We consider the Dirichlet problem for $p(x)$-Laplacian equations of the form

$$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$

The odd nonlinearity $f(x,u)$ is $p(x)$-sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $. Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$.

Large truncated spherical near-field systems with conductive or absorbing floors are typically used in the measurement of the performances of vehicle-installed antennas. The main advantage of conductive floor systems is the ease of accommodation of the vehicle under test, but their performances are affected by the interaction with the reflecting ground floor. Instead, absorbing-based systems emulating free-space conditions minimize the effect of the interaction with the floor, but generally require longer setup times, especially at lower frequencies (70–400 MHz), where bulky absorbers are typically used to improve reflectivity levels. Considering scaled measurements of a vehicle model, the performances of these two typical implementations are analyzed in the 84–1500 MHz range and compared to free-space measurements. Absorbers with different dimensions and reflectivity have been installed in the scaled measurement setup, and measured data have been investigated with proper post-processing to verify the applicability to realistic systems. Figures of merit of interest for automotive applications, like gain and partial radiated powers, have been compared to free-space to evaluate the impact of different scenarios.

Understanding how person-to-person contagious processes spread through a population requires accurate information on connections between population members. However, such connectivity data, when collected via interview, is often incomplete due to partial recall, respondent fatigue, or study design, e.g. fixed choice designs (FCD) truncate out-degree by limiting the number of contacts each respondent can report. Research has shown how FCD affects network properties, but its implications for predicted speed and size of spreading processes remain largely unexplored. To study the impact of degree truncation on predictions of spreading process outcomes, we generated collections of synthetic networks containing specific properties (degree distribution, degree-assortativity, clustering), and used empirical social network data from 75 villages in Karnataka, India. We simulated FCD using various truncation thresholds and ran a susceptible-infectious-recovered (SIR) process on each network. We found that spreading processes on truncated networks resulted in slower and smaller epidemics, with a sudden decrease in prediction accuracy at a level of truncation that varied by network type. Our results have implications beyond FCD to truncation due to any limited sampling from a larger network. We conclude that knowledge of network structure is important for understanding the accuracy of predictions of process spread on degree truncated networks.

We discuss how to fit mixtures of Erlangs to censored and truncated data by iteratively using the EM algorithm. Mixtures of Erlangs form a very versatile, yet analytically tractable, class of distributions making them suitable for loss modeling purposes. The effectiveness of the proposed algorithm is demonstrated on simulated data as well as real data sets.

Many agricultural producers face cash price distributions that are effectively truncated at a lower limit through participation in farm programs designed to support farm prices and incomes. For example, the 1996 Federal Agricultural Improvement Act (FAIR) makes many producers eligible to obtain marketing loans which truncate their cash price realization at the loan rate, while allowing market prices to freely equilibrate supply and demand. This paper studies the effects of truncated cash price distributions on the optimal use of futures and options. The results show that truncation in the cash price distribution facing an individual producer provides incentives to trade options as well as futures. We derive optimal futures and options trading rules under a range of different truncation scenarios. Empirical results highlight the impacts of basis risk and yield risk on the optimal futures and options portfolio.

Exact lower bounds on the exponential moments of min(y, X) and X1{X < y} are provided given the first two moments of a random variable X. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y, X) over the truncation X1{X < y} are demonstrated. An application to option pricing is given.

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

We consider an independent long-range bond percolation on Z2. Horizontal and vertical bonds of length n are independently open with probability p_n ∈ [0, 1]. Given ∑n=1∞∏i=1n(1 − pi) < ∞, we prove that there exists an infinite cluster of open bonds of length less than or equal to N for some large but finite N. The result gives a partial answer to the truncation problem.

Let X1, X2,…, Xn be independent, identically distributed random variables, uniform on [0,1]. We observe the Xk sequentially and must stop on exactly one of them. No recollection of the preceding observations is permitted. What stopping rule τ minimizes the expected rank of the selected observation? This full-information expected-rank problem is known as Robbins' problem. The general solution is still unknown, and only some bounds are known for the limiting value as n tends to infinity. After a short discussion of the history and background of this problem, we summarize what is known. We then try to present, in an easily accessible form, what the author believes should be seen as the essence of the more difficult remaining questions. The aim of this article is to evoke interest in this problem and so, simply by viewing it from what are possibly new angles, to increase the probability that a reader may see what seems to evade probabilistic intuition.

