colouring maps and planar graphs

... are called ‘harmonic maps. ’ In this situation, the harmonic maps are precisely the rational maps and their complex conjugates (see [2, (11.5)]) In particular, being conformal maps from a surface, ... Wloc (R2 \S) as n → ∞ REPULSION AND QUANTIZATION IN ALMOST-HARMONIC MAPS 469 We refer to the map u∞ : S → S as a ‘body’ map, and the maps ω i : S → S as ‘bubble’ maps The points {x1 , , xm } ... k ≥ and sufficiently large n Almost-harmonic maps — the proof of Theorem 1.2 The goal of this section is to understand the structure of maps whose tension field is small when measured in L2 , and...

Ngày tải lên: 28/03/2014, 22:20

... Contents of Fordy Wood file:///tmp/index.htm The twistor approach Twistors, nilpotent orbits and harmonic maps, P.Z Kobak, p 295 Index Index of terms used in the articles, p 323 This page is maintained...

Ngày tải lên: 24/04/2014, 16:50

... [0, ) and (3.14) implies (3.15), and the conclusion of Corollary 3.3 follows from Corollary 3.2 (b) Theorems 3.1-3.4 and Corollaries 3.1 and 3.2 all generalize and improve [5, Theorem 3.4] and ... ∅ Remark 4.2 (a) Corollary 4.3 and Corollary 4.4 are indeed equivalent (b) Theorems 4.1-4.6 and Corollaries 4.1-4.4 all generalize and improve [5, Theorem 2.6] and the primitive Kannan’s fixed ... l.s.c; (w3) for any ε > 0, there exists δ > such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε Recently, Lin and Du introduced and studied τ-functions [5,9,18-22] A function p: X × X ® [0, ∞)...

Ngày tải lên: 20/06/2014, 22:20

... continued by Hussain et al [26], Latif and Al-Mezel [27], and Marín et al [1] In particular, the authors of [27] and [1] have obtained a Rakotch-type and a Bianchini-Grandolfi-type fixed point theorems, ... < t and limr→t+ sup ϕ(r) < t for all t >0, and such that, for each x, y Î X and each u Î Tx there exists v Î Ty with q(u, v) ≤ (q(x, y)) Theorem 2.6 Let (X, d) be a complete T qpm space and ... z Î Tz and q(z, z) = Proof By hypothesis, there is a function  : [0, ∞) ® [0, ∞) satisfying (0) = 0, (t) < t and limr→t+ sup ϕ(r) < t for all t >0, and such that for each x, y Î X and u Î...

Ngày tải lên: 21/06/2014, 02:20

... two-dimensional Fourier transform and inverse transform were used to characterize surfaces, planar defects, and preferred orientation, if any, of the phases Results The phase assemblages and microstructures ... magnetite and the predominant calcite with strong {10" and moderate {11" diffractions due 14} 20} to (hkil)-specific preferred orientation (Fig 1) Optical polarized micrographs under open and crossed ... degenerate planar bending based on previous assignments [18] The shoulder on the lowfrequency side of the 875 and 711 cm-1 band indicates a minor amount of aragonite that has characteristic doublet bands...

Ngày tải lên: 21/06/2014, 08:20

... integers, v ∈ X, and let di j (x) be defined as in (∗ ) Suppose there exist nonnegative real numbers a0 ,a1 ,a2 , ,an−1 and b0 ,b1 ,b2 , Fixed Point Theory and Applications with n −1 i=0 ≤ and ∞ j =0 ... is continuous, and inf x∈X d(x, f (x)) = inf n∈N d(x1n ,x2n ) = inf n∈N 1/n = Let c5 = a7 ≥ be a real number, and let other coefficients , c j , and bk be all 0, then both (A.1) and (A.3) hold ... given (X,d) and f : X → X, since d( f (x), f (y)) ≤ 1, and c5 d( f (x), f (x)) = a7 d( f (x), f (x)) = 1, both (A.2) and (A.4) hold, too However, it is clear that f has no fixed points, and each...

