:∂ 6592:− 6514:− 6505:Δ 6433:¯ 6306:¯ 6290:¯ 6254:¯ 6244:− 6238:¯ 6226:μ 6217:∇ 6190:∈ 6178:∑ 6159:¯ 6137:∇ 6108:¯ 6052:μ 5971:¯ 5921:μ 5889:¯ 5879:− 5873:¯ 5861:μ 5849:¯ 5831:μ 5784:μ 5720:∈ 5681:¯ 5663:μ 5644:∈ 5637:∑ 5624:¯ 5566:∈ 5533:∈ 5527:¯ 5495:¯ 5424:− 5382:− 5339:− 5314:− 5305:− 5262:− 5246:− 5228:− 5203:− 5194:− 5178:− 5169:− 5160:− 5098:× 5093:∇ 5037:− 4998:− 4958:− 4936:− 4927:− 4874:− 4828:− 4812:− 4797:− 4744:− 4704:− 4682:− 4673:− 4606:∇ 4553:− 4517:− 4455:− 4393:− 4346:∇ 4297:× 4292:∇ 4281:γ 4256:∇ 4245:γ 4227:∇ 4213:γ 4209:− 4200:γ 4196:− 4171:γ 4158:γ 4153:∇ 4076:− 4027:γ 3988:− 3952:γ 3949:− 3930:× 3926:∇ 3922:γ 3905:∇ 3898:γ 3895:− 3876:γ 3872:∇ 3806:− 3638:− 3587:− 3558:… 3483:… 3415:… 3230:− 2955:− 2742:− 2584:− 2492:− 2459:→ 2285:− 2213:− 2129:− 2093:− 2033:− 2018:≈ 1997:Δ 1965:Laplacian 1932:− 1749:otherwise 1695:∈ 1688:∑ 1684:− 1662:∈ 1631:β 1627:⁡ 1609:α 1605:⁡ 1506:Δ 1459:× 1436:∈ 1284:β 1254:α 1174:− 1146:β 1142:⁡ 1124:α 1120:⁡ 1105:∑ 1079:≡ 1063:Δ 969:ϕ 932:∑ 922:⁡ 895:ϕ 856:∈ 821:γ 783:ϕ 780:− 768:ϕ 751:γ 713:∑ 694:ϕ 689:γ 685:Δ 655:→ 649:: 646:γ 607:ϕ 604:Δ 581:ϕ 578:Δ 558:ϕ 474:ϕ 471:− 459:ϕ 417:∑ 398:ϕ 395:Δ 369:ϕ 349:Δ 318:→ 312:: 309:ϕ 12122:(2008). 11975:(2010). 11603:See also 11222:⟩ 11171:⟨ 10738:frame2im 10720:getframe 9555:⟩ 9523:⟨ 9107:, since 8873:⟩ 8841:⟨ 8664:To find 6927:to node 6856:, where 6832:Suppose 6620:Theorems 6489:Spectrum 6265:′ 6207:′ 6186:′ 5755:on grid 3625:if 3599:if 3374:D filter 2342:negative 1823:And let 1003:manifold 335:function 218:negative 12055:Bibcode 12014:Bibcode 11846:: 1–16. 11072:is the 10840:imwrite 10783:imwrite 10753:rgb2ind 10696:sprintf 10672:imagesc 10645:reshape 6808:and in 2189:, 1913:. Then 1355:is the 538:is the 157:physics 90:scholar 12136: 12103: 12075: 11983: 11955: 11924: 11916: 11882: 11818: 11747: 11710: 11238:where 11104:posets 10540:'* 10252:index2 10204:index2 10096:length 9583:where 9258:kernel 7394: 7284: 6481:, and 5702:where 4331:where 3681:where 2830:using 1866:whose 1771:Where 1531:. Let 1415:, and 1328:, and 812:where 503:where 301:. Let 210:graphs 132:, the 92: 85: 78: 71: 63: 12073:S2CID 11922:S2CID 11840:(PDF) 11804:(PDF) 11648:(PDF) 11040:Then 10939:be a 10846:imind 10789:imind 10744:frame 10714:frame 10690:title 10684:caxis 10384:zeros 10246:index 10198:<= 