... CHAPTER ** The ** ** Binomial ** ** Distribution ** 1 1 1 6 10 15 10 20 15 Each number in ** the ** triangle is ** the ** sum of ** the ** number above and to ** the ** right of it and ** the ** number above and to ** the ** left of it For example, ** the ** ... number 10 in ** the ** ﬁfth row is found by adding ** the ** and in ** the ** fourth row ** The ** number 15 in ** the ** sixth row is found by adding ** the ** and 10 in ** the ** previous row ** The ** numbers in each row represent ** the ** number ... ** distribution ** CHAPTER ** The ** ** Binomial ** ** Distribution ** A ** binomial ** ** distribution ** is obtained from a probability experiment called a ** binomial ** experiment ** The ** experiment must satisfy these conditions: Each...

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... indicates ** the ** speciﬁc node in ** the ** ﬁnal column of nodes 77 ** The ** ** Binomial ** ** Model ** (D) Assume ** the ** derivative depends only on ** the ** ﬁnal stock price Corresponding to ** the ** stock price at each ﬁnal node, there ... drops by ** the ** amount of ** the ** dividend Unfortunately, this dislocates ** the ** entire tree as shown ** The ** tree is said to have become bushy Let us recall ** the ** original random walk on which ** the ** ** binomial ** ** model ** ... out ** the ** stock value for each node in ** the ** tree but if ** the ** tree is European, we only need ** the ** stock values in ** the ** last column of nodes (C) Corresponding to each of ** the ** ﬁnal nodes at time t = T , there...

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... [50] ** Develop ** ** computer ** ** programs ** ** for ** ** simplifying ** ** sums ** ** that ** ** involve ** ** binomial ** coeﬃcients Exercise 1.2.6.63 in ** The ** ** Art ** ** of ** ** Computer ** ** Programming, ** ** Volume ** ** 1: ** ** Fundamental ** ** Algorithms ** by Donald ... beforehand Let’s give ** the ** ﬂoor to Dave Bressoud [Bres93]: **The ** existence ** of ** ** the ** ** computer ** is giving impetus to ** the ** discovery ** of ** ** algorithms ** ** that ** generate proofs I can still hear ** the ** echoes ** of ** ** the ** ... lucky ** that ** computers had not yet been invented in Jacobi’s time It is possible ** that ** they would have prevented ** the ** discovery ** of ** one ** of ** ** the ** most beautiful theories in ** the ** whole ** of ** mathematics: ** the ** theory...

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... a bit on ** the ** preparation ** of ** ** the ** beholder A proof which is not understood will not produce ** the ** aha! reaction ** Of ** ** the ** ** proofs ** given for ** the ** ** binomial ** ** theorem ** ** the ** induction proof and ** the ** proof using ... proof ** of ** ** the ** Pythagorean ** Theorem)** and others by ** the ** element ** of ** surprise in how their pieces ﬁt together (Euclid’s proof ** of ** ** the ** Pythagorean ** Theorem)** In this paper I propose to consider several ** proofs ** ... ** of ** ** the ** ** proofs ** ** The ** induction proof suggests ** the ** utility ** of ** recurrences It also gives one ** of ** ** the ** most basic examples ** of ** an essential proof technique As such it opens vistas on many parts ** of ** mathematics...

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... terms ** of ** each ** binomial ** is Denote by xp ** the ** g.c.d ** of ** ** the ** ﬁrst term ** of ** fu ** and ** ** the ** ﬁrst term ** of ** fv , by xt ** the ** g.c.d ** of ** ** the ** ﬁrst term ** of ** fu ** and ** ** the ** second term ** of ** fv , by xr ** the ** g.c.d ** of ** ** the ** second ... codimension Proof Since ** the ** ** Rees ** ring R(I) = K[x, T ]/J is ** of ** dimension n + 1, then ** Symmetric ** ** and ** ** Rees ** ** Algebras ** ** of ** ** Some ** ** Binomial ** Ideals 67 codim(K[x, T ]/J ) = (n + 4) − (n + 1) = In addition, ** the ** ... intersection In both cases, ** the ** ** Rees ** algebra ** and ** ** the ** ** Symmetric ** algebra are isomorphic Proof Assume that one ** of ** four monomials xp , xt , xr , xs is a unit Because ** the ** role ** of ** these four monomials is the...

