THE DEGREE DISTRIBUTION OF RANDOM PLANAR GRAPHS
|
|
- Jodie James
- 5 years ago
- Views:
Transcription
1 THE DEGREE DISTRIBUTION OF RANDOM PLANAR GRAPHS Michael Drmota joint work with Omer Giménez and Marc Noy Institut für Diskrete Mathematik und Geometrie TU Wien supported by the Austrian Science Foundation FWF, grant S th LL-Seminar on Graph Theory, Leoben, April 25, 2009
2 Contents Short history of random planar graphs Generating function for random planar graphs Asymptotic enumeration of planar graphs The degree distribution of random planar graphs Outerplanar graphs Series-parallel graphs
3 Random Planar Graphs History R n... labelled planar graphs with n vertices with uniform distribution X n... number of edges is a random planar graph with n vertices Denise, Vasconcellos, Welsh 1996) P{X n > 3 2 n} 1, P{X n < 5 2 n} 1. Note that 0 e 3n for all planar graphs.)
4 Random Planar Graphs History McDiarmid, Steger, Welsh 2005) P{H appears in R n at least αn times} 1 H... any fixed planar graph, α > 0 sufficiently small. H H H H...
5 Random Planar Graphs Consequences: P{There are αn vertices of degree k} 1 k > 0 a given integer, α > 0 sufficiently small. for some C > 1. P{There are C n automorphisms} 1
6 Random Planar Graphs Further Results: P{R n is connected} γ > 0 [McDiarmid+Reed] n... maximum degree in R n E n = Θlog n)
7 Random Planar Graphs The number of planar graphs [Bender, Gao, Wormald 2002)] b n... number of 2-connected labelled planar graphs b n c n 7 2 γ n 2 n!, γ 2 = [Gimenez+Noy 2005)] g n... number of all labelled planar graphs g n c n 7 2 γ n n!, γ =
8 Random Planar Graphs Precise distributional results [Gimenez+Noy 2005)] X n satisfies a central limit theorem: E X n n, V X n c n. P{ X n n > εn} e αε) n Connectedness: P{R n is connected} e ν = number of components of R n =: C n 1 + P oν).
9 Random Planar Graphs Degree Distribution Theorem [D.+Giménez+Noy] Let p n,k be the probability that a random node in a random planar graph R n has degree k. Then the limit p k := lim n p n,k exists. The probability generating function pw) = p k w k k 1 can be explicitly computed; p k c k 1 2q k for some c > 0 and 0 < q < 1. p 1 p 2 p 3 p 4 p 5 p
10 Random Planar Graphs Implicit equation for D 0 y, w): SD0 t 1) + t) 1 + D 0 = 1 + y w ) exp 43t + 1)D 0 + 1) ) D2 0 t4 12t t 9) + D 0 2t 4 + 6t 3 6t t 12) + t 4 + 6t 3 + 9t 2, 4t + 3)D 0 + 1)3t + 1) 1 + 2t where t = ty) satisfies y+1 = 1 + 3t)1 t) exp 1 ) t 2 1 t) t + 5t 2 ) t)1 + 2t)1 + 3t) 2 and S = D 0 t 1) + t)d 0 t 1) 3 + tt + 3) 2 ). Explicit expressions in terms of D 0 y, w) SEVERAL PAGES!!!!): B 0 y, w), B 2 y, w), B 3 y, w) Explict expression for pw): pw) = e B 01,w) B 0 1,1) B 2 1, w) + e B 01,w) B 0 1,1) 1 + B 21, 1) B 3 1, w) B 3 1, 1)
11 Random Planar Graphs Conjecture for maximum degree n n log n 1 log1/q) in probability and E n log n log1/q) where q = appear in the asymptotics of p k c k 1 2q k ; 1/ log1/q) =
12 Degree Distribution X k) n... number of vertices of degree k in a random labelled planar graph of size n p n,k... probability that a random vertex in a random labelled planar graph of size n has degree k ˆp n,k... probability that the root vertex in a random labelled vertex rooted planar graph of size n has degree k p n,k = ˆp n,k E X k) n = n p n,k
13 Degree Distribution Generating functions for counting planar graphs b n,m... number of 2-connected labelled planar graphs with n vertices and m edges, b n = m b n,m Bx, y) = x n b n,m n,m n! ym c n,m... number of connected labelled planar graphs with n vertices and m edges, c n = m c n,m Cx, y) = x n c n,m n,m n! ym g n,m... number of all labelled planar graphs with n vertices and m edges, g n = m g n,m Gx, y) = x n g n,m n,m n! ym
14 Degree Distribution Generating functions for counting planar graphs Gx, y) = exp Cx, y)), Cx, y) B Cx, y) = exp x, y x x x Bx, y) y Mx, D) 2x 2 D )) = x2 1 + Dx, y), y ) 1 + D = log xd2 1 + y 1 + xd, Mx, y) = x 2 y xy y U)2 1 + V ) U + V ) 3 U = xy1 + V ) 2, V = y1 + U) 2., ),
15 Degree Distribution Asymptotic enumeration of planar graphs b n = b ρ n 1 n 7 2n! 1 + O c n = c ρ n 2 n 7 2n! g n = g ρ n 2 n 7 2n! 1 + O 1 + O )) 1 n )) 1 n )) 1 n,, ρ 1 = , ρ 2 = , b = , c = , g =
16 Degree Distribution Generating functions for the degree distribution of planar graphs C = C x... GF for graphs with a root vertex which is not counted w... additional variable that counts the degree of the root vertex Generating functions: G x, y, w) C x, y, w) B x, y, w) T x, y, w) all rooted planar graphs connected rooted planar graphs 2-connected rooted planar graphs 3-connected rooted planar graphs Note that G x, y, 1) = G x, y) etc. x
