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Showing papers in "Random Structures and Algorithms in 2001"


Journal ArticleDOI
TL;DR: Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.
Abstract: Recently, Barabasi and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabasi and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd−γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001

891 citations


Journal ArticleDOI
TL;DR: A simple, linear-time algorithm for the graph l-partition problem is presented and it is shown that if p−r≥n−1/2+ϵ for some constant ϵ, then the algorithm finds the optimal partition with probability 1− exp(−nΘ(e)).
Abstract: The NP-hard graph bisection problem is to partition the nodes of an undirected graph into two equal-sized groups so as to minimize the number of edges that cross the partition. The more general graph l-partition problem is to partition the nodes of an undirected graph into l equal-sized groups so as to minimize the total number of edges that cross between groups. We present a simple, linear-time algorithm for the graph l-partition problem and we analyze it on a random “planted l-partition” model. In this model, the n nodes of a graph are partitioned into l groups, each of size n/l; two nodes in the same group are connected by an edge with some probability p, and two nodes in different groups are connected by an edge with some probability r

428 citations


Journal ArticleDOI
TL;DR: Aldous and DJ as discussed by the authors constructed the optimal matching on the infinite tree, which yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal bipartite matching.
Abstract: Author(s): Aldous, DJ | Abstract: The random assignment (or bipartite matching) problem asks about An = minπ ∑ni=1 c(i, π(i)) where (c(i, j)) is a n × n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. Mezard and Parisi (1987) used the replica method from statistical physics to argue nonrigorously that EAn → ζ(2) = π2/6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost-optimal matching coincides with the optimal matching except on a small proportion of edges. © 2001 John Wiley a Sons, Inc. Random Struct. Alg., 18, 381-418, 2001.

264 citations


Journal ArticleDOI
TL;DR: For additive functions satisfying mild conditions (including the cases of the mean, the entropy, and mutual information), the plug-in estimates of F are universally consistent as mentioned in this paper, and without further assumptions, no rate-of-convergence results can be obtained for any sequence of estimators.
Abstract: Suppose P is an arbitrary discrete distribution on acountable alphabet . Given an i.i.d. sample (X1,…,Xn) drawnfrom P, we consider the problem of estimating the entropy H(P) or some other functional F=F(P) of the unknown distribution P. We show that, for additive functionals satisfying mild conditions (including the cases of the mean, the entropy, and mutual information), the plug-in estimates of F are universally consistent. We also prove that, without further assumptions, no rate-of-convergence results can be obtained for any sequence of estimators. In the case of entropy estimation, under a variety of different assumptions, we get rate-of-convergence results for the plug-in estimate and for a nonparametric estimator based on match-lengths. The behavior of the variance and the expected error of the plug-in estimate is shown to be in sharp contrast to the finite-alphabet case. A number of other important examples of functionals are also treated in some detail. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 163–193, 2001

255 citations


Journal ArticleDOI
TL;DR: Using this order parameter, it is proved that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1 and the values of two other critical exponents are determined, showing that the exponents of 2-SAT are identical to those of the random graph.
Abstract: We consider the random 2-satisfiability (2-SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/nα, the problem is known to have a phase transition at αc=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α−(n,δ), α+(n,δ)) such that the probability of satisfiability is greater than 1−δ for α α+. We show that W(n,δ)=(1−Θ(n−1/3), 1+Θ(n−1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+e)n, where e may depend on n as long as |e| is sufficiently small and |e|n1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(−Θ(ne3)) above the window, and goes to one like 1−Θ(n−1|e|−3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2-SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001

205 citations


Journal ArticleDOI
TL;DR: A class of “universal” phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function are exhibited, related to the coalescence of saddle points and the confluence of singularities of generating functions.
Abstract: A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of “universal” phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this article, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent ) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for a multiple connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the article also presents a general classification of compositional schemas in analytic combinatorics. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 194–246, 2001

168 citations


Journal ArticleDOI
TL;DR: Results are obtained on many of the properties of a random d-regular graph when d=d(n) grows more quickly than .
Abstract: Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random d-regular graph when d=d(n) grows more quickly than . These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, among others. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 346–363, 2001.