10174:<= 10045:index 9955:zeros 9928:zeros 6854:graph 6399:, in 5913:with 333:be a 144:or a 142:graph 97:JSTOR 83:books 12134:ISBN 12101:ISBN 11981:ISBN 11953:ISBN 11914:ISSN 11880:ISBN 11816:ISBN 11366:and 11333:and 11132:by 11120:The 11106:and 11094:The 10907:Let 10837:else 10570:0.05 10528:(:); 10360:diag 10288:diag 10234:newy 10213:newx 10195:newy 10186:> 10183:newy 10171:newx 10162:> 10159:newx 10132:newy 10108:newx 9196:> 7007:and 5775:and 3089:by: 2767:0.25 2757:0.25 2730:0.25 2720:0.25 2402:and 1275:and 339:ring 223:Let 208:for 187:and 163:and 69:news 12063:doi 12051:183 12022:doi 12010:153 11945:doi 11906:doi 11808:doi 11755:doi 11718:doi 11681:doi 11325:if 10897:end 10894:end 10888:0.1 10831:0.1 10819:inf 10765:256 10711:)); 10687:(); 10678:Phi 10651:Phi 10639:Phi 10630:Phi 10618:Phi 10594:exp 10588:C0V 10582:Phi 10555:for 10531:C0V 10339:eig 10327:Adj 10321:Deg 10309:)); 10300:Adj 10294:sum 10282:Deg 10276:end 10273:end 10270:end 10267:end 10240:Adj 10081:for 10027:for 10009:for 9949:Adj 9739:lim 9687:of 9492:lim 9260:of 9119:lim 8954:lim 8590:of 7785:of 7397:deg 7287:deg 6677:lim 6529:on 6068:in 3730:= 0 3700:or 2762:0.5 2750:0.5 2737:0.5 2725:0.5 1624:cot 1602:cot 1422:). 1359:of 1225:of 1139:cot 1117:cot 919:deg 128:In 52:by 12176:: 12126:. 12071:. 12061:. 12049:. 12045:. 12020:. 12008:. 12004:. 11951:. 11920:. 11912:. 11902:22 11900:. 11878:, 11866:^ 11852:^ 11842:. 11814:. 11776:. 11753:. 11741:33 11739:. 11716:. 11704:33 11702:. 11675:. 11650:. 11580:2. 11110:. 11091:. 11080:. 10891:); 10852:cm 10834:); 10795:cm 10777:== 10771:if 10768:); 10759:im 10747:); 10732:im 10729:); 10681:); 10666:); 10612:); 10591:.* 10543:C0 10525:C0 10519:C0 10504:13 10480:C0 10474:10 10465:15 10459:10 10453:15 10447:10 10441:C0 10402:C0 10399:); 10378:C0 10369:); 10348:); 10156:if 10153:); 10150:ne 10144:dy 10129:); 10126:ne 10120:dx 10102:dx 10084:ne 10000:dy 9991:dx 9982:); 9943:); 9913:20 9280:. 9215:if 9180:if 8807:; 7679:0. 7047:, 6824:. 5040:12 3696:, 3694:−1 3288:26 3233:88 3195:26 3105:26 2372:, 2356:, 1820:. 1411:, 1245:, 878:: 871:. 633:v' 152:. 12142:. 12079:. 12065:: 12057:: 12030:. 12024:: 12016:: 11989:. 11961:. 11947:: 11928:. 11908:: 11886:. 11824:. 11810:: 11786:. 11761:. 11757:: 11724:. 11720:: 11687:. 11683:: 11660:. 11597:8 11571:= 11519:, 11513:= 11462:. 