In this paper, we explore two conjectures about Rademacher sequences. Let (ε_{i}) be a Rademacher sequence, i.e., a sequence of independent {-1, 1}-valued symmetric random variables. Set S_{n}=a_{1}ε_{1}+…+a_{n}ε_{n} for a=(a_{1}, …, a_{n}) ∈ R^{n}. The first conjecture says that P (|S_{n}|≤||a||) ≥ 1/2 for all a ∈ R^{n} and n ∈ N. The second conjecture says that P (|S_{n}|≥||a||) ≥ 7/32 for all a ∈ R^{n} and n ∈ N. Regarding the first conjecture, we present several new equivalent formulations. These include a topological view, a combinatorial version and a strengthened version of the conjecture. Regarding the second conjecture, we prove that it holds true when n ≤ 7.
In epidemiological and clinical studies, the restricted mean lifetime is often of direct interest quantity. The differences of this quantity can be used as a basis of comparing several treatment groups with respect to their survival times. When the factor of interest is not randomized and lifetimes are subject to both dependent and independent censoring, the imbalances in confounding factors need to be accounted. We use the mixture of additive hazards model and inverse probability of censoring weighting method to estimate the differences of restricted mean lifetime. The average causal effect is then obtained by averaging the differences in fitted values based on the additive hazards models. The asymptotic properties of the proposed method are also derived and simulation studies are conducted to demonstrate their finite-sample performance. An application to the primary biliary cirrhosis (PBC) data is illustrated.
A matching M of a graph G is an induced matching if no two edges in M are joined by an edge of G. Let iz(G) denote the total number of induced matchings of G, named iz-index. It is well known that the Hosoya index of a graph is the total number of matchings and the Hosoya index of a path can be calculated by the Fibonacci sequence. In this paper, we investigate the iz-index of graphs by using the Fibonacci-Narayana sequence and characterize some types of graphs with minimum and maximum iz-index, respectively.
The paper is devoted to the homogenization of elliptic systems in divergence form. We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C^{1,γ} domain when the coefficients are Dini continuous, inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives. The results extend Avellaneda and Lin's work[Comm. Pure Appl. Math., 40:803-847 (1987)], where Hölder continuity is the main assumption on smoothness of the data.
For graphs F and G, let F → (G, G) denote that any red/blue edge coloring of F contains a monochromatic G. Define Folkman number f(G; t) to be the smallest order of a graph F such that F → (G, G) and ω(F) ≤ t. It is shown that f(G; t) ≤ cn for p-arrangeable graphs with n vertices, where p ≥ 1, c=c(p) and t=t(p) are positive constants.
The admissibility of the initial-boundary data, which characterizes the existence of solution for the initial-boundary value problem, is important. Based on the Fokas method and the inverse scattering transformation, an approach is developed to solve the initial-boundary value problem of the nonlinear Schrödinger equation on a finite interval. A necessary and sufficient condition for the admissibility of the initial-boundary data is given, and the reconstruction of the potential is obtained.
Cost effective sampling design is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the location parameter for location family. A maximum likelihood estimator (MLE) of the location parameter for this family is studied and its properties are obtained. We prove that the MLE is an equivariant estimator under location transformation. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), the asymptotic efficiency of the MLE using MERSS w.r.t. that using SRS is computed for some usual location distributions. The relative results show that the MLE using MERSS can be real competitors to the MLE using SRS.
In this paper, we assume that the pest population is divided into susceptible pests and infected pests, and only susceptible pests do harm to crops. Considering the two methods of spraying pesticides and releasing infected pests and natural enemies to control susceptible pests (the former is applied more frequently), and assuming that only susceptible pests develop resistance to pesticides, a pest control model with resistance development is established. By using the basic theory of impulsive differential systems and analytical methods, the sufficient condition for the global attractiveness of the susceptible pest eradication periodic solution is given. Combined with numerical simulations, the effects of spraying frequency of pesticides on critical threshold conditions for eradicating susceptible pests are discussed. The results confirm that it is not that the more frequently the pesticides are sprayed, the better the result of the pest control is. Two control strategies for eradicating susceptible pests are proposed:switching pesticides and releasing natural enemies elastically. Finally, the parameters in the critical threshold are analyzed from the following two aspects:(1) The key factors affecting pest control are determined by parameter sensitivity analyses. The results indicate that the correlation of the critical threshold concerning the killing efficiency rate and the decay rate of pesticides to susceptible pests varies due to the resistance development of susceptible pests. (2) Three-dimensional graphs and contours of susceptible pest eradication critical threshold with two parameters are simulated, and the effects of the main parameters on the critical threshold are analyzed.
