On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$

Jagdish Saran; Kanwar Sen

Displaying similar documents to “On the distributions of $R_{m n}^{+} (j)$ and $(D_{m n}^{+}, R_{m n}^{+} (j))$ ”

Some distribution results on generalized ballot problems

Jagdish Saran, Kanwar Sen (1985)

Aplikace matematiky

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Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $(a + b a)$ voting sequences are equally probable. Denote by $α_{r}$ and by $β_{r}$ the number of votes registered for $A$ and for $B$ , respectively, among the first $r$ votes recorded, $r = 1, \dots, a + b$ . The purpose of this paper is to derive, for $a \geq b - c$ , the probability distributions of the random variables defined as the number of subscripts $r = 1, \dots, a + b$ for which (i) $α_{r} = β_{r} - c$ , (ii) $α_{r} = β_{r} - c$ but $α_{r - 1} = β_{r - 1} - c \pm 1$ , (iii) $α_{r} = β_{r} - c$ but $α_{r - 1} = β_{r - 1} - c \pm 1$ and $α_{r + 1} = β_{r + 1} - c \pm 1$ , where $c = 0, \pm 1, \pm 2, \dots$ .

Sets Expressible as Unions of Staircase $n$ -Convex Polygons

Marilyn Breen (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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Let $k$ and $n$ be fixed, $k \geq 1$ , $n \geq 1$ , and let $S$ be a simply connected orthogonal polygon in the plane. For $T \subseteq S, T$ lies in a staircase $n$ -convex orthogonal polygon $P$ in $S$ if and only if every two points of $T$ see each other via staircase $n$ -paths in $S$ . This leads to a characterization for those sets $S$ expressible as a union of $k$ staircase $n$ -convex polygons $P_{i}$ , $1 \leq i \leq k$ .

Generalized Schröder matrices arising from enumeration of lattice paths

Lin Yang, Sheng-Liang Yang, Tian-Xiao He (2020)

Czechoslovak Mathematical Journal

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We introduce a new family of generalized Schröder matrices from the Riordan arrays which are obtained by counting of the weighted lattice paths with steps $E = (1, 0)$ , $D = (1, 1)$ , $N = (0, 1)$ , and $D^{'} = (1, 2)$ and not going above the line $y = x$ . We also consider the half of the generalized Delannoy matrix which is derived from the enumeration of these lattice paths with no restrictions. Correlations between these matrices are considered. By way of illustration, we give several examples of Riordan arrays of combinatorial interest....

Stronger bounds for generalized degrees and Menger path systems

R.J. Faudree, Zs. Tuza (1995)

Discussiones Mathematicae Graph Theory

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For positive integers d and m, let $P_{d, m} (G)$ denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that $P_{d, m} (G)$ is satisfied have been investigated. In particular, it has been shown,...

Path functionals over Wasserstein spaces

Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio (2006)

Journal of the European Mathematical Society

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Given a metric space $X$ we consider a general class of functionals which measure the cost of a path in $X$ joining two given points $x_{0}$ and $x_{1}$ , providing abstract existence results for optimal paths. The results are then applied to the case when $X$ is aWasserstein space of probabilities on a given set $Ω$ and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures $μ_{0}$ and $μ_{1}$ by means of finite cost paths are given. ...

Paths of low weight in planar graphs

Igor Fabrici, Jochen Harant, Stanislav Jendrol' (2008)

Discussiones Mathematicae Graph Theory

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The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are: 1. Every 3-connected simple planar graph G that contains a k-path, a path on k vertices, also contains a k-path P such that for its weight (the sum of degrees of its vertices) in G it holds $w_{G} (P) : = \sum_{u \in V (P)} d e g_{G} (u) \leq (3 / 2) k ² + (k)$ 2. Every plane triangulation T that contains a k-path also contains a k-path P such that for its weight in T it holds $w_{T} (P) : = \sum_{u \in V (P)} d e g_{T} (u) \leq k ² + 13 k$ 3. Let G be a 3-connected simple planar graph of...

On the stabilization problem for nonholonomic distributions

Ludovic Rifford, Emmanuel Trélat (2009)

Journal of the European Mathematical Society

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Let $M$ be a smooth connected complete manifold of dimension $n$ , and $Δ$ be a smooth nonholonomic distribution of rank $m \leq n$ on $M$ . We prove that if there exists a smooth Riemannian metric on1for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $Δ$ on $M$ . Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific...

Hamiltonicity of cubic Cayley graphs

Henry Glover, Dragan Marušič (2007)

Journal of the European Mathematical Society

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Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a $(2, s, 3)$ -presentation, that is, for groups $G = 〈 a, b ∣ a^{2} = 1, b^{s} = 1, {(a b)}^{3} = 1, \dots 〉$ generated by an involution $a$ and an element $b$ of order $s \geq 3$ such that their product $a b$ has order $3$ . More precisely, it is shown that the Cayley graph $X = Cay (G, {a, b, b^{- 1}})$ has a Hamilton cycle when $| G |$ (and thus $s$ ) is congruent to 2 modulo 4, and has a...

Displaying similar documents to “On the distributions of R m n + ( j ) and ( D m n + , R m n + ( j ) ) ”

Displaying similar documents to “On the distributions of $R_{m n}^{+} (j)$ and $(D_{m n}^{+}, R_{m n}^{+} (j))$ ”