138 citations


Journal ArticleDOI
TL;DR: It is shown that the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n, which resolves a question of Achlioptas.
Abstract: Let e1, e′1; e2, e′2;…;ei, e′i;⋅⋅⋅ be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from ei,e′i to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. We show that these choices can be made so that whp the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n. This resolves a question of Achlioptas. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 75–85, 2001

131 citations


Journal ArticleDOI
TL;DR: Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle ℤ/Nℤ are given in this article, where the authors show that the diameter of a graph can be bounded by the number of vertices in the graph.
Abstract: Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle ℤ/Nℤ are given. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 102–111, 2001

105 citations


Journal ArticleDOI
TL;DR: This paper considers the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized and proves that with high probability the optimum partition is unique, and that the optimum discrepancy is bounded.
Abstract: We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n, we prove that the problem has a phase transition at κ = 1, in the sense that for κ 1, there are no perfect partitions with probability tending to 1. Moreover, we show that this transition is first-order in the sense the derivative of the so-called entropy is discontinuous at κ = 1. We also determine the finite-size scaling window about the transition point: κn = 1 − � 2n� −1 log2 n + λn/n, by showing that the probability of a perfect partition tends to 1� 0, or some explicitly computable p� λ �∈� 0� 1� , depending on whether λn tends to −∞� ∞ ,o r λ ∈ �−∞ � ∞� , respectively. For λn →− ∞fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λn →∞ , we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is � 2 λn � . Within the window, i.e., if � λnis bounded, we prove that the optimum discrepancy is bounded. Both for λn →∞ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 247-288, 2001

90 citations


Journal ArticleDOI
TL;DR: A uniform approach to describe the phase change of the limiting distribution of space measures in random m-ary search trees: the space requirement, when properly normalized, is asymptotically normally distributed for m≤26 and does not have a fixed limiting distribution for m>26.
Abstract: We propose a uniform approach to describe the phase change of the limiting distribution of space measures in random m-ary search trees: the space requirement, when properly normalized, is asymptotically normally distributed for m≤26 and does not have a fixed limiting distribution for m>26. Our tools are based on the method of moments and asymptotic solutions of differential equations, and are applicable to secondary cost measures of quicksort with median-of-(2t+1) for which the same phase change occurs at t=58. Both problems are essentially special cases of the generalized quicksort of Hennequin in which a sample of m(t+1)−1 elements are used to select m−1 equi-spaced ranks that are used in turn to partition the input into m subfiles. A complete description of the numbers at which the phase change occurs is given. For example, when m is fixed and t varies, the phase change occurs at (m, t)={(2, 58), (3, 19), (4, 10), (5, 6), (6, 4), …}. We also indicate some applications of our approach to other problems including bucket recursive trees. A general framework on “asymptotic transfers” of the underlying recurrence is also given. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 316–358, 2001

Journal ArticleDOI
TL;DR: A general multivariate limit law is proved, which also leads to an approach to asymptotic covariances and correlations of the parameters, and a bivariate limitLaw for the number of key comparisons and exchanges of median-of-(2t+1) Quicksort.
Abstract: The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law, which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application, the asymptotic correlations and a bivariate limit law for the number of key comparisons and exchanges of median-of-(2t+1) Quicksort are given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 498–524, 2001