Ngày tải lên: 22/06/2014, 19:20

... 2 Diametrically contractive maps and fixed points references therin; we will call them shrinking We briefly recall some results and properties of maps in this class: (1) the fixed point, ... 1, 81–82 8 Diametrically contractive maps and fixed points [9] B Sims, Examples of fixed point free mappings, Handbook of Metric Fixed Point Theory (W A Kirk and B Sims, eds.), Kluwer Academic, ... diametrically contractive maps are shrinking; (2) if M is a compact set and T is shrinking, then it is diametrically contractive; (3) there are examples of shrinking maps that are not diametrically...

Ngày tải lên: 22/06/2014, 22:20

... et al [9], Jungck and Sessa [12], Khan and Khan [13], O’Regan and Shahzad [15], Rhoades and Saliga [16], Sahab et al [18], Shahzad [19–21], and Singh [22] Common fixed point and approximation ... -commuting maps and invariant approximations Theorem 2.2 Let I and T be self -maps on a q-starshaped subset M of a p-normed space X Assume that cl(T(M)) ⊂ I(M), q ∈ F(I), and I is affine Suppose that T and ... Al-Thagafi and Shahzad’s [2, Theorem 2.2 ], Dotson’s [3, Theorem 1], Guseman and Peters’s [4, Theorem 2], and Hussain and Berinde’s [7, Theorem 3.6] Theorem 3.1 Let T and I be self -maps on a compact...

Ngày tải lên: 22/06/2014, 22:20

... KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol 218, Marcel Dekker, New York, 1999 Y Q Chen et al [18] S S Zhang and G S Yang, ... theorem and minimax theorem Theorem 4.1 (Ky Fan’s coincidence theorem) Let X and Y be nonempty convex subsets of topological vector spaces E and F, respectively Let A,B : X → 2Y be two maps satisfying ... 1, ,l and t = 1,2, ,k By the convexities of A−1 x and B y, we have y0 ∈ Ax0 and y0 ∈ Bx0 This completes the proof Remark 4.2 The classical Ky Fan’s coincidence theorem assume that both X and Y...

Ngày tải lên: 22/06/2014, 22:20

... C: Mind maps - advanced methods and uses Overview • • • • • Models for the brain Technology and new insights into ourselves The left and right brain and mind mapping Advanced ... the same time The left-hand page should be for mapped information and the right-hand page for more linear or graphic information such as formulas, special lists, and graphs etc See fig 42 When ... colour, images, dimension, symbols, and visual rhythms etc: in other words Mind Maps Fig 40 The left and right brain 110 NOTING Fig 41 A mind map on the uses of mind maps See text page 112 III From...

Ngày tải lên: 22/07/2014, 22:22

... Chapter for yourself In these mind maps key words and images are linked to each other around a main centre (in these cases, the overall theme of a chapter), and a mental picture is built up of ... The mind maps on pages 97-100 represent a new method for noting There are four of them, and they summarise the first four chapters of the book A ... picture is built up of an entire thought structure • The theory and method for making these patterned notes is fully outlined in sections B and C of chapter 4, starting on page 86 • Use the notes for...

Ngày tải lên: 22/07/2014, 22:22

... B: Mind maps for recall and creative thinking Overview • • • • Exercise Linear history of speech and print Contrast: the structure of the brain Advanced note taking and mapping techniques ... other, that these are presented in lines and that we therefore write and note in lines, such linear presentation is not necessary for understanding, and in many instances is a disadvantage The ... is coming out, a continuing and enormously complex process of sorting and selecting is taking place in your 88 NOTING A Normal line structure - sentenced-based B Standard list structure - order-of-importance-based...

Ngày tải lên: 22/07/2014, 22:22

... these words and phrases and not others had been selected Answers usually included the following: 'good image words', 'imaginative', 'descriptive', 'appropriate', 'good for remembering', and 'evocative', ... title cards of some of your stories and challenging you to remember the titles and authors You would have no idea to start with to which story your cards referred, and would have to rely solely on ... words and key creative words, and the way in which they interact after a period of time has passed A key recall word or phrase is one which funnels into itself a wide range of special images, and...

Ngày tải lên: 22/07/2014, 22:22

... Key words Overview • Exercise key words; standard responses • Key words and concepts - creative and recall • Memory - a comparison between standard note and key word noting • Transition from advanced ... Key-Words from them, and to make Mind Maps of each essay In this way you will be reviewing your speed reading skills, developing your note taking and Mind Mapping skills, and establish110 ing ... concentration and memory) and then branch out in an organised fashion around that image, using Key-Words and Key Images As you continue to build up the Mind Map, your brain creates an organised and integrated...

