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1.
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of
scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which
do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered”
posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size
κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.
The authors warmly thank Uri Abraham for his many useful suggestions and comments. Mirna Džamonja thanks EPSRC for their support
on an EPSRC Advanced Fellowship. 相似文献
2.
We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process
corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The
controller can choose the drift coefficient which represents the service rate and the buffer size b>0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A “control cost” related
to the dynamically controlled service rate, a “congestion cost” which depends on the queue length and a “rejection penalty”
for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the
ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size
b
*>0. When the buffer size b>0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl.
Probab. 15, 1145–1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005)). To obtain a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we analyze a family of ordinary differential
equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b
*>0.
A.P. Weerasinghe’s research supported by US Army Research Office grant W911NF0510032. 相似文献
3.
This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the
notion of monotonically κ-monolithic space for any infinite cardinal κ and present the relevant results. We show, among other things, that any σ-product of monotonically κ-monolithic spaces is monotonically κ-monolithic for any infinite cardinal κ; besides, it is consistent that any strongly monotonically ω-monolithic space with caliber ω
1 is second countable. We also study (strong) monotone κ-monolithicity in linearly ordered spaces and subspaces of ordinals. 相似文献
4.
Tokuzo Shiga 《Probability Theory and Related Fields》1997,108(3):417-439
Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf;
d
in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate
the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0.
Received: 21 May 1996 / In revised form: 2 February 1997 相似文献
5.
Stephen H. Hechler 《Israel Journal of Mathematics》1973,15(4):384-395
For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN
κ of κ copies ofN (the discrete space of cardinality ℵ0). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is
consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal
less than or equal to 2ℵ
0. Mrówka has also defined and studied the class ℳ={κ: Exp (N
κ)=κ} whereN
κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot
even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2ℵ
0 is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which
are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2ℵ
0 and that ℳ nevertheless contain all cardinals no greater than 2ℵ
0. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any
normal measure on a measurable cardinal, then μ(ℳ)=0. 相似文献
6.
We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival
phase states, that is, there are many types of arrivals which have different service time distributions. The service process
is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first
passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service
descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis
of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained.
The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary
workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered
in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for
the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes
transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship
between the stationary waiting time distribution and the stationary distribution of the number of customers in the system
at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types
of customers at departures.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
7.
For any α∈(0,2), a truncated symmetric α-stable process in ℝ
d
is a symmetric Lévy process in ℝ
d
with no diffusion part and with a Lévy density given by c|x|−d−α
1{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary
Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that
the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However,
for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper,
we show that, for a large class of not necessarily convex bounded open sets in ℝ
d
called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean
boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly
connected κ-fat open sets.
The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic
Research Promotion Fund) (KRF-2007-331-C00037).
The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. 相似文献
8.
Assaf Rinot 《Archive for Mathematical Logic》2007,46(5-6):457-464
In their paper from 1981, Milner and Sauer conjectured that for any poset , if , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ
κ
antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The
consistency of the negation of this hypothesis is unknown.
相似文献
9.
We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several
classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture
when the bodies are on a geodesic that rotates elliptically or hyperbolically. 相似文献
10.
For κ ⩾ 0 and r0 > 0 let ℳ(n, κ, r0) be the set of all connected, compact n-dimensional Riemannian manifolds (Mn, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r0. We study the relation between the kth eigenvalue λk(M) of the Laplacian associated to (Mn,g), Δ = −div(grad), and the kth eigenvalue λk(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r0) such that for all M ∈ ℳ(n, κ, r0) and X a discretization of
for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r0) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r0 such that
for all
.
Mathematics Subject Classification (2000): 58J50, 53C20
Supported by Swiss National Science Foundation, grant No. 20-101 469 相似文献
11.
The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α
i
} and death rates {β
i
}, where α
i
≥0 and β
i
≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic
proofs of various Stein’s factors for the PBD approximation with α
i
=a and β
i
=i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance
between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution
is much more precise than the approximation by the Poisson or shifted Poisson distributions.
相似文献
12.