17 Degree Distribution more precisely... g n,m,k... number of vertex rooted labelled planar graphs with n + 1 vertices, m edges, where the uncounted and unlabelled) root vertex has degree k. G x, y, w) = g x n n,m,k n,m,k n! ym w k k g n 1,m,k = n g n,m, p n,k = g n 1,m,k n g n,m k 1 p n,k w k = 1 n g n,m k 1 g n 1,m,k wk = [xn 1 ]G x, 1, w) [x n 1 ]G x, 1, 1) = pw) = k 1 p n w k = lim n [x n 1 ]G x, 1, w) [x n 1 ]G x, 1, 1)
18 Degree Distribution G x, y, w) = exp Cx, y, 1)) C x, y, w), C x, y, w) = exp B xc x, y, 1), y, w )), w B x, y, w) w = xyw exp Sx, y, w) + 1 x 2 Dx, y, w) T 1 Dx, y, w) = 1 + yw) exp Sx, y, w) + x 2 Dx, y, w) T Dx, y, w) x, Dx, y, 1), Dx, y, 1) Sx, y, w) = xdx, y, 1) Dx, y, w) Sx, y, w)), T x, y, w) = x2 y 2 w wy xy 1 u + 1) 2 w 1 u, v, w) + u w + 1) x, Dx, y, 1), )) 1 2wvw + u 2 + 2u + 1)1 + u + v) 3 ux, y) = xy1 + vx, y)) 2, vx, y) = y1 + ux, y)) 2. w 2 u, v, w) Dx, y, w) Dx, y, 1) ), ))
19 Degree Distribution with polynomials w 1 = w 1 u, v, w) and w 2 = w 2 u, v, w) given by w 1 = uvw 2 + w1 + 4v + 3uv 2 + 5v 2 + u 2 + 2u + 2v 3 + 3u 2 v + 7uv) + u + 1) 2 u + 2v v 2 ), w 2 =u 2 v 2 w 2 2wuv2u 2 v + 6uv + 2v 3 + 3uv 2 + 5v 2 + u 2 + 2u + 4v + 1) + u + 1) 2 u + 2v v 2 ) 2.
20 Planar Maps vs. Planar Graphs Whitney s Theorem Every 3-connected planar graph has a unique embedding into the plane. = The counting problem of rooted 3-connected planar maps is equivalent to the counting problem of rooted labelled) 3-connected planar graphs despite of a factor n 1)!) Furthermore, the counting problem of rooted 3-connected planar maps is equivalent to the counting problem of rooted simple quadrangulations.
21 3-Connected Maps 3-connected map simple quadrangulations
22 3-Connected Maps 3-connected map simple quadrangulations
23 3-Connected Maps 3-connected map simple quadrangulations
24 3-Connected Maps 3-connected map simple quadrangulations
25 3-Connected Maps 3-connected map simple quadrangulations
26 3-Connected Maps 3-connected map simple quadrangulations
27 3-Connected Maps q ijk... number of simple quadrangulations with i + 1 vertices of type 1 ), j + 1 vertices of type 2 ), and with root vertex of degree k + 1 Qx, y, w) = i,j,k q i,j,k x i y j w k Theorem [Mullin+Schellenberg, D+Gimenez+Noy] Qx, y, w) = xyw with wy x 1 ) UV W R, S, w) 1 + U + V ) 3
28 3-Connected Maps with algebraic function U = Ux, y), V = V x, y) given by U = xv + 1) 2, V = yu + 1) 2 and W U, V, w) = w 1U, V, w) + U w + 1) w 2 U, V, w) 2V + 1) 2 V w + U 2 + 2U + 1) with polynomials w 1 = w 1 U, V, w) and w 2 = w 2 U, V, w) given by w 1 = UV w 2 + w1 + 4V + 3UV 2 + 5V 2 + U 2 + 2U + 2V 3 + 3U 2 V + 7UV ) + U + 1) 2 U + 2V V 2 ), w 2 =U 2 V 2 w 2 2wUV 2U 2 V + 6UV + 2V 3 + 3UV 2 + 5V 2 + U 2 + 2U + 4V + 1) + U + 1) 2 U + 2V V 2 ) 2.
29 3-Connected Planar Graphs Corollary By Whitney s theorem: T x, y, w) = xw 2 Qxy, y, w).
30 2-Connected Planar Graphs Planar networks A network N is a multi-)graph with two distinguished vertices, called its poles usually labelled 0 and ) such that the multi-)graph ˆN obtained from N by adding an edge between the poles of N is 2- connected. Let M be a network and X = N e, e EM)) a system of networks indexed by the edge-set EM) of M. Then N = MX) is called the superposition with core M and components N e and is obtained by replacing all edges e EM) by the corresponding network N e and, of course, by identifying the poles of N e with the end vertices of e accordingly). A network N is called an h-network if it can be represented by N = MX), where the core M has the property that the graph ˆM obtained by adding an edge joining the poles is 3-connected and has at least 4 vertices. Similarly N = MX) is called a p-network if M consists of 2 or more edges that connect the poles, and it is called an s-network if M consists of 2 or more edges that connect the poles in series.
31 2-Connected Planar Graphs Planar networks Trakhtenbrot s canonical network decomposition theorem: any network with at least 2 edges belongs to exactly one of the 3 classes of h-, p- or s-networks. Furthermore, any h-network has a unique decomposition of the form N = MX), and a p-network or any s- network) can be uniquely decomposed into components which are not themselves p-networks or s-networks).