Journal ArticleDOI
TL;DR: This paper investigates adaptive allocation schemes that achieve optimal tradeoffs between the maximum load, the maximum allocation time, and the average allocation time and provides a tight analysis of a natural class of processes that each time a ball is placed in one of d randomly chosen bins may move balls among these d bins arbitrarily.
Abstract: Many dynamic resource allocation and on-line load balancing problems can be modeled by processes that sequentially allocate balls into bins. The balls arrive one by one and are to be placed into bins on-line without using a centralized controller. If n balls are sequentially placed into n bins by placing each ball in a randomly chosen bin, then it is widely known that the maximum load in bins is ln n /ln ln n⋅(1+o(1)) with high probability. Azar, Broder, Karlin, and Upfal extended this scheme, so that each ball is placed sequentially into the least full of d randomly chosen bins. They showed that the maximum load of the bins reduces exponentially and is ln ln n/In d+Θ(1) with high probability, provided d<2. In this paper we investigate various extensions of these schemes that arise in applications in dynamic resource allocation and on-line load balancing. Traditionally, the main aim of allocation processes is to place balls into bins to minimize the maximum load in bins. However, in many applications it is equally important to minimize the number of choices performed (the allocation time). We study adaptive allocation schemes that achieve optimal tradeoffs between the maximum load, the maximum allocation time, and the average allocation time. We also investigate allocation processes that may reallocate the balls. We provide a tight analysis of a natural class of processes that each time a ball is placed in one of d randomly chosen bins may move balls among these d bins arbitrarily. Finally, we provide a tight analysis of the maximum load of the off-line process in which each ball may be placed into one of d randomly chosen bins. We apply this result to competitive analysis of on-line load balancing processes. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 297–331, 2001

Journal ArticleDOI
TL;DR: In this article, the authors give asymptotically almost sure bounds on the number of edges of a random maximal H-free graph with girth greater than and chromatic number n*y1/( −1+o(1) ).
Abstract: Given a graph H, a random maximal H-free graph is constructed by the following random greedy process. First assign to each edge of the complete graph on n vertices a birthtime which is uniformly distributed in [0, 1]. At time p=0 start with the empty graph and increase p gradually. Each time a new edge is born, it is included in the graph if this does not create a copy of H. The question is then how many edges such a graph will have when p=1. Here we give asymptotically almost sure bounds on the number of edges if H is a strictly 2-balanced graph, which includes the case when H is a complete graph or a cycle. Furthermore, we prove the existence of graphs with girth greater than and chromatic number n*y1/(-1)+o(1), which improves on previous bounds for >3. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 61–82, 2001

Journal ArticleDOI
TL;DR: This article investigates, from a probabilistic point of view, the first empty part, the maximum part size and the distribution of the number of distinct part sizes of the classical composition of an integer.
Abstract: Compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. In this article, we use as measure of distinctness the number of distinct parts (or components). We investigate, from a probabilistic point of view, the first empty part, the maximum part size and the distribution of the number of distinct part sizes. We obtain asymptotically, for the classical composition of an integer, the moments and an expression for a continuous distribution F, the (discrete) distribution of the number of distinct part sizes being computable from F. We next analyze another composition: the Carlitz one, where two successive parts are different. We use tools such as analytical depoissonization, Mellin transforms, Markov chain potential theory, limiting hitting times, singularity analysis and perturbation analysis. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 407–437, 2001

Journal ArticleDOI
TL;DR: This work studies the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution.
Abstract: The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique fixed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution. We demonstrate geometrically fast convergence for various metrics and discuss some implications for numerical calculations of the limiting Quicksort distribution. Finally, we give companion lower bounds which show that the convergence is not faster than geometric. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 376–406, 2001

Journal ArticleDOI
Svante Janson1
TL;DR: Four different methods are employed for different ranges of the parameters; together they yield a complete description of the construction cost, measured as the total displacement, for hash tables using linear probing.
Abstract: We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola [On the analysis of linear probing hashing, Algorithmica 22 (1998), 490–515]. The average cost of unsuccessful searches is considered too. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 438–471, 2001