11459:) 11456:v 11453:( 11448:w 11440:= 11437:) 11431:; 11428:w 11425:, 11422:v 11419:( 11416:G 11413:) 11404:H 11401:( 11388:v 11354:V 11348:w 11335:0 11331:w 11329:= 11327:v 11323:1 11307:v 11304:w 11296:= 11293:) 11290:v 11287:( 11282:w 11251:w 11216:w 11207:| 11195:H 11191:1 11186:| 11180:v 11167:= 11164:) 11158:; 11155:w 11152:, 11149:v 11146:( 11143:G 11085:H 11060:P 11057:+ 11051:= 11048:H 11025:. 11022:) 11019:v 11016:( 11010:) 11007:v 11004:( 11001:P 10998:= 10995:) 10992:v 10989:( 10986:) 10980:P 10977:( 10945:P 10927:R 10921:V 10915:P 10885:, 10879:, 10873:, 10867:, 10861:, 10855:, 10849:, 10843:( 10828:, 10822:, 10816:, 10810:, 10804:, 10798:, 10792:, 10786:( 10780:0 10774:t 10762:, 10756:( 10750:= 10741:( 10735:= 10726:1 10723:( 10717:= 10708:t 10705:, 10699:( 10693:( 10675:( 10663:N 10660:, 10657:N 10654:, 10648:( 10642:= 10633:; 10627:* 10624:V 10621:= 10609:t 10606:* 10603:D 10600:- 10597:( 10585:= 10576:5 10573:: 10567:: 10564:0 10561:= 10558:t 10546:; 10537:V 10534:= 10522:= 10516:; 10513:7 10510:= 10507:) 10501:: 10498:8 10495:, 10492:5 10489:: 10486:2 10483:( 10477:; 10471:= 10468:) 10462:: 10456:, 10450:: 10444:( 10438:; 10435:5 10432:= 10429:) 10426:5 10423:: 10420:2 10417:, 10414:5 10411:: 10408:2 10405:( 10396:N 10393:, 10390:N 10387:( 10381:= 10366:D 10363:( 10357:= 10354:D 10345:L 10342:( 10336:= 10330:; 10324:- 10318:= 10315:L 10306:2 10303:, 10297:( 10291:( 10285:= 10264:; 10261:1 10258:= 10255:) 10249:, 10243:( 10237:; 10231:+ 10228:N 10225:* 10222:) 10219:1 10216:- 10210:( 10207:= 10201:N 10189:0 10177:N 10165:0 10147:( 10141:+ 10138:y 10135:= 10123:( 10117:+ 10114:x 10111:= 10105:) 10099:( 10093:: 10090:1 10087:= 10078:; 10075:y 10072:+ 10069:N 10066:* 10063:) 10060:1 10057:- 10054:x 10051:( 10048:= 10042:N 10039:: 10036:1 10033:= 10030:y 10024:N 10021:: 10018:1 10015:= 10012:x 10006:; 10003:= 9997:; 9994:= 9979:N 9976:* 9973:N 9970:, 9967:N 9964:* 9961:N 9958:( 9952:= 9940:N 9937:, 9934:N 9931:( 9925:= 9922:A 9916:; 9910:= 9907:N 9835:. 