32 2-Connected Planar Graphs Planar networks Lwt Dx, y, w) and Sx, y, w), respectively, the GFs of planar) networks and series networks, with the same meaning for the variables x, y and w: Then by a variant of [Walsh 1982)] Dx, y, w) = 1 + yw) exp Sx, y, w) + Sx, y, w) = xdx, y, 1) Dx, y, w) Sx, y, w)), 1 x 2 Dx, y, w) T x, Ex, y), Dx, y, w) Dx, y, 1) )
33 2-Connected Planar Graphs A planar network with non-adjacent poles is obtained by distinguishing, orienting and then deleting any edge of an arbitrary 2-connected planar graph: w B x, y, w) w = xyw exp Sx, y, w) + 1 x 2 Dx, y, w) T x, Dx, y, 1), Dx, y, w) Dx, y, 1) ))
34 Connected Planar Graphs C x, y, w) = e B xc x,y,1),x,w) xc B B B xc xc xc xc xc xc
35 All Planar Graphs G x, y, w) = exp Cx, y, 1)) C x, y, w) C C C C C G
36 Asymptotics for Random Planar Graphs Functional equations Suppose that Ax, u) = Φx, u, Ax, u)), where Φx, u, a) has a power series expansion at 0, 0, 0) with non-negative coefficients and Φ aa x, u, a) 0. Let x 0 > 0, a 0 > 0 inside the region of convergence) satisfy the system of equations: a 0 = Φx 0, 1, a 0 ), 1 = Φ a x 0, 1, a 0 ). Then there exists analytic function gx, u), hx, u), and ρu) such that locally Ax, u) = gx, u) hx, u) 1 x ρu).
37 Asymptotics for Random Planar Graphs Asymptotics for coefficients Ax) = gx) hx) 1 x ρ + some technical conditions) = [x n ] Ax) = hρ) 2 π ρ n n O )) 1 n. Similarly: Ax, u) = gx, u) hx, u) 1 x ρu) + some technical conditions) = [x n ] Ax, u) = hρu), u) 2 π ρu) n n O )) 1 n.
38 Asymptotics for Random Planar Graphs Asymptotics for coefficients and Ax) = gx) + hx) 1 x ρ) α + some technical conditions) = [x n ] Ax) = hρ) Γ α) ρ n n α O 1 n )).
39 Asymptotics for Random Planar Graphs Singular expansion Ax) = gx) hx) 1 x ρ = g 0 + g 1 x ρ) + g 2 x ρ) 2 + ) + h 0 + h 1 x ρ) + h 2 x ρ) 2 + ) 1 x ρ = a 0 + a 1 1 x )1 2 + a2 1 x )2 2 + a3 1 x 2 + ρ ρ ρ)3 = a 0 + a 1 X + a 2 X 2 + a 3 X 3 + with X = 1 x ρ.
40 Asymptotics for Random Planar Graphs Ux, y) = xy1 + V x, y)) 2, V x, y) = y1 + Ux, y)) 2 = Ux, y) = xy1 + y1 + Ux, y)) 2 ) 2 = Ux, y) = gx, y) hx, y) = V x, y) = g 2 x, y) h 2 x, y) 1 y τx) 1 y τx) Mx, y) = x 2 y xy y U)2 1 + V ) 2 ) 1 + U + V ) 3!!! = Mx, y) = g 3 x, y) + h 3 x, y) 1 y τx) due to cancellation of the 1 y/τx)-term )3 2
41 Asymptotics for Random Planar Graphs Mx, D) 2x 2 D = log 1 + D 1 + y!!! = Dx, y) = g 4 x, y) + h 4 x, y) due to interaction of the singularities!!! ) xd2 1 + xd 1 x Ry) )3 2 Bx, y) y = x Dx, y), 1 + y!!! = Bx, y) = g 5 x, y) + h 5 x, y) = b n b R1) n n 7 2n! 1 x Ry) )5 2
42 Asymptotics for Random Planar Graphs B x, y) = g 6 x, y) + h 6 x, y) C x, y) = e B xc x,y),y),!!! = C x, y) = g 7 x, y) + h 7 x, y) due to interaction of the singularities!!! 1 x Ry) 1 x ry) )3 2, )3 2 = Cx, y) = g 8 x, y) + h 8 x, y) 1 x ry) )5 2 = c n c r1) n n 7 2n!
43 Asymptotics for Random Planar Graphs Cx, y) = g 8 x, y) + h 8 x, y) 1 x ry) = Gx, y) = e Cx,y) = g 9 x, y) + h 9 x, y) )5 2 1 x ry) )5 2. = g n g r1) n n 7 2n!
44 Asymptotic Degree Distribution 3-connected planar graphs T x, y, w) = x2 y 2 w wy xy 1 U + 1) 2 w 1 U, V, w) + U w + 1) 2wV w + U 2 + 2U + 1)1 + U + V ) 3 w 2 U, V, w) ), ũ 0 y) = y, ry) = ũ 0 y) y1 + y1 + ũ 0 y)) 2 ) 2, X = 1 x ry) = T x, y, w) = T 0 y, w) + T 2 y, w) X 2 + T 3 y, w) X 3 + O X 4 ) due to cancellation of the 1 x/rz)-term.
