共查询到20条相似文献,搜索用时 31 毫秒
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Yun S. Song 《Annals of Combinatorics》2003,7(3):365-379
We study subtree-prune-and-regraft (SPR) operations on
leaf-labelled rooted binary trees, also known as rooted binary phylogenetic trees.
This study is motivated by the problem of
graphically representing evolutionary histories of biological sequences subject to recombination.
We investigate some basic properties of the induced SPR-metric on the space
of leaf-labelled
rooted binary trees with n leaves. In contrast to the case of
unrooted trees, the number |U(T)| of trees in
which are one SPR operation away from a given tree
depends on the
topology of T. In this paper, we construct recursion relations which allow one to determine the
unit-neighbourhood size |U(T)| efficiently for any tree topology. In fact, using the recursion
relations we are able to derive a simple closed-form formula for the unit-neighbourhood size.
As a corollary, we construct sharp upper and lower bounds on the size of unit-neighbourhoods
and investigate the diameter of
.
Lastly, we consider an enumeration problem relevant to population genetics.AMS Subject Classification: 05C05, 92D15. 相似文献
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Let T
n
be the complete binary tree of height n, with root 1
n
as the maximum element. For T a tree, define and . We disprove a conjecture of Kubicki, Lehel and Morayne, which claims that for any fixed n and arbitrary rooted trees T
1
T
2. We show that A(n; T) is of the form where l is the number of leaves of T, and each q
j
is a polynomial. We provide an algorithm for calculating the two leading terms of q
l
for any tree T. We investigate the asymptotic behaviour of the ratio A(n; T)/C(n; T) and give examples of classes of pairs of trees T
1, T
2 where it is possible to compare A(n; T
1)/C(n; T
1) and A(n; T
2)/C(n; T
2). By calculating these ratios for a particular class of pairs of trees, we show that the conjecture fails for these trees,
for all sufficiently large n. Kubicki, Lehel and Morayne have proved the conjecture when T
1, T
2 are restricted to being binary trees. We also look at embeddings into other complete trees, and we show how the result can
be viewed as one of many possible correlation inequalities for embeddings of binary trees. We also show that if we consider
strict order-preserving maps of T
1, T
2 into T
n
(rather than embeddings) then the corresponding correlation inequalities for these maps also generalise to arbitrary trees. 相似文献
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L. Sanguiao Sande 《Geometriae Dedicata》2011,151(1):305-321
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Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on C such that F :=∩t≥0 Fix(T(t)) ≠ 0, and f : C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:{yn=αnxn+(1-αn)T(tn)xn,xn=βnf(xn)+(1-βn)yn
{u0∈C,vn=anun+(1-an)T(tn)un,un+1=bnf(un)+(1-bn)vn
We prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:
〈(I-f)q,j(q-u)〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51-60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751-757 (2007)]. 相似文献
{u0∈C,vn=anun+(1-an)T(tn)un,un+1=bnf(un)+(1-bn)vn
We prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:
〈(I-f)q,j(q-u)〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51-60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751-757 (2007)]. 相似文献
11.
Yun S. Song 《Annals of Combinatorics》2006,10(1):147-163
We study some properties of subtree-prune-and-regraft (SPR) operations on leaflabelled rooted binary trees in which internal
vertices are totally ordered. Since biological events occur with certain time ordering, sometimes such totally-ordered trees must be used to avoid
possible contradictions in representing evolutionary histories of biological sequences. Compared to the case of plain leaf-labelled
rooted binary trees where internal vertices are only partially ordered, SPR operations on totally-ordered trees are more constrained and therefore more difficult to study. In this paper,
we investigate the unit-neighbourhood U(T), defined as the set of totally-ordered trees one SPR operation away from a given totally-ordered tree T. We construct a recursion relation for | U(T) | and thereby arrive at an efficient method of determining | U(T) |. In contrast to the case of plain rooted trees, where the unit-neighbourhood size grows quadratically with respect to
the number n of leaves, for totally-ordered trees | U(T) | grows like O(n3). For some special topology types, we are able to obtain simple closed-form formulae for | U(T) |. Using these results, we find a sharp upper bound on | U(T) | and conjecture a formula for a sharp lower bound. Lastly, we study the diameter of the space of totally-ordered trees
measured using the induced SPR-metric.
Received May 18, 2004 相似文献
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James East 《Semigroup Forum》2010,81(2):357-379
The (full) transformation semigroup Tn\mathcal{T}_{n} is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group Sn í Tn{\mathcal{S}_{n}\subseteq \mathcal{T}_{n}} is the group of all permutations on {1,…,n} and is the group of units of Tn\mathcal{T}_{n}. The complement Tn\Sn\mathcal{T}_{n}\setminus \mathcal{S}_{n} is a subsemigroup (indeed an ideal) of Tn\mathcal{T}_{n}. In this article we give a presentation, in terms of generators and relations, for Tn\Sn\mathcal{T}_{n}\setminus \mathcal{S}_{n}, the so-called singular part of Tn\mathcal{T}_{n}. 相似文献
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In this note, we construct a 2-basic set of the alternating group
\mathfrakAn{\mathfrak{A}_n}. To do this, we construct a 2-basic set of the symmetric group
\mathfrakSn{\mathfrak{S}_n} with an additional property, such that its restriction to
\mathfrakAn{\mathfrak{A}_n} is a 2-basic set. We adapt here a method developed by Brunat and Gramain (J. Reine Angew. Math., to appear) for the case
when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by Külshammer et al. (Invent.
Math. 151, 513–552, (2003)). 相似文献
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For x = (x 1, x 2, ..., x n ) ∈ ℝ+ n , the symmetric function ψ n (x, r) is defined by $\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }}
{{x_{i_j } }}} } ,$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }}
{{x_{i_j } }}} } , 相似文献
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Horst Alzer 《Proceedings Mathematical Sciences》2010,120(2):131-137
Let n ≥ 1 be an integer and let P
n
be the class of polynomials P of degree at most n satisfying z
n
P(1/z) = P(z) for all z ∈ C. Moreover, let r be an integer with 1 ≤ r ≤ n. Then we have for all P ∈ P
n
:
$
\alpha _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt} \leqslant \int_0^{2\pi } {|P^r (e^{it} )|^2 dt} \leqslant \beta _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt}
$
\alpha _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt} \leqslant \int_0^{2\pi } {|P^r (e^{it} )|^2 dt} \leqslant \beta _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt}
相似文献
19.
We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples.
Theorem A.Let 0<θ<1/2and let {a
n
}be a sequence of complex numbers satisfying the inequality
for N = 1,2,3,…,also for n = 1,2,3,…let α
n
be real and |αn| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function
in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T
0(θ,δ)a large positive constant.
Theorem B.In the above theorem we can relax the condition on a
n
to
and |aN| ≤ (1/2-θ)-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided
for every ε > 0.
Dedicated to the memory of Professor K G Ramanathan 相似文献
20.
Limit theorems for the ratio of the empirical distribution function to the true distribution function 总被引:4,自引:0,他引:4
Jon A. Wellner 《Probability Theory and Related Fields》1978,45(1):73-88
Summary We consider almost sure limit theorems for
and
where
n
is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda
n
0. It is shown that (1) ifna
n
/log2
n then both
and
converge to 1 a.s.; (2) ifna
n
/log2
n=d>0 (d>1) then
has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna
n
/log2
n0 then
and
have limit superior + almost surely. Similar results are established for the inverse function
n
–1
.Supported by the National Science Foundation under MCS 77-02255 相似文献
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