Simon [J. Approxim. Theory,
127, 39–60 (2004)] proved that the maximal operator σα,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H
p
to the space L
p
for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σα,κ,* is bounded from the martingale Hardy space H
1/(1+α) to the space weak- L
1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H
p
such that the maximal operator σα,κ,*
f does not belong to the space L
p
. 相似文献
13.
Mladen Savov 《Journal of Theoretical Probability》2010,23(1):209-236
We specify a function b
0(t) in terms of the Lévy triplet such that lim sup
t→0
X
t
/b
0(t)∈[1,1.8] a.s. iff
ò01[` \varPi ](+)(b0(t)) dt < ¥\int_{0}^{1}\overline{ \varPi }^{(+)}(b_{0}(t))\,dt<\infty
for any Lévy process X with unbounded variation and a Brownian component σ=0. We show with an example that there are cases where lim sup
t→0
X
t
/b(t)=1 a.s. but b(t) is not asymptotically equivalent to b
0(t) as t tends to 0. We achieve this by introducing an integral criterion which checks whether lim sup
t→0
X
t
/b(t) is 0, infinity, or a finite positive value for b(t) satisfying very mild conditions and any Lévy process. 相似文献
14.
Saharon Shelah 《Israel Journal of Mathematics》1996,95(1):61-114
We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the
cofinality of (S
≤ℵ
0(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ+,2) in more cases even when cf(λ)=ℵ0 (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure
I had already written but did not find (Section 6).
Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by
the Basic Research Fund, Israel Academy of Sciences. Publication number 430. 相似文献
15.
Vladimir V. Tkachuk 《Central European Journal of Mathematics》2012,10(2):456-465
Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ
+ if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ
+. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ
+ for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ
+ for compact spaces. 相似文献
16.
We prove that the SLE
κ
trace in any simply connected domain G is continuous (except possibly near its endpoints) if κ < 8. We also prove an SLE analog of Makarov’s Theorem about the support of harmonic measure. 相似文献
17.
In this paper we consider the least squares (LS) and total least squares (TLS) problems for a Michaelis–Menten enzyme kinetic
model f(x; a, b) = ax/(b + x), a, b > 0. In various applied research such as biochemistry, pharmacology, biology and medicine there are lots of different applications
of this model. We will systematize some of our results pertaining to the existence of the LS and TLS estimate, which were
proved in Hadeler et al. (Math Method Appl Sci 30:1231–1241, 2007) and Jukić et al. (J Comput Appl Math 201:230–246, 2007).
Finally, we suggest a choice of good initial approximation and give one numerical example.
相似文献
18.
In the present paper, for a boundary value problem with noncoordinated degeneration of the data and a singularity in the solution,
we show that the R
ν
-generalized solution belongs to the weighted space W
2,ν+gb
2+κ+1/κ+2 (Ω, δ)(κ > 0).
Original Russian Text ? V.A. Rukavishnikov, E.V. Kuznetsova, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45,
No. 6, pp. 894–898. 相似文献
19.
Avner Landver 《Israel Journal of Mathematics》1993,81(3):289-296
Letκ be a regular cardinal. Consider the Bair numbers of the spaces (2θ)κ for variousθ≥κ. Letl be the number of such different Baire numbers. Models of set theory withl=1 orl=2 are known and it is also known thatl is finite. We show here that ifκ>ω, thenl could be any given finite number. 相似文献
20.
Zhao Zengqin 《高校应用数学学报(英文版)》1998,13(1):15-24
In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R+ ×R
+
; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that
.
Then a necessary and sufficient condition for the equation u′’+f(κ,u,u) = 0 on the boundary condition au(.a)-βu′ (a) = 0, γ(b)+δu′(b) = 0 to have C1(I) nonzero solutions is that
where α,β,γ,δ are nonnegative real numbers, Δ= (b-a)αγ + αγ+βδ+βγ>0, e(κ) =G(κ,κ), G(κ,y) is Green’s function of above mentioned boundary value problem (when f(κ,u,υ)≡0).
Project supported by the Natural Science Foundation of Shandong Province. 相似文献