45 Asymptotic Degree Distribution Planar networks Dx, y, w) = 1 + yw) exp Sx, y, w) + T x, Dx, y, 1), 1 x 2 Dx, y, w) Sx, y, w) = xdx, y, 1) Dx, y, w) Sx, y, w)) Dx, y, w) Dx, y, 1) )) 1 τx)... inverse function of ry) DRy), y, 1) = τry)) X = 1 x Ry) = Dx, y, w) = D 0 y, w) + D 2 y, w)x 2 + D 3 y, w)x 3 + OX 4 ),
46 Asymptotic Degree Distribution 2-connected planar graphs w B x, y, w) w = xyw exp Sx, y, w) + 1 x 2 Dx, y, w) T x, Dx, y, 1), Dx, y, w) Dx, y, 1) )) = B x, y, w) = B 0 y, w) + B 2 y, w)x 2 + B 3 y, w)x 3 + OX 4 ) Remark. All these functions B j y, w) can be explicitly computed.
47 Asymptotic Degree Distribution connected planar graphs C x, 1, w) = exp B xc x), 1, w ))???
48 Asymptotic Degree Distribution Lemma fx) = n 0 a n x n n! = f 0 + f 2 X 2 + f 3 X 3 + OX 4 ), X = 1 x ρ, Hx, z, w) = h 0 x, w) + h 2 x, w)z 2 + h 3 x, w) Z 3 + OZ 4 ), Z = 1 z fρ), f H x) = Hx, fx), w) = n 0 b n w) xn n! = lim n b n w) a n = h 2ρ, w) f 0 + h 3ρ, w) f 3 f 2 f 0 ) 3/2.
49 Asymptotic Degree Distribution connected planar graphs C x, 1, w) = exp B xc x), 1, w )) Application of the lemma with and = pw) = lim n b n w) a n fx) = xc x) Hx, z, w) = xe B z,1,w). = e B 01,w) B 0 1,1) B 2 1, w) + e B 01,w) B 0 1,1) 1 + B 21, 1) B 3 1, w) B 3 1, 1)
50 Random Planar Graphs Classes of planar graphs Outerplanar graphs: all vertices are on the infinite face equivalently no K 4 and no K 2,3 as a minor). Series-parallel graphs: series-parellel extension of a tree or forest equivalently no K 4 as a minor). Planar graphs. no K 5 and no K 3,3 as a minor) Remark. outerplanar series-parallel planar
51 Outerplanar Graphs All vertices are on the infinite face.
52 Outerplanar Graphs Theorem 1 Let p n,k be the probability that a random node in a random 2-connected, connected or unrestricted outerplanar graph with n vertices has degree k. Then the limit p k := lim n p n,k exists. The probability generating function pw) = p k w k k 1 can be explicitly computed.
53 Outerplanar Graphs pw) = p k w k k 1 2-connected pw) = )w )w) 2 connected or unrestricted: pw) = c 1w 2 1 c 2 w) 2 exp with certain constants c 1, c 2, c 3, c 4 > 0). c 3 w + c 4w 2 1 c 2 w) )
54 Outerplanar Graphs Theorem 2 X k) n... number of vertices of degree k in random 2-connected, connected or unrestricted labelled outerplanar graphs with n vertices. = X k) n satisfies a central limit theorem with E X k) n µ k n and V X k) n σ 2 k n. Remark. µ k = p k.
55 Outerplanar Graphs Theorem 3 n... maximum degree in random 2-connected, connected or unrestricted labelled outerplanar graphs with n vertices. = n log n c in probability E n c log n, where c = 1/ log1/q) and 1/q in radius of convergence of pw).
56 Series-Parallel Graphs Series-parallel extension of a tree or forest Series-extension: Parallel-extension:
57 Series-Parallel Graphs Theorem 1 Let p n,k be the probability that a random node in a random 2-connected, connected or unrestricted series-parallel graph with n vertices has degree k. Then the limit p k := lim n p n,k exists. The probability generating function pw) = p k w k k 1 can be explicitly computed.
58 Series-Parallel Graphs We just mention the case of 2-connected series-parallel graphs pw) = k 1 p k w k : pw) = B 11, w) B 1 1, 1), where B 1 y, w) is given by the following procedure...
59 Series-Parallel Graphs E 0 y) 3 E 0 y) 1 = log 1 + E ) 2 0y) 1 + Ry) E 0y), 1 1/E0 y) 1 Ry) =, E 0 y) ) 2Ry)E 0 y) Ry)E 0 y)) E 1 y) =, 2Ry)E 0 y) + Ry) 2 E 0 y) 2 ) 2 + 2Ry)1 + Ry)E 0 y)) Ry)E 0y) 1+Ry)E D 0 y, w) = 1 + yw)e D0y,w) 0 y) 1, D 1 y, w) = 1 + D 0y, w)) Ry)E 1y)D 0 y,w) 1+Ry)E 0 y), D 0 y, w)) Ry)E 0y)D 0 y,w) 1+Ry)E 0 y) B 0 y, w) = Ry)D 0y, w) 1 + Ry)E 0 y) Ry)2 E 0 y)d 0 y, w) Ry)E 0 y)), B 1 y, w) = Ry)D 1y, w) 1 + Ry)E 0 y) Ry)2 E 0 y)d 0 y, w)d 1 y, w) 1 + Ry)E 0 y) Ry)2 E 1 y)d 0 y, w)1 + D 0 y, w)/2) 1 + Ry)E 0 y)) 2.
60 Series-Parallel Graphs Theorem 2 X k) n... number of vertices of degree k in random 2-connected, connected or unrestricted labelled series-parallel graphs with n vertices. = X k) n satisfies a central limit theorem with E X k) n µ k n and V X k) n σ 2 k n. Remark. µ k = p k.