9823:) 9820:0 9817:( 9812:i 9808:c 9802:N 9797:1 9794:= 9791:i 9781:N 9778:1 9773:= 9770:) 9767:t 9764:( 9759:j 9743:t 9715:j 9673:j 9645:] 9642:1 9639:, 9633:, 9630:1 9627:, 9624:1 9621:[ 9615:N 9611:1 9606:= 9600:1 9596:v 9565:1 9561:v 9548:1 9544:v 9539:, 9536:) 9533:0 9530:( 9527:c 9519:= 9516:) 9513:t 9510:( 9496:t 9468:N 9448:) 9445:0 9442:( 9439:c 9416:0 9413:= 9390:k 9366:k 9344:1 9339:v 9317:0 9314:= 9309:j 9306:i 9302:L 9296:j 9268:L 9234:0 9231:= 9226:i 9209:, 9206:1 9199:0 9191:i 9174:, 9171:0 9165:{ 9160:= 9155:t 9150:i 9142:k 9135:e 9123:t 9095:0 9092:= 9087:i 9060:t 9055:i 9047:k 9040:e 9036:) 9033:0 9030:( 9025:i 9021:c 9017:= 9014:) 9011:t 9008:( 9003:i 8999:c 8978:) 8975:t 8972:( 8958:t 8925:L 8901:L 8886:. 8867:i 8862:v 8857:, 8854:) 8851:0 8848:( 8837:= 8834:) 8831:0 8828:( 8823:i 8819:c 8793:i 8788:v 8766:) 8763:0 8760:( 8737:) 8734:0 8731:( 8708:i 8688:) 8685:0 8682:( 8677:i 8673:c 8659:t 8645:) 8642:0 8639:( 8634:i 8630:c 8607:i 8592:L 8576:i 8548:. 8543:t 8538:i 8530:k 8523:e 8519:) 8516:0 8513:( 8508:i 8504:c 8500:= 8497:) 8494:t 8491:( 8486:i 8482:c 8451:, 8448:) 8445:t 8442:( 8437:i 8433:c 8427:i 8419:k 8416:+ 8410:t 8407:d 8402:) 8399:t 8396:( 8391:i 8387:c 8383:d 8373:= 8370:0 8359:] 8353:i 8348:v 8341:i 8333:) 8330:t 8327:( 8322:i 8318:c 8314:k 8311:+ 8306:i 8301:v 8293:t 8290:d 8285:) 8282:t 8279:( 8274:i 8270:c 8266:d 8259:[ 8253:i 8239:= 8229:] 8223:i 8218:v 8213:L 8210:) 8207:t 8204:( 8199:i 8195:c 8191:k 8188:+ 8183:i 8178:v 8170:t 8167:d 8162:) 8159:t 8156:( 8151:i 8147:c 8143:d 8136:[ 8130:i 8116:= 8106:) 8100:i 8095:v 8090:) 8087:t 8084:( 8079:i 8075:c 8069:i 8060:( 8056:L 8053:k 8050:+ 8044:t 8041:d 8035:) 8029:i 8024:v 8019:) 8016:t 8013:( 8008:i 8004:c 7998:i 7989:( 7985:d 7975:= 7972:0 7943:i 7938:v 7926:L 7910:. 7905:i 7900:v 7895:) 7892:t 7889:( 7884:i 7880:c 7874:i 7866:= 7863:) 7860:t 7857:( 7832:i 7827:v 7820:i 7812:= 7807:i 7802:v 7797:L 7787:L 7771:i 7766:v 7715:2 7700:L 7676:= 7670:L 7667:k 7664:+ 7658:t 7655:d 7647:d 7614:, 7608:L 7605:k 7599:= 7586:) 7583:A 7577:D 7574:( 7571:k 7565:= 7555:t 7552:d 7544:d 7507:. 7502:j 7493:) 7488:j 7485:i 7481:L 7477:( 7471:j 7463:k 7457:= 7445:j 7436:) 7430:j 7427:i 7423:A 7416:) 7411:i 7407:v 7403:( 7389:j 7386:i 7377:( 7371:j 7363:k 7357:= 7346:) 7340:j 7330:j 7327:i 7323:A 7317:j 7306:) 7301:i 7297:v 7293:( 7279:i 7270:( 7266:k 7260:= 7249:) 7243:j 7233:j 7230:i 7226:A 7220:j 7207:j 7204:i 7200:A 7194:j 7184:i 7175:( 7171:k 7165:= 7154:) 7148:j 7135:i 7126:( 7120:j 7117:i 7113:A 7107:j 7099:k 7093:= 7083:t 7080:d 7073:i 7065:d 7035:k 7015:j 6995:i 6973:j 6960:i 6935:j 6915:i 6891:i 6869:i 6789:. 