Ngày tải lên: 22/07/2014, 22:22

... two -graphs (the set of unbalanced triangles of a signed complete graph) and on even and odd polygons and paths in graphs and digraphs Nevertheless, it was not always easy to decide what belongs I ... even-cycle and bicircular matroids; bidirected graphs; binary clutters (which are equivalent to signed binary matroids); some of the literature on two -graphs and double covering graphs • And some ... η : V → {±1}, for signed planar and toroidal graphs and subgraphs of 3-dimensional grids Analyzed structurally, in terms of perfect matchings in a modified dual graph, and algorithmically The last...

Ngày tải lên: 07/08/2014, 05:23

... Switching; Negation; Subgraphs and contractions; Subdivision and splitting Relations 26 Line Graphs 27 Covering or Derived Graphs 28 Matrices 29 Matroids 29 Topology (of signed graphs) 31 Coloring ... Gain, and Biased Graphs 12 Basic Concepts of Signed Graphs 12 Aspects of balance Clusterability Additional Basic Concepts for Gain and Biased Graphs ... is, IV and IE are incidence relations between, respectively, I and V (this is the “vertex incidence relation”) and I and E (this is the “edge incidence relation”) The members of V , E , and I...

Ngày tải lên: 07/08/2014, 05:23

... ring and let A and A be two n × n matrices Then rankR (A + A ) ≤ rankR (A) + rankR (A ) Proof: Let A = BC and A = B C , where B is an n × r and C is an r × n matrix, while B is an n × r and C ... of the columns of B and B , and C is an (r + r ) × n matrix, formed from the union of rows of C and C The following theorem shows, that for any prime p, the co-triangle (and, consequently, the ... result of Ramsey [7] and Erd˝s and Szekeres [4], every n-vertex graph has o either a clique on k, or an anti-clique on vertices, if n≥ If we set k = log n log log n k+ −2 k−1 , and = log2 n , then...

Ngày tải lên: 07/08/2014, 06:20

... Bollob´s, Colouring lattices, Alg Universalis (1977), 313–314 a [3] B Bollob´s, Random Graphs, Academic Press 1985 a [4] B Bollob´s, G Brightwell and J Neˇetˇil, Random Graphs and Covering Graphs ... Solution March 1948 [6] P.Erd˝s and A.R´nyi, On the evolution of random graphs, Publ Math Inst o e Hungar Acad Sci (1960) 17-61 [7] T.Luczak, The chromatic number of random graphs, Combinatorica 11 ... was introduced by Bollob´s, a Brightwell, and Neˇetˇil [4] s r First consider dense random graphs It is shown in [4] that for arbitrary integer l ≥ and p = p(n) = o n(l−2)/(l−1) , c(Gn,p ) ≤...

Ngày tải lên: 07/08/2014, 06:20

... have been proved for planar graphs However, even restricted to planar graphs, Jaeger’s conjecture remains largely open For planar graphs, the flow problem can be dualized to a colouring problem For ... of nearly eulerian graphs and vertex splitting, manuscript, 2000 [20] C.Q.Zhang, Integer flows and cycle covers of graphs, Marcel Dekker Inc New York, (1997) [21] X Zhu, Planar graphs with circular ... graphs, manuscript, 2000 [13] J Neˇetˇil and X Zhu, On Bounded Tree-Width Duality of Graphs, J Graph Theory, s r 23 (1996), 151-162 [14] E Steffen and X Zhu, On the star chromatic numbers of graphs, ...

Ngày tải lên: 07/08/2014, 06:22

... submaps and minors Forbidden submaps In this subsection we study the topological minor inclusion on maps We define for each g ≥ a finite set Fg of forbidden maps, and we use it to characterize maps ... of the considered graphs Such an informative monadic second-order expressibility has been established by Courcelle [6] for 3-connected planar graphs and for ordered planar graphs We consider ... to the given graph and yields the desired “informative” characterization by a monadic second-order formula 1.1 Preliminaries Graphs and maps All graphs will be undirected and finite They may have...

Ngày tải lên: 07/08/2014, 07:21