An expression
x=a0+ b1
a1+ b2
a2+ b3 a3+· · ·
is called a continued fraction, all aj, bj ∈ Z, all denominators are non-zero.
If all bj := 1, we write x = cfrac(a0, a1, a2, . . .). This representation as a continued fraction is unique1. There are many beautiful classical results about continued fractions (see, e.g., [50,115]), e.g., the sequenceaj is periodic iffx∈R\Qandxis algebraic of degree 2, i.e., there is a polynomialP ∈Q[T] with deg(P) ≤ 2 and P(x) = 0. We will only use that continued fractions can be used for (optimum) rational approximations of real numbers.
Lemma A.5.1 A continued fractionx= cfrac(a0, . . . , ak)is finite iffx∈Q. Proof: We give a proof because it implies a constructive method to obtain a rational approximation of any real number. If the continued fraction is finite, clearly, x ∈ Q. So, let w.l.o.g. x > 0 be a rational number. The following (extended) version of the Euclidean algorithm, Algorithm 13, computes the finite continued fraction cfrac(a0, a1, . . . , ak).
Apart from coefficients a0, . . . , ak, the extended Euclidean algorithm also computes rational approximationssj/tj withs0/t0 =bxcand sk/tk=x >0.
This also works if x = u0/u1 ∈ R+\Q. In that case we obtain infinite sequences with limj→∞sj/tj =x. In general, forj ≥0
|x−sj/tj|<1/t2j . (A.3)
1It is unique up to cfrac(a0, a1, . . . , ak,1) = cfrac(a0, a1, . . . , ak+ 1).
Algorithm 13: Extended Euclidean algorithm
Input: Rational number x=u0/u1, where w.l.o.g. u0, u1 ∈N+. Output: Finite continued fraction x= cfrac(a0, a1, . . . , ak), and
rational approximations sj/tj for 0≤j ≤k.
begin
letj := 0, s−2 := 0, t−2 := 1, s−1 := 1, t−1 := 0 be initial values while uj+1 6= 0 do
aj ← buj/uj+1c ∈N (quotient of uj, uj+1) uj+2 ←uj −ajuj+1 ∈N (remainder ofuj, uj+1) sj ←ajsj−1+sj−2
tj ←ajtj−1+tj−2
j ←j+ 1 end
Suppose that x = s/t with s, t ∈ N+ and t ≤ tmax is bounded. The set M :={s/t : s, t ∈N+ and t ≤tmax}, i.e., all positive rational numbers with denominators bounded by tmax, is discrete. This follows easily from the fact that for s/t 6= ˆs/tˆ∈M
s t −sˆ
tˆ
=
sˆt−stˆ ttˆ
≥ 1
t2max . (A.4)
Hence, if during the extended Euclidean algorithm (for x ∈ M) tk > tmax, then (A.3) and (A.4) implyx=sk/tk. Theorem 5.4.3makes use of this fact.
Note that the minimum k with that property is in O(logtmax).
Since this thesis comprises topics from a lot of different areas, all having their standard symbols, we decided to use some symbols with multiple meanings instead of redefining traditionally reserved symbols. For example, we use symbols ρ, I, each of them with two meanings. While a physicist usually thinks of electrical charge density and current, in numerical analysis, these symbols denote spectral radius and identity matrix. We point out some more particularities of our notation. We usef both for functions onRnand on the vertices V ={v1, . . . , vn} of a graph. The symbol n is thereby used both for specifying the dimension inRnand the number of vertices. Anyhow, from the context, or finally because we state here thatn has these different meanings, it should be clear what is meant. About the symbol C, that either denotes continuous functions or a color space, there should also be no confusion. Γ is both a Green’s function and the Euler function, uj and λj are nearly always used for unit eigenvectors and eigenvalues of the Laplacian matrix L (or a submatrix of L) or the Laplacian operator, but uand λ(C) also may denote a vertex and the number of neighbors of vertices in C ⊆V that are inV \C, respectively. Finally, τ(G) and τ(v) denote the number of spanning trees in G and throughput of vertexv in an electrical network, respectively.
We completely avoid implicitly used standard inner products like, e.g.,
∇2 = ∇ · ∇ = ∆ or (∇h)·(∇h), but write (∇h)T∇h instead. Hence each symbol · either is multiplication with at least one real value, or is used for functions without specifying the independent variable, e.g., f(·, x), or Mj,· to denote row j of a matrixM.