6782:2 6773:] 6770:) 6761:x 6758:( 6755:F 6749:) 6746:x 6743:( 6740:F 6737:[ 6731:] 6728:) 6725:x 6722:( 6719:F 6713:) 6707:+ 6704:x 6701:( 6698:F 6695:[ 6687:0 6673:= 6665:2 6661:x 6652:F 6647:2 6604:] 6601:1 6598:, 6595:1 6589:[ 6569:] 6566:2 6563:, 6560:0 6557:[ 6537:Z 6517:M 6511:I 6508:= 6465:n 6453:/ 6430:r 6407:n 6385:k 6381:f 6360:f 6340:K 6313:k 6303:r 6296:= 6287:r 6279:| 6274:) 6271:) 6262:k 6251:r 6235:r 6229:( 6221:2 6213:( 6204:k 6199:f 6193:K 6183:k 6174:= 6171:) 6166:k 6156:r 6149:( 6146:f 6141:2 6114:) 6105:r 6099:( 6096:f 6076:n 6030:T 6026:) 6020:n 6016:x 6012:. 6009:. 6006:. 6001:2 5997:x 5993:, 5988:1 5984:x 5980:( 5977:= 5968:r 5945:n 5901:) 5896:k 5886:r 5870:r 5864:( 5858:= 5855:) 5846:r 5840:( 5835:k 5810:K 5788:k 5763:K 5743:f 5723:R 5715:k 5711:f 5687:) 5678:r 5672:( 5667:k 5657:k 5653:f 5647:K 5641:k 5633:= 5630:) 5621:r 5615:( 5612:f 5589:f 5569:R 5563:f 5541:n 5537:R 5524:r 5501:) 5492:r 5486:( 5483:f 5455:. 5452:) 5447:0 5444:, 5441:0 5438:, 5435:0 5431:f 5427:8 5419:1 5416:+ 5413:, 5410:1 5407:+ 5404:, 5401:1 5398:+ 5394:f 5390:+ 5385:1 5379:, 5376:1 5373:+ 5370:, 5367:1 5364:+ 5360:f 5356:+ 5351:1 5348:+ 5345:, 5342:1 5336:, 5333:1 5330:+ 5326:f 5322:+ 5317:1 5311:, 5308:1 5302:, 5299:1 5296:+ 5292:f 5288:+ 5283:1 5280:+ 5277:, 5274:1 5271:+ 5268:, 5265:1 5258:f 5254:+ 5249:1 5243:, 5240:1 5237:+ 5234:, 5231:1 5224:f 5220:+ 5215:1 5212:+ 5209:, 5206:1 5200:, 5197:1 5190:f 5186:+ 5181:1 5175:, 5172:1 5166:, 5163:1 5156:f 5152:( 5147:4 5144:1 5139:= 5134:0 5131:, 5128:0 5125:, 5122:0 5118:) 5114:f 5109:2 5102:3 5089:( 5068:, 5065:) 5060:0 5057:, 5054:0 5051:, 5048:0 5044:f 5032:1 5029:+ 5026:, 5023:1 5020:+ 5017:, 5014:0 5010:f 5006:+ 5001:1 4995:, 4992:1 4989:+ 4986:, 4983:0 4979:f 4975:+ 4970:1 4967:+ 4964:, 4961:1 4955:, 4952:0 4948:f 4944:+ 4939:1 4933:, 4930:1 4924:, 4921:0 4917:f 4913:+ 4908:1 4905:+ 4902:, 4899:0 4896:, 4893:1 4890:+ 4886:f 4882:+ 4877:1 4871:, 4868:0 4865:, 4862:1 4859:+ 4855:f 4851:+ 4846:1 4843:+ 4840:, 4837:0 4834:, 4831:1 4824:f 4820:+ 4815:1 4809:, 4806:0 4803:, 4800:1 4793:f 4789:+ 4784:0 4781:, 4778:1 4775:+ 4772:, 4769:1 4766:+ 4762:f 4758:+ 4753:0 4750:, 4747:1 4741:, 4738:1 4735:+ 4731:f 4727:+ 4722:0 4719:, 4716:1 4713:+ 4710:, 4707:1 