a tour of checkerboard worlds

... Martin Gardner and Ian Stewart, two scintillating stars in the universe of recreational mathematics and mathematics education, are always a source of inspiration Martin Gardner, a mathematician, ... Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Singapore Taipei ... Einstein, Ramanujan, Hawking 66 33 A Ranking of the Most Influential Female Mathematicians 69 34 A Ranking of the Saddest Mathematical Scandals 35 The 10 Most Important Unsolved Mathematical Problems...

Ngày tải lên: 28/10/2014, 23:31

... has many articles last since last August My sister read read many articles night (since last August) My friend has worked at a pizza place since 2003 My friend worked at a pizza place a year ago ... delete any past tense time phrases Maria was in the garden yesterday (since a. m.) Maria has been in the garden since a. m Bill Gates has written emails yesterday 1975 Bill Gates wrote many many emails ... several books and has given billions of dollars to schools For the last several years, he and his wife Melinda have donated a substanial amount of money to organizations working in global health...

Ngày tải lên: 19/08/2013, 14:15

... is an acceptable Experiential perfect The Resultative reading is not available, as (35c) and (35d) illustrate (35) a Maria vinagi e pristigala v Maria always is arrive-IMPERF in ‘Maria has always ... Maria.’ b Ivan obic!as!e Maria Ivan love-IMPERF.PAST Maria ‘Ivan loved Maria.’ (23) a Ivan pisna (*dva c!asa) Ivan scream-PERF.PAST two hours ‘Ivan started screaming (*for two hours).’ b Ivan ... pristignala sega Maria is arrive-PERF now ‘Maria has arrived now.’ d Maria e pristignala i s!te sedi utre Maria is arrive-PERF and will stay till tomorrow ‘Maria has arrived and will stay until...

Ngày tải lên: 18/10/2013, 14:15

... carta Tengo escrita la carta I have written the letter I’ve gotten the letter written 49 A STUDENT GRAMMAR OF SPANISH Similarly: Han abandonado la casa Tienen abandonada la casa Hemos alquilado ... aprovechar las siguientes palabras: Llegar rancho de un amigo Abrir barrera Entrar casa Platicar ranchero Presentar familia Introducir sal´ n Visitar corral Ver los burros, otros animales Subir a caballo ... familia, su ´ ´ capacidad intelectual, su a cion a los deportes, su ropa, su casa Aprovecha los ´ vocablos de abajo como gu a: ı ser, tocar m´ sica, ejercer abogado (a) /contable, trabajar una...

Ngày tải lên: 01/11/2013, 06:20

... began raining two hours ago and it is still raining Trời m a Trời m a hai m a How long has it been raining? It has been raining for two hours Trời m a hai tiếng đồng hồ Chúng ta dùng present perfect ... been talking to Carol about the problem and she thinks that … Mình v a đề cập đến vấn đề Carol cô nghĩ … B It has been raining for two hours Xét ví dụ sau: It is raining now It began raining ... How many pages of that book have you read? Bạn đọc trang sách rồi? Mary has written ten letters today Hôm Mary viết mười thư They’ve played tennis three times this week Họ chơi quần vợt ba lần...

Ngày tải lên: 19/01/2014, 17:20

... từ lúc Chúng ta dùng simple để hỏi hay nói how much, how many hay how many times (cho hành động kết thúc) How many pages of that book have you read? Bạn đọc trang sách rồi? Mary has written ten ... động xảy ra) How long have you been reading that book? Bạn đọc sách rồi? Mary is still writing letters She’s been writing letters all day Mary viết thư Cô viết thư ngày hôm They’ve been playing tenis ... It’s nice to meet you again What have you been doing since we last met? Rất mừng gặp lại bạn Bạn làm từ gặp lần sau cùng? Where’s the book I gave you? What have you done with it? Cuốn sách...

Ngày tải lên: 22/06/2014, 21:20

... (laughs) Oh well, what are we going to do? D Let`s just make a note of their good points and bad points again Let`s take Bill… A dictation of names and places My name`s Jameson That`s J -A- M-E-S-O-N I`ll ... capon lin`d, With eyes severe, and beard of formal cut, Full of wise saws and modern instances; And so he plays his part The sixth age shifts Into the lean and slipper`d pantaloon With spectacles ... experience of the stage and the television He`s been with the Royal Shakespeare Company for years, and I think we could probably work on the appearance side, you know Make-up would age him D Harry...

Ngày tải lên: 12/07/2014, 17:20

... a partition With the above construction of SQ(µ) = A ∪ A1 ∪ A2 and R, we have  (1)  h(x) x∈R In particular,  h(x) = sµ (1, · · · , 1) x A1 A x A1 A h(x) divides µ1 x∈R h(x) [Note that A ... 13 · · · · = 144 Note that the hook numbers in SR(µ) are the same as those in the area A1 A of SQ(µ) As usual, µ1 = (µ) is the number of nonzero parts of µ Recall that sµ (x1 , x2 , · · ·) is ... electronic journal of combinatorics no.1 (1997), #R22 §1 Introduction and the main results Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric...

Ngày tải lên: 07/08/2014, 06:20

... [1] A pagoda function is a function P (x) that assigns a real value to each position on the board, with the property that the sum of the values of all peg positions can never increase by a legal ... empty Then the range of the pegs is bounded, even if we allow diagonal moves Upper bounds for small values of n can be found in Table Table Unreachable levels of Diapegs in Zn n unreachable level ... paper, we will study how far pegs can reach when diagonal moves are allowed, such as r e e r e r - - 4.1 The pagoda function We will use the standard tool for this kind of problems: the pagoda...