Stochastic gradient descent (SGD) is one of the most common optimization algorithms used in pattern recognition and machine learning. This algorithm and its variants are the preferred algorithm while optimizing parameters of deep neural network for their advantages of low storage space requirement and fast computation speed. Previous studies on convergence of these algorithms were based on some traditional assumptions in optimization problems. However, the deep neural network has its unique properties. Some assumptions are inappropriate in the actual optimization process of this kind of model. In this paper, we modify the assumptions to make them more consistent with the actual optimization process of deep neural network. Based on new assumptions, we studied the convergence and convergence rate of SGD and its two common variant algorithms. In addition, we carried out numerical experiments with LeNet-5, a common network framework, on the data set MNIST to verify the rationality of our assumptions.
We investigate a hyperbolic system of one-dimensional isothermal fluid with liquid-vapor phase transition. The refraction-reflection phenomena are intensively analyzed when elementary waves travel across the two-phase interface. We apply the characteristic method and hodograph transform of Riemann to reduce the nonlinear PDEs to a concise form. Specially for the case of incident rarefaction wave, reduced linear equations are convenient to solve by Laplace transform. Then an integral formula in wave interaction region is derived in this paper, instead of the hypergeometric functions solutions for non-isothermal polytropic gases. It is also observed that when incident waves travel from the vapor phase to the liquid phase, the refracted waves must be accelerated and move forward.
This paper deals with an initial boundary value problem for a class of nonlinear wave equation with nonlinear damping and source terms whose solution may blow up in finite time. An explicit lower bound for blow up time is determined by means of a differential inequality argument if blow up occurs.
In this paper, we concern the Klein-Gordon-Maxwell system with steep potential well ???20210112??? Without global and local compactness, we can tell the difference of multiple solutions from their norms in L^{p}(R^{3}).
In this paper, we propose a new nonmonotone trust region Barzilai-Borwein (BB for short) method for solving unconstrained optimization problems. The proposed method is given by a novel combination of a modified Metropolis criterion, BB-stepsize and trust region method. The new method uses the reciprocal of BB-stepsize to approximate the Hessian matrix of the objective function in the trust region subproblems, and accepts some bad solutions according to the modified Metropolis criterion based on simulated annealing idea. Under some suitable assumptions, the global convergence of the new method is established. Some preliminary numerical results indicate that, the new method is more efficient compared with the existing trust region BB method.
A nonincreasing sequence π=(d_{1}, …, d_{n}) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π. Given a graph H, a graphic sequence π is potentiallyH-graphic if π has a realization containing H as a subgraph. For graphs G_{1} and G_{2}, the potential-Ramsey numberr_{pot}(G_{1}, G_{2}) is the smallest integer k such that for every k-term graphic sequence π, either π is potentially G_{1}-graphic or the complementary sequence π=(k-1-d_{k}, …, k-1-d_{1}) is potentially G_{2}-graphic. For 0 ≤ k ≤ ⎣ t/2 」, denote K_{t}^{-k} to be the graph obtained from K_{t} by deleting k independent edges. If k=0, Busch et al. (Graphs Combin., 30(2014)847-859) present a lower bound on r_{pot}(G, K_{t}) by using the 1-dependence number of G. In this paper, we utilize i-dependence number of G for i ≥ 1 to give a new lower bound on rpot(G, K_{t}^{-k}) for any k with 0 ≤ k ≤ ⎣ t/2 」. Moreover, we also determine the exact values of r_{pot}(K_{n}, K_{t}^{-k}) for 1 ≤ k ≤ 2.
Identity by descent (IBD) sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history. In this paper we provide a framework to estimate IBD sharing for a demographic model called two-population model with migration. We adopt the structured coalescent theory and use a continuous-time Markov jump process {X(t), t ≥ 0} to describe the genealogical process in such model. Then we apply Kolmogorov backward equation to calculate the distribution of coalescence time and develop a formula for estimating the IBD sharing. The simulation studies show that our method to estimate IBD sharing for this demographic model is robust and accurate.
An m×k matrix is said to be a d-row (column) antimagic matrix if its row-sums (column-sums) form an arithmetic progression with a difference d. The goal of this paper is to obtain the existence theorems and construction methods of some d-row (column) antimagic matrices. Using these results we give the necessary and sufficient condition for the existence of an (m, d)-partition of[1, mk].