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... integer ≥ ** The ** idea ** of ** his proof was to compare ** the ** actual asymptotic behavior ** of ** ** the ** given ** sum,** for ﬁxed s and n → ∞, with ** the ** asymptotic behavior ** of ** a hypothetical ** closed ** form, and to show that ** the ** ... Recurrences for sums ** of ** powers ** of ** ** binomial ** ** coeﬃcients,** J Comb Theory Ser A 52 (1989), 77–83 [McI] Richard J McIntosh, Recurrences for alternating sums ** of ** powers ** of ** ** binomial ** ** coeﬃcients,** J Comb Theory Ser ... r ** the ** argument would work, but without further human input it could not produce a general proof, i.e., a proof for all p, r This is somewhat analogous to ** the ** sums ** of ** ** the ** pth powers ** of ** all ** of ** the...

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... ** A ** ** BINOMIAL ** ** COEFFICIENT ** ** IDENTITY ** ** ASSOCIATED ** ** TO ** ** A ** ** CONJECTURE ** ** OF ** BEUKERS Scott Ahlgren, Shalosh B Ekhad, Ken Ono and Doron Zeilberger Using the WZ method, ** a ** ** binomial ** coeﬃcient ** identity ** ... sum satisﬁes ** a ** certain (homog.) third order linear recurrence equation ** To ** ﬁnd the recurrence, and its proof, download the Maple package EKHAD and the Maple program zeilWZP from http://www.math.temple.edu/~ ... zeilWZP(k*(n+k)!**2/k!**4/(n-k)!**2,F,G,k,n,N): References **[A-** O] [B] [Z] S Ahlgren and K Ono, ** A ** Gaussian hypergeometric series evaluation and Ap´ry number congruences (in prepae ration) F Beukers, Another congruence for Ap´ry numbers, J...

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... the ** q-binomial ** ** theorem ** ** and ** Theorems 1.2 ** and ** 1.3 Our combinatorial proof of the ** q-binomial ** ** theorem ** is based on ** Theorem ** 2.1, ** and ** is essentially the same as that of Alladi [2] or Pak [8] Proof of ** Theorem ** ... on the right-hand side is equal to q |µ| aodd(µ) µ∈P1 (µ)≤n The proof then follows from the involution σ in the proof of ** Theorem ** 2.1 Proof of ** Theorem ** 1.2 Replacing q ** and ** z by q ** and ** −zq, respectively, ... σ has the required properties ** and ** ** Theorem ** 2.1 is proved Note that the above involution σ on P1 also preserves odd(λ) Combinatorial Proofs of Theorems 1.1, 1.2, ** and ** 1.3 In this section, we give...

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... Ekhad - K Ono - D Zeilberger, A ** binomial ** ** coeﬃcient ** ** identity ** ** associated ** to a ** conjecture ** of Beukers, The Electronic J Combinatorics (1998), #R10 [3] F Beukers, Another congruence for ** Ap´ry ** ** numbers,** ... Recently, Ahlgren and Ono [1] have shown that this ** conjecture ** is implied by the following beautiful ** binomial ** ** identity ** n n k k=1 2 n+k k + 2kHn+k + 2kHn−k − 4kHk = (1) which has been conﬁrmed successfully ... of the Theorem the electronic journal of combinatorics 11 (2004), #N15 References [1] S Ahlgren - K Ono, A Gaussian hypergeometric series evaluation and ** Ap´ry ** number e congruences, J Reine Angew...