Results obtained by the indirect zonal isotropic uniform random (IUR) estimation were compared with those obtained by the direct point and interception counting methods on vertical (VS) or IUR sections in a stereological study of bovine articular cartilage collagen fibrils at the ultrastructural level. Besides comparisons between the direct and indirect estimations (direct IUR vs indirect IUR estimations) and between different sampling methods (VS vs IUR sampling), simultaneous comparison of the 2 issues took place (direct VS vs indirect IUR estimation). Using the direct VS method, articular cartilage superficial zone collagen volume fraction (Vv 41%) was 67% and fibril surface density (Sv 0.030 nm2/nm3) 15% higher (P<0.05) than values obtained by the indirect IUR method (Vv 25% and Sv 0.026 nm2/nm3). The same was observed when the direct IUR method was used: collagen volume fraction (Vv 40%) was 63% and fibril surface density (Sv 0.032 nm2/nm3) 21% higher (P<0.05) than those obtained by the indirect IUR technique. Similarly, in the deep zone of articular cartilage direct VS and direct IUR methods gave 50 and 55% higher (P<0.05) collagen fibril volume fractions (Vv 43 and 44% vs 29%) and the direct IUR method 25% higher (P<0.05) fibril surface density values (Sv 0.025 vs 0.020 nm2/nm3) than the indirect IUR estimation. On theoretical grounds, scrutiny calculations, as well as earlier reports, it is concluded that the direct VS and direct IUR methods systematically overestimated the Vv and Sv of collagen fibrils. This bias was due to the overprojection which derives from the high section thickness in relation to collagen fibril diameter. On the other hand, factors that during estimation tend to underestimate Vv and Sv, such as profile overlapping and truncation (‘fuzzy’ profiles), seemed to cause less bias. As length density (Lv ) and collagen fibril diameter are minimally biased by the high relative section thickness, the indirect IUR method, based on utilisation of these estimates, is here regarded as representing a ‘gold standard’. The sensitivity of these 3 methods was also tested with cartilage from an in vitro loading experiment which caused tissue compression. In the superficial zone of articular cartilage Vv and Sv of collagen fibrils increased (P<0.05). This difference in the stereological estimates was only detected by the indirect IUR estimation but not by the direct VS or direct IUR methods. This indicated that the indirect IUR estimation was more sensitive than the direct VS or direct IUR estimations. On the basis of these observations, the indirect zonal IUR estimation can be regarded as the technique of choice in the electron microscopic stereology of cartilage collagen.

The calculation of mean claim sizes, in the presence of a deductible, is usually achieved through numerical integration. In case of a Lognormal or Gamma distribution, the quantities of interest can easily be expressed as functions of the cumulative distribution function, with modified parameters. This also applies to the F-distribution, where the incomplete Beta function enters the scene; see for instance the appendix in Hogg and Klugman (1984).

The purpose of this paper is to derive an explicit formula for the first two moments of the Inverse Gaussian distribution, in the presence of censoring. For reasons of completeness we also consider truncation of the Inverse Gaussian distribution by an upper limit.

The tractability of the derivation depends in a crucial way on two properties of the Inverse Gaussian distribution. Firstly, the cumulative distribution function of the Inverse Gaussian can be written as a simple function using the Normal probability integral. Secondly, the moment generating function of a censorized Inverse Gaussian distribution boils down to an expression containing the cumulative Inverse Gaussian distribution. This manifests itself most clearly in case of life insurance where the quantity of interest is the expectation of a present value. In case of non-life insurance, where the dimension of the Inverse Gaussian random variable is money instead of time, a further step is required: differentiation of the moment generating function.

So, a natural order of this paper is to address ourselves first to the derivation for the life case and afterwards tackling the more laborious derivation for the non-life case.

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.