... progressions with the difference ρ Precisely ϕr of them have length n/ρ and ϕ − ϕr are of length n/ρ Since these progressions are pairwise disjoint, there are at least (ϕ + ϕr )2 n/ρ the electronic ... comments Due to their suggestions we were able to prove the main result ofthe note in its present sharp form the electronic journal of combinatorics (2000), #R30 References [1] N Alon, Independent ... 68 (1998), 151–160 [6] P J Cameron, P Erd˝s, Onthenumberof sets of integers with various o properties, in R A Mollin (ed.), Number Theory: Proc First Conf Can Number Th Ass., Banff, 1988, de...
... descendants, and the claim in the proposition follows Onthe other hand, let j be thenumberof comparisons made between the j th element and other elements, during the partition where the j th element ... LBSTs, then the proposition applies for the variant of quickselect that uses the median of a random sample of three elements as the pivot in each partitioning phase The study ofthenumberof descendants ... define thenumberof descendants Dn,j as the size ofthe subtree rooted at the j th node, so we count the j th node as a descendant of itself Thenumberof ascendants An,j is thenumberof internal...
... both cases the bounds are “close” to the expected numberof in the random graph Gn,d The results here are strongly related to the result of Alon, R¨dl and Ruci´ ski [2] o n They considered bipartite ... First of all, part (b) of Theorem improves Corollary 2.9 of Thomason [9] which estimates thenumberof Hamilton cycles in a pseudo-random graph Secondly, if G is dn-regular and the second eigenvalue ... the electronic journal of combinatorics (2000), #R57 Hamilton Cycles A Hamilton cycle is the union of two perfect matchings and so h(G) ≤ m(G)2 and the upper bound in part (b) of Theorem follows...
... mn/4 They conjectured (the case k = of Conjecture in [3]) that b(m, n) ≤ 2mn/3 In this paper we prove this conjecture, and together with a well-known construction obtain that it is sharp Furthermore, ... M2 Denote by M3 the block consisting ofthe rows i = |A+ | + 1, , m and the columns j = 1, , ρn Denote by M4 the block consisting ofthe rows i = |A+ | + 1, , m and the columns j = 2ρn+ ... Denote by M5 the block consisting ofthe rows i = 1, , |A+ | and the columns j = 2ρn + 1, , n Define β = |A+ |/m Figure visualizes these terms Let c(s, k) denote thenumberof entries of M equal...
... algebra, the radical of an ideal I is the set of all elements ofthe ring some positive power of which belongs to I; in the present situation, q(m) is the positive generator ofthe radical ofthe ... The function m → κm 4.3 Statement and proof ofthe main theorem The situation is the following: we look for the asymptotics ofthe coefficients pn (m) ofthe formal series Pm = Cm Rm where the coefficients ... dn The conclusion follows now from the dominated convergence theorem the electronic journal of combinatorics (2002), #R3 Figure shows the first thousand values ofthe function m → πm , with m on...
... in the region F GM, given by aY (F1 , F2 ), times thenumberof configurations in the region AKJMGDE, times thenumberof configurations in the region F MLKBC In order to compute thenumberof configurations ... of FPL configurations in the region ABCLK which respect the matching X Clearly, the region ABCLK to the right of ξ1 does not dependon m We denote thenumberof FPL configurations of that region ... and the line connecting A and K We state the result ofthe application of Lemma 3.1 to the current case in form ofthe following lemma Lemma 6.1 The region of fixed edges ofthe FPL configurations...
... is not strong and that no sets A and B satisfy the lemma Let t be thenumberof strong components of T − v and let Wi be the set of vertices in the ith strong component If |W1 | ≥ 3, then choose ... other arcs are ofthe form vi → vj with i < j In other words, reversing the arc → v1 gives the transitive tournament of order n As noted by Moon [1], there is a bijection between partitions of ... apply the induction hypothesis using the paths P and Q to obtain a path R satisfying the lemma Next, we repeat the above argument with the portion of R beginning at um−1 and the vertex um Theorem...
... answers a conjecture of J Benton, R Snow, and N Wallach in [1] the electronic journal of combinatorics 13 (2006), #N8 Start ofthe proof Let N = {0, 1, } Define the weight of a matrix N to be the ... y1 ofthe vector y We will need a subsidiary function Let N(λ; t) be thenumberof solutions of (i) and (ii) with y1 = t By definition, N(λ) = t≥0 N(λ; t) We need one more definition to state the ... we have the following lemma Lemma Let a in Nn+1 satisfy |a| = m Then thenumberof x in Nm such that a(x) = a is m a Let λ = (λ1 , , λn ) = λ(x) be the column sums ofthe matrix R constructed...
... This implies that thenumberofthenumberof k-tuples of k mutually orthogonal vectors in E is also |E|k −(k) (1 + o(1)) q , k! completing the proof of Theorem 1.2 Acknowledgments The research is ... correspond to the equivalence classes ofthe set of all non-zero vectors x = (x1 , , xd ) over q , where two vectors are equivalent if one is a multiple ofthe other by an element ofthe field ... induced copies of H Alon extended this result to (n, d, λ)-graphs He proved that every large subset ofthe set of vertices of an (n, d, λ)-graph contains the “correct” numberof copies of any fixed...
