Types Of Partial Orders Research Articles

A New Class of Partial Orders

Let [Formula: see text] be a unital [Formula: see text]-ring. For any [Formula: see text], we apply the [Formula: see text]-core inverse to define a new class of partial orders in [Formula: see text], called the [Formula: see text]-core partial order. Suppose that [Formula: see text] are [Formula: see text]-core invertible. We say that [Formula: see text] is below [Formula: see text] under the [Formula: see text]-core partial order if [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the [Formula: see text]-core inverse of [Formula: see text]. Characterizations of the [Formula: see text]-core partial order are given, and its relationships with several types of partial orders are also considered. In particular, we show that the core partial order coincides with the [Formula: see text]-core partial order, and the star partial order coincides with the [Formula: see text]-core partial order.

An efficient construction of polar codes based on the general partial order

Traditionally, the construction of polar codes requires intense computations to sort all bit channels. In previous works, two types of partial orders (POs) of polar codes were proposed to decrease the computations in the construction process. In this paper, a procedure is presented to employ POs with complexity O(N2), which is lower than the existing procedures, where N=2n is the code length (n≥1). To further reduce the computation complexity, in this paper, we propose a general partial order (GPO), which works at a lower dimension (nu<n) to sort bit channels. The principle of the GPO is to divide the binary expansion of bit channels into two parts: the upper part and the lower part. First, the relationships among bit channels formed from the upper part (with length nu) are completely determined. Existing algorithms are called in this new level nu when needed. Second, we prove that the ordering at this lower dimension nu can be combined with the PO ordering of bit channels formed from the lower part. Therefore, more intrinsic relationships among bit channels are obtained. Working at the lower dimension nu<n, the complexity of GPO is inherently smaller than the existing sorting algorithms. The studies show that for n=10 and the code rate R=0.5 (the worst code rate from the empirical studies), the GPO can order 82% of bit channels, which is 32% larger than that obtained from using only POs. The proposed GPO in this paper can significantly reduce computations in the construction of polar codes.

Nonparametric estimation of a distribution with type I bias with applications to competing risks

A random variable X has a symmetric distribution about a if and only if X − a and −X + a are identically distributed. By considering various types of partial orderings between the distributions of X − a and −X + a, one obtains various types of partial skewness or one-sided bias. For example, F has Type I bias about a if F¯(a + x) ≥ F((a − x)−) for all x > 0; here F¯ = 1 − F. In this article we assume that a = 0, and propose a nonparametric estimator of a continuous distribution function F under the restriction that it has Type I bias. We derive the weak convergence of the resulting process which is used to test for symmetry against that type of bias. The new estimator is then compared with the nonparametric likelihood estimator (NPMLE), [Fcirc] n , of F in terms of mean squared error. A simulation study seems to indicate that the new estimator outperforms the NPMLE uniformly at all the quantiles of the distributions that we have investigated. It turns out that the results developed here could be used to compare two risks in a competing risks problem. We show how this can be done and illustrate the theory with a real life example.

Likelihood ratio tests for symmetry against one-sided alternatives

A random variableX is said to have a symmetric distribution (about 0) if and only ifX and −X are, identically distributed. By considering various types of partial orderings between the distributions ofX and −X, one obtains various notions of skewness or one-sided bias. In this paper we study likelihood ratio tests for testing the symmetry of a discrete distribution about zero against the alternatives, (i)X is stochastically greater than −X; and (ii) pr(X=j)≥pr(X=−j) for allj>0. In the process, we obtain maximum likelihood estimators of the distribution function under the above alternatives. The asymptotic null distributions of the test statistics have been obtained and are of the chi-bar square type. A simulation study was performed to compare the powers of these tests with other tests.

A family of bounds for the transient behavior of a Jackson network

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

An Invariance Property of Solutions to Second Order Differential Inequalities in Ordered Banach Spaces

Let B be a Banach space, K be a cone in B and $I = [a,b]$. Conditions are imposed on ${\bf f}(x,{\bf u},{\bf u}')$, ${\bf f}:I \times B \times B \to B$ such that the following invariance property holds: ${\bf u}'' + {\bf f}(x,{\bf u},{\bf u}') \in - K$, ${\bf u}(x_1 ) \in K$, ${\bf u}(x_2 ) \in K$, $a \leqq x_1 < x_2 \leqq b$ implies ${\bf u}(x) \in K$ for all $x \in [x_1 ,x_2 ]$. From this property, comparison theorems for solutions to differential inequalities are obtained. These comparison results give sufficient conditions for $C^2 $ functions to be subfunctions with respect to two point boundary value problems. These results extend earlier results in $R^2 $ by Heimes to more general types of partial orderings in both infinite and finite dimensional spaces. The approach taken in this paper relaxes certain coupling restrictions present in earlier results of this type.