61 Series-Parallel Graphs Theorem 3 n... maximum degree in random 2-connected, connected or unrestricted labelled series-parallel graphs with n vertices. = n log n c in probability E n c log n, where c = 1/ log1/q) and 1/q in radius of convergence of pw).
62 References Articles O. Giménez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. to appear). M. Drmota, O. Giménez, and M. Noy, The number of vertices of given degree in series-parallel graphs Random Structures and Algorithms to appear). M. Drmota, O. Giménez, and M. Noy, Degree distribution in random planar graphs, manuscript. M. Drmota, O. Giménez, and M. Noy, The maximum degree of planar graphs, draft.
63 General References Books Michael Drmota, Random Trees, Springer, Wien-New York, Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press,
64 Thank You!
GENERATING FUNCTIONS AND CENTRAL LIMIT THEOREMS
GENERATING FUNCTIONS AND CENTRAL LIMIT THEOREMS Michael Drmota Institute of Discrete Mathematics and Geometry Vienna University of Technology A 1040 Wien, Austria michael.drmota@tuwien.ac.at http://www.dmg.tuwien.ac.at/drmota/
More informationDegree Distribution in Random Planar Graphs
ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria Degree Distribution in Random Planar Graphs Michael Drmota Omer Gim Marc Noy Vienna, Preprint
More informationASYMPTOTIC ENUMERATION AND LIMIT LAWS OF PLANAR GRAPHS
ASYMPTOTIC ENUMERATION AND LIMIT LAWS OF PLANAR GRAPHS OMER GIMÉNEZ AND MARC NOY Abstract. We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate g n g n
More informationExtremal Statistics on Non-Crossing Configurations
Extremal Statistics on Non-Crossing Configurations Michael Drmota Anna de Mier Marc Noy Abstract We analye extremal statistics in non-crossing configurations on the n vertices of a convex polygon. We prove
More information10 Steps to Counting Unlabeled Planar Graphs: 20 Years Later
10 Steps to : 20 Years Later Manuel Bodirsky October 2007 A005470 Sloane Sequence A005470 (core, nice, hard): Number p(n) of unlabeled planar simple graphs with n nodes. Initial terms: 1, 2, 4, 11, 33,
More informationThe degree distribution in unlabelled 2-connected graph families
AofA 0 DMTCS proc. AM, 00, 453 47 The degree distribution in unlabelled -connected graph families Veronika Kraus Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-0/04, 040
More informationA LINEAR APPROXIMATE-SIZE RANDOM SAMPLER FOR LABELLED PLANAR GRAPHS
A LINEAR APPROXIMATE-SIZE RANDOM SAMPLER FOR LABELLED PLANAR GRAPHS ÉRIC FUSY Abstract. This article introduces a new algorithm for the random generation of labelled planar graphs. Its principles rely
More informationCombinatorial Enumeration. Jason Z. Gao Carleton University, Ottawa, Canada
Combinatorial Enumeration Jason Z. Gao Carleton University, Ottawa, Canada Counting Combinatorial Structures We are interested in counting combinatorial (discrete) structures of a given size. For example,
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
More informationTHE NUMBER OF INDEPENDENT DOMINATING SETS OF LABELED TREES. Changwoo Lee. 1. Introduction
Commun. Korean Math. Soc. 18 (2003), No. 1, pp. 181 192 THE NUMBER OF INDEPENDENT DOMINATING SETS OF LABELED TREES Changwoo Lee Abstract. We count the numbers of independent dominating sets of rooted labeled
More informationarxiv:math/ v1 [math.co] 19 Dec 2005
arxiv:math/512435v1 [math.co] 19 Dec 25 ENUMERATION AND LIMIT LAWS OF SERIES-PARALLEL GRAPHS MANUEL BODIRSKY, OMER GIMÉNEZ, MIHYUN KANG, AND MARC NOY Abstract. We show that the number g n of labelled series-parallel
More informationConnectivity of addable graph classes
Connectivity of addable graph classes Paul Balister Béla Bollobás Stefanie Gerke January 8, 007 A non-empty class A of labelled graphs that is closed under isomorphism is weakly addable if for each graph
More informationConnectivity of addable graph classes
Connectivity of addable graph classes Paul Balister Béla Bollobás Stefanie Gerke July 6, 008 A non-empty class A of labelled graphs is weakly addable if for each graph G A and any two distinct components
More informationk-protected VERTICES IN BINARY SEARCH TREES
k-protected VERTICES IN BINARY SEARCH TREES MIKLÓS BÓNA Abstract. We show that for every k, the probability that a randomly selected vertex of a random binary search tree on n nodes is at distance k from
More informationIntegration of Rational Functions by Partial Fractions
Title Integration of Rational Functions by MATH 1700 MATH 1700 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 2 / 11 Rational functions A rational function is one of the form where P and Q are
More informationIntegration of Rational Functions by Partial Fractions
Title Integration of Rational Functions by Partial Fractions MATH 1700 December 6, 2016 MATH 1700 Partial Fractions December 6, 2016 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 Partial Fractions
More informationPower Series Solutions to the Legendre Equation
Department of Mathematics IIT Guwahati The Legendre equation The equation (1 x 2 )y 2xy + α(α + 1)y = 0, (1) where α is any real constant, is called Legendre s equation. When α Z +, the equation has polynomial
More informationInfinite Systems of Functional Equations and Gaussian Limiting Distributions
AofA 12 DMTCS proc. AQ, 2012, 453 478 Infinite Systems of Functional Equations and Gaussian Limiting Distributions Michael Drmota 1, Bernhard Gittenberger 1 and Johannes F. Morgenbesser 2 1 Institut für
More informationarxiv: v1 [math.nt] 16 Jan 2018
ROOTED TREE MAPS AND THE KAWASHIMA RELATIONS FOR MULTIPLE ZETA VALUES HENRIK BACHMANN AND TATSUSHI TANAKA arxiv:1801.05381v1 [math.nt] 16 Jan 2018 Abstract. Recently, inspired by the Connes-Kreimer Hopf
More informationDecomposing planar cubic graphs
Decomposing planar cubic graphs Arthur Hoffmann-Ostenhof Tomáš Kaiser Kenta Ozeki Abstract The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree,
More informationAsymptotic properties of some minor-closed classes of graphs
FPSAC 2013 Paris, France DMTCS proc AS, 2013, 599 610 Asymptotic properties of some minor-closed classes of graphs Mireille Bousquet-Mélou 1 and Kerstin Weller 2 1 CNRS, LaBRI, UMR 5800, Université de
More informationOn the number of matchings of a tree.
