Báo cáo toán học: "Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe" pps

... means of series–rearrangements and finite differences, elementar y proofs are presented for the well–know n convolution identities of Abel and Hagen–Rothe. 1 Introduction There are numerous identities ... [16], this short paper will present elementary proofs for the convolution identities of Abel and Hagen–Rothe. It may be unexp ected that these proofs are su...

Ngày tải lên: 08/08/2014, 11:20

... knowledge of the state σ(B) and C. Proof. For the state σ(B)=(s 1 , ,s m ) and a subset of edges C of P m we construct a graph with vertices {1, 2}×{1, ,m} and edges B  ∪ C,whereB  is the set of edges joining ... number of acyclic orientations of G and by f(G) the number of spanning forests of G. ∗ Partially supported by projects SEUI-PB98-0933 and by CUR Gen. Ca...

Ngày tải lên: 07/08/2014, 07:21

... gives a more direct and elegant proof of the generating function. Moreover it also gives a more explicit expression for  V (d, n)z n . They even gave a sketch of a presentation of the first transfer-matrix ... the equation just before Example 4.7.17 in Stanley [6]. We now g ive a proof of Theorem 1. Proof. For x ∈ X, let A x be the infinite matrix (a i,j ) be defined by a i,j = 0...

Ngày tải lên: 08/08/2014, 01:20

... construction and give a bijective proof using inclusion–exclusion. 2 Exposition of identities and proofs The origin of this paper was the attempt to give a bijective proof of the following identity for ... 1) and ends at ( λ i − i, N) (see the left picture of Figure 2 for an illustration), and where the weight w (P) of such an r -tuple P is defined as before. 4 Biject...

Ngày tải lên: 07/08/2014, 06:22

... in ΠΣ-fields, the algorithms of [5] directly extend to this more general class of summands. Given a summand f(m, n, k) = h(m, n, k)S(n, k) as specified above and given a finite set of pairs S ⊆ Z 2 , the ... triangular recurrence of the form S(n, k) = a 1 (n, k)S(n + α, k + β) + a 2 (n, k)S(n + γ, k + δ) (4) with α, β, γ, δ ∈ Z and    α β γ δ    = ±1 and coefficients a 1 (n, k)...

Ngày tải lên: 07/08/2014, 15:22

Ngày tải lên: 05/08/2014, 09:46

... The proofs of these theorems include algorithms for expanding elements of these modules in terms of the respective bases. M. Haiman recently announced a proof showing that the dimension of S [X, ... ST λ denotes the collection of standard tableaux of shape λ, Υ ψ S is a collection of sequences defined in equation (3.4), and |C ρ,1 (M)| and |C ρ,2 (M)| denote the sum of...

Ngày tải lên: 07/08/2014, 07:21

... + m)/2 edges, and hence f(n) ≥3n/2 for n ≥ 7. Combined results of Clark, Entringer and Shapiro [3], [4] and Lin, Jiang, Zhang and Yang [7] show that f(n)=3n/2 for n ≥ 19 and for n =6, 10, ... minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n) ≥ 3n−2 2 −2forn ≥ 20 and g(n) ≤ 3n−2 2  for n ≥ 5...

Ngày tải lên: 07/08/2014, 13:21

... Let G j be the label of node u j and let g j be the label of node v j (so that g 0 = f). Then (i) G d ∩ {f} = ∅, and (ii) G d ∩ G i = ∅, for every 0 ≤ i ≤ d − 1. Proof. The proof is by induction ... holds for the base case, d = 1, let (v 0 , u 1 ) and (v 0 , u  1 ) be two different paths in T f,1 so that u 1 = u  1 . Let G 1 and G 2 be the labels of the nodes u 1 and u ...

Ngày tải lên: 07/08/2014, 21:21

... ‘dual Cauchy formula’ obtained in [14, (6.17)] and which is a particular case of the Cauchy identity for the double Schubert polynomials [12]. An independent proof of a version of (1.4) for the shifted ... transposition of the tableaux with respect to the main diagonal and swap i and i  for each i. Note that [24] also contains an equivalent combinatorial formula for Σ θ...

Ngày tải lên: 07/08/2014, 21:21