Journal ArticleDOI
TL;DR: In this paper, the relationship between pyrolisis and physical and chemical characteristics of the jatoba (Himenea courbaril L.) charcoal was verified and the results indicated that the difference between the heartwood and sapwood charcoal is caused by differences anatomical structure and chemical composition of the materials.
Abstract: The purpose of this study was to verify the relationship between the pyrolisis and physical and chemical characteristics of the jatoba (Himenea courbaril L.) charcoal. It was also studied the behavior of the heartwood and sapwood with respect to the carbonization process and the effects of its chemical composition to the heat of combustion of the fuelwood. In addition, some physical-chemical characteristics of the wood charcoal were analyzed. The results suggest that the difference between the heartwood and sapwood charcoal is caused by differences anatomical structure and chemical composition of the materials. It was also determined that the charcoal specific gravity pass over a minimum point at 660o C for both heartwood and sapwood. It was not observed any change in the fixed yield carbon of charcoal obtained from heartwood and sapwood with the increase of the final carbonization temperature. Finally, it was determined a positive relationship between the heat of combustion of the wood charcoal with its fixed carbon and ash contents and negative relationship with the volatile material contents.

Journal ArticleDOI
TL;DR: Using probabilistic techniques, it is proved that the minimum number of edges in an (r; l)-system that is not k-colorable is 1.
Abstract: An (r; l)-system is an r-uniform hypergraph in which every set of l vertices lies in at most one edge. Let mk(r; l) be the minimum number of edges in an (r; l)-system that is not k-colorable. Using probabilistic techniques, we prove that

Journal ArticleDOI
TL;DR: An almost sure large deviations theorem for the depth of the external nodes of binary search trees (BSTs) is established and a parametric family of martingales is introduced to get asymptotic results on the number of external nodes at deepest level.
Abstract: We establish an almost sure large deviations theorem for the depth of the external nodes of binary search trees (BSTs). To achieve this, a parametric family of martingales is introduced. This family also allows us to get asymptotic results on the number of external nodes at deepest level. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 112–127, 2001

Journal ArticleDOI
TL;DR: In this article, the bisection width of a graph G is the minimum over all partitions of the number of "cross edges" between the parts of the graph G with edge probability c n.
Abstract: Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ‘‘cross edges’’ between the parts. We are interested in sparse random graphs Ž . G with edge probability c n. We show that, if c ln 4, then the bisection width is n n, c n with high probability; while if c ln 4, then it is equal to 0 with high probability. There are corresponding threshold results for partitioning into any fixed number of parts. 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 31 38, 2001

Journal ArticleDOI
TL;DR: Digital trees, such as tries, and Patricia tries are data structures routinely used in a variety of computer and communication applications including dynamic hashing, partial match retrieval, searching and sorting, conflict resolution algorithms for communication broadcast, data compression, and so forth.
Abstract: Digital trees, such as tries, and Patricia tries are data structures routinely used in a variety of computer and communication applications including dynamic hashing, par- tial match retrieval, searching and sorting, conflict resolution algorithms for communication broadcast, data compression, and so forth. Here, we consider tries and Patricia tries built from n words emitted by a probabilistic dynamical source. Such sources encompass classical and many models such as memoryless sources and finite Markov chains. The probabilistic behav- ior of its main parameters, namely, the size and the path length, appears to be determined by some intrinsic characteristics of the source, such as Shannon entropy and entropy-like con- stants, that depend on the spectral properties of specific transfer operators of Ruelle type. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 289-315, 2001



Journal ArticleDOI
TL;DR: Algorithms that solve the unranking problem for a large collection of labeled combinatorial classes, those that can be built using operators like unions (+), products (⋆), sequences, sets, cycles, and substitutions are designed and analyzed.
Abstract: In this article, we design and analyze algorithms that solve the unranking problem (i.e., generating a combinatorial structure of size, n given its rank) for a large collection of labeled combinatorial classes, those that can be built using operators like unions (+), products (⋆), sequences, sets, cycles, and substitutions. We also analyze the performance of these algorithms and show that the worst-case is (n2) ((n log n) if the so-called boustrophedonic order is used), and provide an algebra for the analysis of the average performance and higher-order moments together with a few examples of its application. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 472–497, 2001