On the number of matchings of a tree. Stephan G. Wagner Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-800 Graz, Austria Abstract In a paper of Klazar, several counting examples
More informationOn the mean connected induced subgraph order of cographs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 71(1) (018), Pages 161 183 On the mean connected induced subgraph order of cographs Matthew E Kroeker Lucas Mol Ortrud R Oellermann University of Winnipeg Winnipeg,
More informationarxiv: v1 [math.co] 16 Oct 2007
On the growth rate of minor-closed classes of graphs arxiv:0710.2995v1 [math.co] 16 Oct 2007 Olivier Bernardi, Marc Noy Dominic Welsh October 17, 2007 Abstract A minor-closed class of graphs is a set of
More informationThe Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University
The Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University These notes are intended as a supplement to section 3.2 of the textbook Elementary
More information2 Series Solutions near a Regular Singular Point
McGill University Math 325A: Differential Equations LECTURE 17: SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS II 1 Introduction Text: Chap. 8 In this lecture we investigate series solutions for the
More informationCounting bases of representable matroids
Counting bases of representable matroids Michael Snook School of Mathematics, Statistics and Operations Research Victoria University of Wellington Wellington, New Zealand michael.snook@msor.vuw.ac.nz Submitted:
More informationGraceful Tree Conjecture for Infinite Trees
Graceful Tree Conjecture for Infinite Trees Tsz Lung Chan Department of Mathematics The University of Hong Kong, Pokfulam, Hong Kong h0592107@graduate.hku.hk Wai Shun Cheung Department of Mathematics The
More informationAlso available at ISSN (printed edn.), ISSN (electronic edn.)
Also available at http://amc-journal.eu ISSN 855-3966 printed edn., ISSN 855-3974 electronic edn. ARS MATHEMATICA CONTEMPORANEA 9 205 287 300 Levels in bargraphs Aubrey Blecher, Charlotte Brennan, Arnold
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationChapter 2 First Order Differential Equations
Chapter 2 First Order Differential Equations 2.1 Problem A The plan for this book is to apply the theory developed in Chapter 1 to a sequence of more or less increasingly complex differential equation
More informationON `-TH ORDER GAP BALANCING NUMBERS
#A56 INTEGERS 18 (018) ON `-TH ORDER GAP BALANCING NUMBERS S. S. Rout Institute of Mathematics and Applications, Bhubaneswar, Odisha, India lbs.sudhansu@gmail.com; sudhansu@iomaorissa.ac.in R. Thangadurai
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009.
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK Summer Examination 2009 First Engineering MA008 Calculus and Linear Algebra
More informationARCS IN FINITE PROJECTIVE SPACES. Basic objects and definitions
ARCS IN FINITE PROJECTIVE SPACES SIMEON BALL Abstract. These notes are an outline of a course on arcs given at the Finite Geometry Summer School, University of Sussex, June 26-30, 2017. Let K denote an
More information23 Elements of analytic ODE theory. Bessel s functions
23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2
More informationA REFINED ENUMERATION OF p-ary LABELED TREES
Korean J. Math. 21 (2013), No. 4, pp. 495 502 http://dx.doi.org/10.11568/kjm.2013.21.4.495 A REFINED ENUMERATION OF p-ary LABELED TREES Seunghyun Seo and Heesung Shin Abstract. Let T n (p) be the set of
More informationProperties of Random Graphs via Boltzmann Samplers
Discrete Mathematics and Theoretical Computer Science (subm.), by the authors, 1 rev Properties of Random Graphs via Boltzmann Samplers Konstantinos Panagiotou and Andreas Weißl Institute of Theoretical
More information4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu**
4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN Robin Thomas* Xingxing Yu** School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA May 1991, revised 23 October 1993. Published
More informationMath 2: Algebra 2, Geometry and Statistics Ms. Sheppard-Brick Chapter 4 Test Review
Chapter 4 Test Review Students will be able to (SWBAT): Write an explicit and a recursive function rule for a linear table of values. Write an explicit function rule for a quadratic table of values. Determine
More informationCalculus I (Math 241) (In Progress)
Calculus I (Math 241) (In Progress) The following is a collection of Calculus I (Math 241) problems. Students may expect that their final exam is comprised, more or less, of one problem from each section,
More informationContinuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationDiophantine equations for second order recursive sequences of polynomials
Diophantine equations for second order recursive sequences of polynomials Andrej Dujella (Zagreb) and Robert F. Tichy (Graz) Abstract Let B be a nonzero integer. Let define the sequence of polynomials
More informationHomework 8 Solutions to Selected Problems
Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x
More informationNonlinear Integral Equation Formulation of Orthogonal Polynomials