4700:f 4696:+ 4691:0 4688:, 4685:1 4679:, 4676:1 4669:f 4665:( 4660:4 4657:1 4652:= 4647:0 4644:, 4641:0 4638:, 4635:0 4631:) 4627:f 4622:2 4615:3 4611:+ 4602:( 4581:, 4576:0 4573:, 4570:0 4567:, 4564:0 4560:f 4556:6 4548:1 4545:+ 4542:, 4539:0 4536:, 4533:0 4529:f 4525:+ 4520:1 4514:, 4511:0 4508:, 4505:0 4501:f 4497:+ 4492:0 4489:, 4486:1 4483:+ 4480:, 4477:0 4473:f 4469:+ 4464:0 4461:, 4458:1 4452:, 4449:0 4445:f 4441:+ 4436:0 4433:, 4430:0 4427:, 4424:1 4421:+ 4417:f 4413:+ 4408:0 4405:, 4402:0 4399:, 4396:1 4389:f 4385:= 4380:0 4377:, 4374:0 4371:, 4368:0 4364:) 4360:f 4355:2 4350:7 4342:( 4316:, 4313:) 4308:2 4301:3 4285:2 4277:+ 4272:2 4265:3 4261:+ 4249:1 4241:+ 4236:2 4231:7 4222:) 4217:2 4204:1 4193:1 4190:( 4187:= 4182:2 4175:2 4167:, 4162:1 4123:] 4117:2 4113:/ 4109:1 4104:0 4099:2 4095:/ 4091:1 4084:0 4079:2 4071:0 4064:2 4060:/ 4056:1 4051:0 4046:2 4042:/ 4038:1 4032:[ 4024:+ 4019:] 4013:0 4008:1 4003:0 3996:1 3991:4 3983:1 3976:0 3971:1 3966:0 3960:[ 3955:) 3946:1 3943:( 3940:= 3935:2 3919:+ 3914:2 3909:5 3901:) 3892:1 3889:( 3886:= 3881:2 3842:. 3837:] 3831:1 3826:1 3821:1 3814:1 3809:8 3801:1 3794:1 3789:1 3784:1 3778:[ 3773:= 3768:2 3763:y 3760:x 3755:D 3738:n 3732:. 3727:i 3723:x 3717:i 3711:s 3706:i 3702:1 3698:0 3688:i 3684:x 3651:0 3644:, 3641:1 3635:n 3632:= 3629:s 3619:1 3612:, 3609:n 3606:= 3603:s 3593:n 3590:2 3580:{ 3576:= 3569:n 3565:x 3561:, 3555:, 3550:2 3546:x 3542:, 3537:1 3533:x 3528:a 3506:, 3501:2 3494:n 3490:x 3486:, 3480:, 3475:2 3471:x 3467:, 3462:1 3458:x 3452:D 3426:n 3422:x 3418:, 3412:, 3407:2 3403:x 3399:, 3394:1 3390:x 3385:a 3372:n 3365:. 3351:] 3345:2 3340:3 3335:2 3328:3 3323:6 3318:3 3311:2 3306:3 3301:2 3295:[ 3285:1 3261:] 3255:3 3250:6 3245:3 3238:6 3225:6 3218:3 3213:6 3208:3 3202:[ 3192:1 3168:] 3162:2 3157:3 3152:2 3145:3 3140:6 3135:3 3128:2 3123:3 3118:2 3112:[ 3102:1 3080:. 3066:] 3060:0 3055:0 3050:0 3043:0 3038:1 3033:0 3026:0 3021:0 3016:0 3010:[ 2986:] 2980:0 2975:1 2970:0 2963:1 2958:6 2950:1 2943:0 2938:1 2933:0 2927:[ 2903:] 2897:0 2892:0 2887:0 2880:0 2875:1 2870:0 2863:0 2858:0 2853:0 2847:[ 2816:2 2811:z 2808:y 2805:x 2800:D 2787:, 2773:] 2745:3 2714:[ 2709:= 2704:2 2699:y 2696:x 2691:D 2655:2 2650:y 2647:x 2642:D 2629:. 