Ngày tải lên: 07/08/2014, 06:20

... Jerrum, A Sinclair and E Vigoda, A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries, to appear [7] A V Kostochka, The number of spanning trees in graphs ... proof shows that the number of Hamilton cycles is within a polynomial factor of the number of two factors of G Therefore one can generate a (near) random Hamilton cycle of G by generating (near) ... number of 2-factors of G can be efficiently approximated to within arbitrary accuracy (Jerrum, Sinclair and Vigoda [6]) we see that we can efficiently estimate the number of Hamilton cycles of G to...

Ngày tải lên: 07/08/2014, 06:22

... Math Soc 13 (1976), 351–359 [14] S Thomass´ and A Yeo, Total domination of graphs and small transversals of hye pergraphs To appear in Combinatorica [15] Z Tuza, Covering all cliques of a graph ... edges in a graph G are independent if they are not adjacent in G A set of pairwise independent edges of G is called a matching in G, while a matching of maximum cardinality is a maximum matching ... have the following result Observation Every two degree-4 vertices in G are at distance at most apart Proof Suppose that G contains two degree-4 vertices, say u and v, at distance at least apart...

Ngày tải lên: 07/08/2014, 13:21

... Proof By Lemma 2.2, mr(MCS ) = C Let A be a real realization of CS1 of minimum rank We claim that rank (A) ≥ Suppose to the contrary that rank (A) ≤ Let W be a real matrix whose columns form a basis ... Computation of minimum rank The question of the decidability of the minimum rank of a graph over a field F was raised at the 2006 American Institute of Mathematics workshop,“Spectra of Families of ... Proof The proof proceeds as that of Theorem 3.1 Let A be a matrix whose graph is G2 Thus, A has the form (1) where D and E are diagonal matrices, and B has pattern CS2 We claim that if A is...

Ngày tải lên: 07/08/2014, 15:23

... sake of completeness In the following problem, a planarization of G is a graph, obtained from a drawing of G by replacing every crossing with a vertex A crossing minimal planarization is a planarization, ... other hand, is far from clear, as the edges of the two bundles can possibly cross each other arbitrarily often in an optimal drawing of the zip product, and optimal drawings of the original graphs ... Hk,0 is a subgraph of Hk,r , we have cr(Hk,r ) cr(Hk,0) On the other hand, one can alter an optimal drawing of Hk,0 to a drawing of Hk,r with the same number of crossings, as in Figure 1 (a) , concluding...

Ngày tải lên: 08/08/2014, 12:22

... annihilation a and creation operator a satisfying the commutation relation aa† − a a = of the Weyl algebra Since the normal ordered form has many desirable properties simplifying many calculations, ... of the conventional Stirling number of second kind find a simple analogue For example, the interpretation of S(n, k) as a rook number of a staircase Ferrers board generalizes in a beautiful fashion ... formula, Ramanujan J., to appear [22] M .A Mendez, P Blasiak and K .A Penson, Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem, J Math Phys 46 (2005) Article...

Ngày tải lên: 08/08/2014, 14:23

... transformation [FS78] The most natural statistic that can be associated with “imaj” is then “ides.” It is again remarkable that D´sir´ Andr´’s set-up will also provide the appropriate combinatorial ... in an alternating way The set of alternating (resp falling alternating) permutations of order n is denoted by Tn (resp by T′ ) n D´sir´ Andr´’s main result was to show that tangent and secant ... of the traditional q-tangent and q-secant numbers T2n+1 (q), En (q) defined in (1.1)q and (1.2)q The order of the variables t, q matters, as other authors have spoken of (q, t)-analogs, in particular...

Ngày tải lên: 08/08/2014, 14:23

... Catch! (page 30) This activity can be played as a practical activity as part of a PE lesson or in the classroom, passing around an object rather than throwing a ball SUGGESTED QUESTION: • What ... developing an awareness of how numbers relate to each other This awareness underpins all mental calculation and it is very important that children have a wide range of experience of comparing and ordering ... interpreting the denominator as the parts of a whole and the numerator as the number of parts; identify and estimate fractions of shapes; use diagrams to compare fractions and establish equivalents 652 325...

Ngày tải lên: 31/10/2014, 13:36

Ngày tải lên: 23/05/2015, 07:54

... children admitted the money a having taken b have taken c having take d having been taken 30 twice, he didn’t want to try again a Having fail b Have failed c Having failed d T o have failed ... been happened 14 Last year I studied abroad I appreciate _ the opportunity to live and study in a foreign country a having been had b having had c to have had d to have been had 15 _ all ... calls on the office phone a having made b to have made c to have been made d having been made 24 hid own boss for such a long time, he found it hard to accept orders from another a To have...

Ngày tải lên: 03/06/2015, 09:09

... wholesale and retail trade sector, where the number of women rose from about 550 000 in 1975 to almost 800 000 two decades later The number of men in this sector remained stable over the period, at ... 000 men and 250 000 women worked in the communications sector Twenty years later, though the number of men remained unchanged, the number of women rose to 550 000 A similar situation was seen ... the period, at around 700 000 Women also made gains in both the finance/banking industries and in the defence-related public sector Whereas some 125 000 women worked in finance and banking institutions...

Ngày tải lên: 04/10/2012, 10:02

Ngày tải lên: 25/10/2012, 10:02