Nonlinear Integral Equation Formulation of Orthogonal Polynomials Eli Ben-Naim Theory Division, Los Alamos National Laboratory with: Carl Bender (Washington University, St. Louis) C.M. Bender and E. Ben-Naim,
More informationMathematics Notes for Class 12 chapter 7. Integrals
1 P a g e Mathematics Notes for Class 12 chapter 7. Integrals Let f(x) be a function. Then, the collection of all its primitives is called the indefinite integral of f(x) and is denoted by f(x)dx. Integration
More informationRandom Boolean expressions
Random Boolean expressions Danièle Gardy PRiSM, Univ. Versailles-Saint Quentin and CNRS UMR 8144. Daniele.Gardy@prism.uvsq.fr We examine how we can define several probability distributions on the set of
More informationChapter 4: Interpolation and Approximation. October 28, 2005
Chapter 4: Interpolation and Approximation October 28, 2005 Outline 1 2.4 Linear Interpolation 2 4.1 Lagrange Interpolation 3 4.2 Newton Interpolation and Divided Differences 4 4.3 Interpolation Error
More informationOn the Sequence A and Its Combinatorial Interpretations
1 2 47 6 2 11 Journal of Integer Sequences, Vol. 9 (2006), Article 06..1 On the Sequence A079500 and Its Combinatorial Interpretations A. Frosini and S. Rinaldi Università di Siena Dipartimento di Scienze
More informationMapping the Discrete Logarithm: Talk 2
Mapping the Discrete Logarithm: Talk 2 Joshua Holden Joint work with Daniel Cloutier, Nathan Lindle (Senior Theses), Max Brugger, Christina Frederick, Andrew Hoffman, and Marcus Mace (RHIT REU) Rose-Hulman
More informationGenerating Functions
Semester 1, 2004 Generating functions Another means of organising enumeration. Two examples we have seen already. Example 1. Binomial coefficients. Let X = {1, 2,..., n} c k = # k-element subsets of X
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More informationMTH310 EXAM 2 REVIEW
MTH310 EXAM 2 REVIEW SA LI 4.1 Polynomial Arithmetic and the Division Algorithm A. Polynomial Arithmetic *Polynomial Rings If R is a ring, then there exists a ring T containing an element x that is not
More informationSubdivisions of Large Complete Bipartite Graphs and Long Induced Paths in k-connected Graphs Thomas Bohme Institut fur Mathematik Technische Universit
Subdivisions of Large Complete Bipartite Graphs and Long Induced Paths in k-connected Graphs Thomas Bohme Institut fur Mathematik Technische Universitat Ilmenau Ilmenau, Germany Riste Skrekovski z Department
More informationarxiv: v1 [math.co] 24 Jul 2013
ANALYTIC COMBINATORICS OF CHORD AND HYPERCHORD DIAGRAMS WITH k CROSSINGS VINCENT PILAUD AND JUANJO RUÉ arxiv:307.6440v [math.co] 4 Jul 03 Abstract. Using methods from Analytic Combinatorics, we study the
More informationLattice paths with catastrophes
Lattice paths with catastrophes Lattice paths with catastrophes SLC 77, Strobl 12.09.2016 Cyril Banderier and Michael Wallner Laboratoire d Informatique de Paris Nord, Université Paris Nord, France Institute
More informationLinear DifferentiaL Equation
Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace
More informationMathematics for Business and Economics - I. Chapter 5. Functions (Lecture 9)
Mathematics for Business and Economics - I Chapter 5. Functions (Lecture 9) Functions The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set,
More informationAlmost all trees have an even number of independent sets
Almost all trees have an even number of independent sets Stephan G. Wagner Department of Mathematical Sciences Stellenbosch University Private Bag X1, Matieland 7602, South Africa swagner@sun.ac.za Submitted:
More informationHamming Weight of the Non-Adjacent-Form under Various Input Statistics. Clemens Heuberger and Helmut Prodinger
FoSP Algorithmen & mathematische Modellierung FoSP Forschungsschwerpunkt Algorithmen und mathematische Modellierung Hamming Weight of the Non-Adjacent-Form under Various Input Statistics Clemens Heuberger
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)
More informationGenerating Orthogonal Polynomials and their Derivatives using Vertex Matching-Partitions of Graphs
Generating Orthogonal Polynomials and their Derivatives using Vertex Matching-Partitions of Graphs John P. McSorley, Philip Feinsilver Department of Mathematics Southern Illinois University Carbondale,
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
More informationChapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)
Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the
More informationFunctions and Equations
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN
More informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More informationCOUNTING WITH AB-BA=1
COUNTING WITH AB-BA=1 PHILIPPE FLAJOLET, ROCQUENCOURT & PAWEL BLASIAK, KRAKÓW COMBINATORIAL MODELS OF ANNIHILATION-CREATION ; IN PREP. (2010) 1 A. = + + B. = 2 Annihilation and Creation A = Annihilate
More informationLECTURE 5, FRIDAY
LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we
More information2 Linear Differential Equations General Theory Linear Equations with Constant Coefficients Operator Methods...