2615:] 2609:0 2604:1 2599:0 2592:1 2587:4 2579:1 2572:0 2567:1 2562:0 2556:[ 2551:= 2546:2 2541:y 2538:x 2533:D 2520:, 2506:] 2500:1 2495:2 2487:1 2481:[ 2476:= 2471:2 2466:x 2456:D 2374:y 2370:x 2368:( 2366:f 2358:y 2354:x 2352:( 2350:f 2338:h 2321:. 2318:} 2315:) 2312:h 2309:+ 2306:y 2303:, 2300:x 2297:( 2294:, 2291:) 2288:h 2282:y 2279:, 2276:x 2273:( 2270:, 2267:) 2264:y 2261:, 2258:h 2255:+ 2252:x 2249:( 2246:, 2243:) 2240:y 2237:, 2234:x 2231:( 2228:, 2225:) 2222:y 2219:, 2216:h 2210:x 2207:( 2204:{ 2191:y 2187:x 2183:h 2166:, 2159:2 2155:h 2150:) 2147:y 2144:, 2141:x 2138:( 2135:f 2132:4 2126:) 2123:h 2120:+ 2117:y 2114:, 2111:x 2108:( 2105:f 2102:+ 2099:) 2096:h 2090:y 2087:, 2084:x 2081:( 2078:f 2075:+ 2072:) 2069:y 2066:, 2063:h 2060:+ 2057:x 2054:( 2051:f 2048:+ 2045:) 2042:y 2039:, 2036:h 2030:x 2027:( 2024:f 2015:) 2012:y 2009:, 2006:x 2003:( 2000:f 1940:C 1935:1 1928:M 1924:= 1921:L 1899:i 1895:A 1874:i 1854:M 1831:M 1808:i 1788:) 1785:i 1782:( 1779:N 1743:0 1736:, 1733:j 1730:= 1727:i 1720:k 1717:i 1713:C 1707:) 1704:i 1701:( 1698:N 1692:k 1677:, 1674:) 1671:i 1668:( 1665:N 1659:j 1651:j 1648:i 1643:) 1638:j 1635:i 1621:+ 1616:j 1613:i 1599:( 1594:2 1591:1 1583:{ 1578:= 1573:j 1570:i 1566:C 1539:C 1517:i 1513:) 1509:u 1503:( 1500:= 1497:u 1494:L 1471:| 1467:V 1463:| 1455:| 1451:V 1447:| 1441:R 1433:L 1387:i 1367:i 1341:i 1337:A 1316:j 1313:i 1291:j 1288:i 1261:j 1258:i 1233:i 1213:j 1190:, 1187:) 1182:i 1178:u 1169:j 1165:u 1161:( 1158:) 1153:j 1150:i 1136:+ 1131:j 1128:i 1114:( 1109:j 1096:i 1092:A 1088:2 1084:1 1074:i 1070:) 1066:u 1060:( 1037:i 1017:u 981:. 978:) 975:w 972:( 964:1 961:= 958:) 955:v 952:, 949:w 946:( 943:d 939:: 936:w 925:v 915:1 910:= 907:) 904:v 901:( 898:) 892:M 889:( 859:E 853:v 850:w 828:v 825:w 796:] 792:) 789:w 786:( 777:) 774:v 771:( 764:[ 758:v 755:w 745:1 742:= 739:) 736:v 733:, 730:w 727:( 724:d 720:: 717:w 709:= 706:) 703:v 700:( 697:) 681:( 658:R 652:E 619:) 616:v 613:( 610:) 601:( 544:v 526:) 523:v 520:, 517:w 514:( 511:d 487:] 483:) 480:w 477:( 468:) 465:v 462:( 455:[ 449:1 446:= 443:) 440:v 437:, 434:w 431:( 428:d 424:: 421:w 413:= 410:) 407:v 404:( 401:) 392:( 321:R 315:V 289:E 269:V 249:) 246:E 243:, 240:V 237:( 234:= 231:G 119:) 113:( 108:) 104:( 94:· 87:· 80:· 73:· 46:. 23:.

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Knowledge

Discrete Laplace operator

Index