... is the intersection of {E1 , E2 , .,Ek+1 } and the points of one ofthe e-graphs, and point Ek+1 is the intersection of {E1 , E2 , , Ek+1 } and the points ofthe other ofthe e-graphs Theorem ... EF the outermost edges of it Also, call the set of crossed edges ofthe e-graph the set of parallel edges ofthe e-graph and call the set of edges ofthe e-graph that are uncrossed the set of ... offspring of C by definition Proposition The genus ofthe reduced form T3 of a u-c-tree T is one if and only if the genus of T is one Proof Let T2 be the pre-reduced form of u-c-tree T Then T2...
... 3n−6 edges ofthe triangulation and the edges ofthe outer triangle must be bidirected because ofthe rule of vertices Theorem says, that the edge orientations together with the colors ofthe special ... The crossover boxes defined in [8] the electronic journal of combinatorics 15 (2008), #R77 32 connection edges of Hk We call a configuration ofthe connection edges of Hk valid if exactly one of ... from the source to the sink of ∆ We say that ∆ is a + triangle of B if looking along the direct source-sink edge the triangle is onthe left Otherwise, if the triangle is onthe right ofthe edge...
... side of (4.10) Divide this numberof choices by thenumberof permutations of {1, , rn} to deduce the lemma Using the methods in the proof of Theorem 4.4 we get the Corollary 4.6 p log E2 ... r Theorem 2.2 yields that the validity ofthe conjecture (6.2) for r = See [4] for the asymptotic version ofthe conjectured inequality (6.2) 7.1 Computational results The Lower Matching Conjecture ... and only if G is either a union of copies of C , or a union of copies of C4 and a copy of Ci for i = 5, 6, 7, respectively Equalities in (2.16-2.18) hold if and only if G is either a union of...
... An interesting paper of Wagner [5] looks at thenumberof independent sets modulo m Wagner showed that the proportion of trees on n vertices with thenumberof independent sets divisible ... that the Fibonacci numbers (of the paths) form a complete system of residues if and only if m = t · 5k , t = 1, 2, 4, 6, 7, 14, 3j where k 0, j Thenumberof matchings in a tree In this section, ... note a recursion between the Fibonacci numberof a rooted tree and its subtrees Suppose r1 , r2 , · · · , rj are the neighbours of r, let (Tk , rk ) be the subtree of T rooted at rk Then we have...
... quantitative aspect of non-unique factorizations: the Narkiewicz constants, International Journal ofNumber Theory, to appear [13] W.D Gao and J.T Peng, Onthenumberof zero-sum subsequences of restricted ... sequences which attain the lower bound for some groups Notation and lower bound Our notation and terminology are consistent with [10] We briefly gather some notions and fix the notation concerning sequences ... shall prove the theorem by induction on m = |S| The case of m ≤ D(G) − is clear We now consider the case of m ≥ D(G) Choose a subsequence T |S of minimum length with σ(T ) = g, and a nonempty zero-sum...
... of different energies onthenumberof monitor units applied Influence of different energies onthenumberof monitor units applied Mean ofthenumberof monitor units of all fields against the ... Influence of different dose rates onthenumberof monitor units applied Influence of different dose rates onthenumberof monitor units applied Mean ofthenumberof monitor units of all fields ... velocities onthenumberof monitor units applied Influence Influence of different maximum leaf velocities onthenumberof monitor units applied Mean ofthenumberof monitor units of all fields...
... discussion ofthe modeling issues is deferred here ently of one another In addition, the model depends crucially onthe concept that the cessation of proliferation and entry into meiotic arrest is controlled ... NTJ: The elements of stochastic processes with applications to the natural sciences John Wiley & Sons, New York; 1964 Yule GU: A mathematical theory of evolution, based onthe conclusions of Dr ... from the analysis The solid line represents the expected numberof oocytes at each age in the postembryonic stages based onthe model given these data The shaded area is the pointwise 80% confidence...
... Proof In each partition of natural number n, for thenumberof summands 1, there exist n + cases: 0, 1, , n By definition of P1 (i; n), the proof is completed ✷ Example 3.1 Find thenumberof ... be thenumberof partitions of a natural number n such that in every partition we have i summands 1, exactly Now P1 (i; n) = P1 (0; n − i), i = 0, 1, , n 1641 Onthenumberof partitions of ... cannot partition the remaining n elements unconditionally, n because when the set is partitioned in two subsets with n2 elements, 12 [ n2 ] 1637 Onthenumberof partitions of sets partitions are counted...
... plasma drug concentration intercept ofthe regression line ofthe absorption phase, Ka is the absorption rate constant, β is the overall elimination rate constant, t is time and e is the natural ... treatment during the absorption phase The findings ofthe present study in goats therefore ruled out the possibility of vasodilatation of enteric vessels [2] during absorption The findings ofthe present ... Based onthe MIC data, only a marginal reduction in the dose (6-15%) was possible following trikatu administration Nonetheless, trikatu administration enhanced the duration of antimicrobial action...