Journal ArticleDOI
TL;DR: A property of self-duality is exploited to show that, when a+b=1, the process is, in a sense to be made precise, either critical or supercritical, but not subcritical.
Abstract: The square lattice is used to generate an oriented graph in which a rightward or upward arrow is present on each edge with probability a, and a leftward or downward arrow with probability b. Independence between different edges of the square lattice is assumed, but nothing is assumed concerning the dependence between the two possible orientations at any given edge. A property of self-duality is exploited to show that, when a+b=1, the process is, in a sense to be made precise, either critical or supercritical, but not subcritical. This observation enables progress with the percolation problem in which each horizontal edge is oriented rightward with probability p and otherwise leftward, and each vertical edge is oriented upward with probability p and otherwise downward. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 257–266, 2001

Journal ArticleDOI
TL;DR: In this article, it was shown that the maximum number of directed Hamiltonian paths in a tournament on n vertices is at mostc · n3/2· n!/2n−1, wherec is a positive constant independent of n.
Abstract: Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n3/2· n!/2n−1, wherec is a positive constant independent ofn.

Journal ArticleDOI
TL;DR: In this paper, a sistema de duplo proposito for cereais de inverno was used to support forragem verde no periodo critico de carencia alimentar, alem de aumentar a estabilidade da receita da producao pela melhoria na qualidade e produtividade dos graos dos cereais.
Abstract: RESUMO - A utilizacao de cereais de inverno no sistema de duplo proposito permite fornecer aos animais forragem verde no periodo critico de carencia alimentar, alem de aumentar a estabilidade da receita da producao pela melhoria na qualidade e produtividade dos graos dos cereais de inverno. O experimento foi conduzido no periodo de abril de 1994 a setembro de 1996 em Guarapuava, Parana, a fim de avaliar o potencial de utilizacao para forragem e graos de aveia branca (Avena sativa L.), trigo (Triticum aestivum), triticale (X. Triticosecale Witt.), aveia preta (Avena strigosa Schreb), centeio (Secale cereale L.) e cevada (Hordeum vulgare L.), visando sua utilizacao em condicoes de duplo proposito. Utilizou-se o delineamento experimental de blocos ao acaso, com tratamentos distribuidos em parcelas subdivididas, em tres repeticoes. Nas parcelas foram estudados os sistemas de producao (sem corte, um e dois cortes) e nas subparcelas, os genotipos. O sistema de dois cortes foi superior aos demais quanto ao rendimento de materia seca, principalmente para a aveia. Para o rendimento de graos, os sistemas sem corte e um corte foram superiores, apesar da maior producao dos genotipos de aveia sob dois cortes. Em todos os genotipos, houve melhoria do peso do hectolitro e reducao da massa de mil sementes, quando se realizaram cortes. Sob condicoes de manejo adequadas, pode-se alcancar consideravel producao de forragem, sem afetar a posterior producao de graos para cereais de inverno.

Journal ArticleDOI
TL;DR: This work evaluates alternating sums involving q-binomial coefficients using a contour integration technique and rederives very easily some results due to Van Hamme, Uchimura, Dilcher, Andrews, Crippa, and Simon.
Abstract: We evaluate alternating sums involving q-binomial coefficients using a contour integration technique. In this way, we rederive very easily some results due to Van Hamme, Uchimura (simplified!), Dilcher, Andrews, Crippa, and Simon. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 552–557, 2001

Journal ArticleDOI
TL;DR: It is shown that for any positive constant c and bipartite graph G=(U, V; E) of order n where the maximum degree of vertices in U is at most, the Ramsey number of the n-cube Qn satisfies which improves the bound 2cn log n due to Graham, Rodl, and Rucinski.
Abstract: Let R(G) be the least integer p such that for all bicolorings of the edges of complete graph Kp, at least one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U, V; E) of order n where the maximum degree of vertices in U is at most In particular, this shows that the Ramsey number of the n-cube Qn satisfies which improves the bound 2cn log n due to Graham, Rodl, and Rucinski. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 99–101, 2001