We give a direct combinatorial proof of their result by characterizing when a product of chains is strictly unimodal and then applying O'Hara's structure theorem for the partition lattice L(m,n) L ( m , n ) mathContainer Loading Mathjax . In fact, we prove a stronger result: if m,n...
Explain how to read a binomial table and what it demonstrates. Prove: Every tree is a bipartite graph by induction on the number of vertices n. You must give a proof by induction. Prove, using mathematical induction, that for any n\geq 1 : 1^{2}+2^{2}+...+n^{2}=\frac{n(n+...
Our proof of Theorem 1.1, given in Section 3.1, casts the statement into the language of cones in McMullen's polytope algebra [14]. In order to further popularize the polytope algebra we give a brief, tailor-made introduction. For the discrete volume, Corollary 1.2 can also be obtained by...
deg (gj ) n i=1 hi gi , and completes the proof. 2 Proof of Theorem 1.2. Clearly we may assume that |Si | = ti +1 for all i. Suppose the result is false, Theorem 1.1 there are polynomials h1 , . . . , hn ∈ F [x1 , . . . , xn ] so that n f= i=1 hi gi ....
variant of \(\textsf{p}\) it should be possible to adapt the proof of theorem 1.1 to show that there exists \(\beta _{\textsf{p}}^b>0\) such that $$\begin{aligned} \beta _{\textsf{p}}^b \int _{\omega } \rho ^{1-\frac{p}{d}}\le \liminf _{n \rightarrow \...
The proof of the following statement is ira- mediate. Corollary 1. (a) I / L has a cross-cut with one element, then #(0, 1) = 0. (b) 1 / L has a cross-cut with two elements, then the only two possible values o[ #(0, l) are 0 and 1. (e) I / L has a cross-...
This formulation of f p (x, y, z), which is believed to be novel, establishes the new identity. However, it appears to offer no new insights into a possible elementary proof of FLT. 362 JOSEPH SINYOR ET AL. To discover the identity, note that r p (x, y, z) =p 2m l=0 ...
[ICALP 1998]. For our proof we give a simple lemma which allows us to convert closeness in Kolmogorov (cdf) distance to closeness in statistical distance. As a corollary of our technique, we give an alternative proof of a powerful variant of the classical central limit theorem showing ...
We,rst established the unimodality of the sequence {bl,m} in[8]by a complicated argu-ment. The proof of Theorem 3.1 given in [6] is completely elementary. The identity (2.7)shows that the unimodality of bl,m follows from it. Theorem 3.1. If P(x) is a polynomial with positive nondecr...
Using the definition of Pn, Fn and Ln, we can derive the following theorem. Theorem 2.1 Let Ln be the matrix as in (3). For the Pascal matrix Pn and the Fibonacci matrix Fn, we have Pn=FnLn. Proof Since the matrix Fn is invertible, we will prove Fn−1Pn=Ln. Let Fn−1=[...