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... (3) and (4) are obviously equivalent Recently, an elegant combinatorial ** proof ** ** of ** (4) was given by Shattuck [12], and ** a ** little complicated combinatorial ** proof ** ** of ** (2) was provided by Chen and Pang ... combinatorial proofs for q = 1, we propose ** a ** combinatorial ** proof ** ** of ** (5) within the framework ** of ** partition theory by applying an algorithm due to Zeilberger [3] the electronic journal ** of ** combinatorics ... such that bk − ik as−ik **(a0** = +∞) and bk − ik becomes ** a ** part ** of ** λ and ik becomes ** a ** positive part ** of ** µ The ** proof ** ** of ** (5) By the inverse ** of ** Algorithm Z, the relation (8) holds and therefore (7) may...

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... version ** of ** ** the ** ** theorem,** ** and ** then use analytic methods (in ** the ** form ** of ** ** the ** Identity ** Theorem)** to prove ** the ** full version We also prove three somewhat unusual summation formulae, ** and ** use these to ... n=0 ** the ** electronic journal ** of ** combinatorics 18 (2011), #P60 (4.3) ** and ** (4.2) will then follow from ** the ** Identity ** Theorem,** by an argument similar to that used in ** the ** proof ** of ** ** the ** ** q-Binomial ** ** Theorem ** ... partition theory ** and ** elliptic modular functions – their ** proofs ** – interconnection with various ** other ** topics in ** the ** theory ** of ** numbers ** and ** some generalizations thereon, PhD thesis (1970), University of...

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... purpose ** of ** this paper is to give a ** q-analogue ** ** of ** (1.6) and (1.7) as follows: ⌊n/2⌋ k=0 ⌊n/4⌋ k=0 m+k k m+k k q2 n−2k m+n m+1 , q( ) = n n − 2k q q q4 n−4k m+1 q( ) = n − 4k q where the q **-binomial ** ** coeﬃcient ** ... k < We shall give two proofs ** of ** (1.8) and (1.9) One is combinatorial and the other algebraic the electronic journal ** of ** combinatorics 18 (2011), #P78 2 Bijective proof ** of ** (1.8) Recall that a partition ... sequence ** of ** nonnegative integers (λ1 , λ2 , , λr ) in decreasing order λ1 λ2 · · · λr A nonzero λi is called a part ** of ** λ The number ** of ** parts ** of ** λ, denoted by ℓ(λ), is called the length ** of ** λ Write...

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... best ** model ** between ** hierarchical ** ** binomial ** ** logit ** ** model ** and binary ** logit ** ** model,** respectively Preselection ** of ** variables is also prepared in this chapter so that application ** of ** ** hierarchical ** ** binomial ** ** logit ** ... use ** hierarchical ** ** binomial ** ** logit ** models to predict ** crash ** ** severity ** ** of ** different ** crash ** types at rural intersections, while (Huang et al (2008) found the impacts ** of ** risk factors on ** severity ** ** of ** drivers’ ... level ** of ** the hierarchy ** of ** ** crash ** injury In addition, the features ** of ** crashes have higher levels because the same ** crash ** may have different effects on the ** severity ** ** of ** drivers A hierarchy ** of ** ** crash ** severity...

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... different ** methods ** are available: solution assays such as colorimetric assays for hexose and sialic acid; membrane-based ** methods ** such as slot-blotting and staining with periodic acid-Schiff reagent; ... correct volume of 10 Davies and Carlstedt ice-cold dry propan-1-ol to give a 100 mM solution DFP is unstable in water but can be stored at –20°C in propan-1-ol After dilution, the vial as well as ... putative cell membrane-associated mucin Biochem J 338, 325–333 Desseyn, J.-L., Guyonet-Dupérat, V., Porchet, N., Aubert, J.-P., and Laine, A (1997) Human mucin gene MUC5B, the 10.7-kb large central...

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