MA322 Ordinary Differential Equations Wong Yan Loi 2 Contents First Order Differential Equations 5 Introduction 5 2 Exact Equations, Integrating Factors 8 3 First Order Linear Equations 4 First Order Implicit
More informationCALCULUS JIA-MING (FRANK) LIOU
CALCULUS JIA-MING (FRANK) LIOU Abstract. Contents. Power Series.. Polynomials and Formal Power Series.2. Radius of Convergence 2.3. Derivative and Antiderivative of Power Series 4.4. Power Series Expansion
More informationOn the number of F -matchings in a tree
On the number of F -matchings in a tree Hiu-Fai Law Mathematical Institute Oxford University hiufai.law@gmail.com Submitted: Jul 18, 2011; Accepted: Feb 4, 2012; Published: Feb 23, 2012 Mathematics Subject
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
More informationAn Asymptotic Formula for Goldbach s Conjecture with Monic Polynomials in Z[x]
An Asymptotic Formula for Goldbach s Conjecture with Monic Polynomials in Z[x] Mark Kozek 1 Introduction. In a recent Monthly note, Saidak [6], improving on a result of Hayes [1], gave Chebyshev-type estimates
More information2326 Problem Sheet 1. Solutions 1. First Order Differential Equations. Question 1: Solve the following, given an explicit solution if possible:
2326 Problem Sheet. Solutions First Order Differential Equations Question : Solve the following, given an explicit solution if possible:. 2 x = 4x3, () = 3, ( ) The given equation is a first order linear
More informationGaussian integrals and Feynman diagrams. February 28
Gaussian integrals and Feynman diagrams February 28 Introduction A mathematician is one to whom the equality e x2 2 dx = 2π is as obvious as that twice two makes four is to you. Lord W.T. Kelvin to his
More informationIndefinite Integration
Indefinite Integration 1 An antiderivative of a function y = f(x) defined on some interval (a, b) is called any function F(x) whose derivative at any point of this interval is equal to f(x): F'(x) = f(x)
More informationAnalytic combinatorics of chord and hyperchord diagrams with k crossings
Analytic combinatorics of chord and hyperchord diagrams with k crossings Vincent Pilaud, Juanjo Rué To cite this version: Vincent Pilaud, Juanjo Rué. Analytic combinatorics of chord and hyperchord diagrams
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
More informationFirst Order Differential Equations
Chapter 2 First Order Differential Equations Introduction Any first order differential equation can be written as F (x, y, y )=0 by moving all nonzero terms to the left hand side of the equation. Of course,
More information1. Definition of a Polynomial
1. Definition of a Polynomial What is a polynomial? A polynomial P(x) is an algebraic expression of the form Degree P(x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 3 x 3 + a 2 x 2 + a 1 x + a 0 Leading
More informationCIRCULAR CHROMATIC NUMBER AND GRAPH MINORS. Xuding Zhu 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 4, No. 4, pp. 643-660, December 2000 CIRCULAR CHROMATIC NUMBER AND GRAPH MINORS Xuding Zhu Abstract. This paper proves that for any integer n 4 and any rational number
More informationLecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2
More informationSeparation of Variables in Linear PDE: One-Dimensional Problems
Separation of Variables in Linear PDE: One-Dimensional Problems Now we apply the theory of Hilbert spaces to linear differential equations with partial derivatives (PDE). We start with a particular example,
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
More informationRoots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
More informationDIGITAL EXPANSION OF EXPONENTIAL SEQUENCES
DIGITAL EXPASIO OF EXPOETIAL SEQUECES MICHAEL FUCHS Abstract. We consider the q-ary digital expansion of the first terms of an exponential sequence a n. Using a result due to Kiss und Tichy [8], we prove
More informationRemarks on BMV conjecture
Remarks on BMV conjecture Shmuel Friedland April 24, 2008 arxiv:0804.3948v1 [math-ph] 24 Apr 2008 Abstract We show that for fixed A, B, hermitian nonnegative definite matrices, and fixed k the coefficients
More informationOn uniquely 3-colorable plane graphs without prescribed adjacent faces 1
arxiv:509.005v [math.co] 0 Sep 05 On uniquely -colorable plane graphs without prescribed adjacent faces Ze-peng LI School of Electronics Engineering and Computer Science Key Laboratory of High Confidence
More informationOptimal global rates of convergence for interpolation problems with random design
Optimal global rates of convergence for interpolation problems with random design Michael Kohler 1 and Adam Krzyżak 2, 1 Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289
More informationMA 123 (Calculus I) Lecture 3: September 12, 2017 Section A2. Professor Jennifer Balakrishnan,
What is on today Professor Jennifer Balakrishnan, jbala@bu.edu 1 Techniques for computing limits 1 1.1 Limit laws..................................... 1 1.2 One-sided limits..................................
More informationHAMILTONIAN CYCLES AVOIDING SETS OF EDGES IN A GRAPH
HAMILTONIAN CYCLES AVOIDING SETS OF EDGES IN A GRAPH MICHAEL J. FERRARA, MICHAEL S. JACOBSON UNIVERSITY OF COLORADO DENVER DENVER, CO 8017 ANGELA HARRIS UNIVERSITY OF WISCONSIN-WHITEWATER WHITEWATER, WI
More informationMath Practice Exam 3 - solutions
Math 181 - Practice Exam 3 - solutions Problem 1 Consider the function h(x) = (9x 2 33x 25)e 3x+1. a) Find h (x). b) Find all values of x where h (x) is zero ( critical values ). c) Using the sign pattern
More informationEXPLICIT EVALUATIONS OF SOME WEIL SUMS. 1. Introduction In this article we will explicitly evaluate exponential sums of the form
EXPLICIT EVALUATIONS OF SOME WEIL SUMS ROBERT S. COULTER 1. Introduction In this article we will explicitly evaluate exponential sums of the form χax p α +1 ) where χ is a non-trivial additive character
More informationTwo hours. To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER. 29 May :45 11:45
Two hours MATH20602 To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER NUMERICAL ANALYSIS 1 29 May 2015 9:45 11:45 Answer THREE of the FOUR questions. If more
More information