∆ Laplacian operator, 18
L, A, B, D Laplacian, adjacency, incidence, and degree matrix, 38 N,N+ natural numbers N:={0,1,2, . . . ,},N+ :=N\ {0}
0,1 vectors 0 := (0, . . . ,0)T and 1 := (1, . . . ,1)T in Rn, 0 may also denote a zero matrix of appropriate dimensions
Ω (bounded) domain in Rn, 18
λj(M), λj(∆) eigenvalue wrt to matrix M, wrt operator ∆, 35
u1, u2, . . . orthonormal eigenvectors of L (submatrix of L), 42, and
∆, 35
(·,·)F,(·,·)M inner product in function space F,23, wrt matrix M, 164
#M number of elements in a set M
◦
M interior of a set M ⊆Rn
◦
V a subset of V, interior ofV,59
Ck(Ω),Ck,α k-times continuously differentiable functions, 18, 19 C0∞(Ω) test functions on Ω, 22
χ embedding into a color space C,138 deg(P),deg(v) degree of a polynomial, of a vertex, 38 degmax maximum degree of a vertex in G, 42 δ Dirac δ-distribution, 23
δjk Kronecker symbol, equal to 1 if j =k, 0 otherwise
divF divergence Pn
j=1∂jFj of a vector valued function F, 16 Γ(·,·) Green’s function, 25
κ(C) number of edges with vertices both in C and in V \C,42 κ(M) condition number of matrix M, 164
λ(C) number of vertices in C with a neighbor in V \C, 46 Lp(Ω),Lploc(Ω) Lebesgue spaces, 22
adj(M) adjugate matrix of matrix M, 41 n normal vector wrt to a given set, 19
∇ Nabla operator, 18
ω(e) weights on the edge set E of a graph, 38
ωn area of the unit sphere surface in Rn, 20
Ω closure of Ω, 18
G complementary graph of G,43
∂Ω, ∂V boundary of Ω, 18, a subset of V, boundary of V, 59
∂β derivatives wrt multi-index β, 19
∂xj =∂j, ∂n partial and normal derivatives, 18, 19 ρ(M) spectral radius of a matrix M
rot F rotation of a vector field; symbolically written, the cross product ∇ ×F,16
σ, Gσ, Eσ orientationσyields a directed graphGσ with edge setEσ,37 τ(G) number of spanning trees of G, 40
τ(v) throughput of a vertex in an electrical network, 84 h¨olα(f,Ω) H¨older value of f in Ω wrt α, 19
tr(M) trace of matrix M, 41 ϕε(x) Friedrichs’ Mollifier, 22
LeW omitting rows and columns of L wrt W ⊆V,42 Wk,p(Ω),W0k,p(Ω) Sobolev spaces, 28, 73
b(·,·) barrier function, 26
B(x, r) ball at x∈Rn with radius r, 18
B(x, r1, r2) cut ball at x∈Rn with inner, outer radius r1, r2, 18 D(h) Dirichlet integral wrt h, 27
E(h) energy wrt embedding h, 63
f(x), f(v), x(v) functions onRn, and V, respectively, 38
f(x)∈ O(g(x)),o(g(x)),Θ(g(x)),Ω(g(x)) Landau symbols,Omay also refer to asymptotical behavior at zero
f|∂V restriction of f to the boundary∂V,59 f|∂Ω restriction of f to the boundary∂Ω,18 hn(x) fundamental solution in Rn, 24
Kr Kelvin transform, 17
pC barycentric embedding, 116
Q orthogonal matrix, 43
Reff(s, t) effective resistance between vertices s and t in an electrical network, 85
U, I, p, ρ, c potential (also on Ω), current, potential difference, supply (density on Ω,), conductance in electrical networks, 57 xT transpose of a vector x∈Rn
x1⊥x2 x1, x2 orthogonal, xT1x2 = 0
A
color refinement, Voronoi . . . .145
current-flow betweenness . . . . .88
current-flow closeness . . . .89
min vertex widths on trees. . . .53
approximation current-flow betweenness . . . . .91
Dirichlet problem . .63, 64,71, 72 solution of wave equation . . . . .69
current-flow betweenness . . . . .85
current-flow closeness . . . .85
information centrality . . . .86
random-walk betweenness . . . .85
shortest-path betweenness . . . .84
shortest-path closeness . . . 84
CGMsee conjugate gradient method chromaticity diagram . . . .139
classical Dirichlet problem17,21, 36, 65
condition number, κ(M) . . . . .90, 164
conformal mapping . . . .26
conjugate gradient method . . 90, 164 connected components . . . .40, 105
of Jacobi iteration . .61, 120, 129 Coulomb’s Law . . . .16
Dirichlet boundary conditions .17, 34 Dirichlet integral, D(h) . . . .27, 70 essentially diagonally dominant . . .60
Euclidean algorithm . . . .168
current-flow betweenness . . . . .93
F
Friedrichs’ mollifier, ϕε. . . .22, 23 Gabriel Graph, restricted, GGG. .117
Gauß integration theorem . . . .19
Gauß’ Law . . . .18
Gauß-Seidel iteration . . . .61, 163 GBFR . . . .see greedy barycentric FR GBR .see greedy barycentric routing generalized Dirichlet problem . . . . .28
geographic routing . . . .114
greedy barycentric face routing . .123
greedy barycentric routing. . . .122
inhomogeneous wave equation . . . .36
on graphs . . . .68
JI . . . .see Jacobi iteration K
KCL . . .see Kirchhoff’s Current Law
k-connected graph . . . .66
Kelvin transform, Kr. . . .17, 26 Kirchhoff’s Current Law . . . .57, 83 Kirchhoff’s Potential Law . . . . .57, 83 Kirchhoff’s theorem . . . 40, 126 KPL . .see Kirchhoff’s Potential Law k-times continuously differentiable functions . . . .18
k-times weakly differentiable . . . 28
L
Laplacian-, L 38, 45,83, 96, 116, 141 orthogonal-, Q. . . .43
stiffness- . . . .70
matrix-tree theorem . . . .40, 126 max quadratic assignment prob. .140
maximum principle . . . .21
on graphs . . . .61
maximum vertex bisection . . . .49
N P-completeness of- . . . .49
Maxwell’s equations . . . .18
MDS . . .see multidimensional scaling meanvalue property . . . .20
multidimensional scaling . . . .142, 162 N nabla operator, ∇. . . .18
neighborhood . . . 18
of a vertex. . . .37
network analysis. . . .81
Neumann boundary conditions . . .17
Newton’s Law of Gravitation . . . . .16
O
partial differential equation . . .15, 75 PDE . . . .see partial differential eq.
restricted Gabriel Graph, GGG . .117
restriction of a function, f|∂Ω . . . .18
of Laplacian operator . . . .44
weighted Laplacian matrix, L38, 83, 96,141