JOINT HOLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.

A Comparison of Statistical Methods for Analyzing Discrete Hierarchical Data: A Case Study of Family Data on Alcohol Abuse

Although hierarchical correlated data are increasingly available and are being used in evidence-based medical practices and health policy decision making, there is a lack of information about the strengths and weaknesses of the methods of analysis with such data. In this paper, we describe the use of hierarchical data in a family study of alcohol abuse conducted in Edmonton, Canada, that attempted to determine whether alcohol abuse in probands is associated with abuse in their first-degree relatives. We review three methods of analyzing discrete hierarchical data to account for correlations among the relatives. We conclude that the best analytic choice for typical correlated discrete hierarchical data is by nonlinear mixed effects modeling using a likelihood-based approach or multilevel (hierarchical) modeling using a quasilikelihood approach, especially when dealing with heterogeneous patient data.

Dynamical behaviors of a constant prey refuge ratio-dependent prey-predator model with Allee and fear effects

In this paper,we consider a nonlinear ratio-dependent prey-predator model with constant prey refuge in the prey population.Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population.The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail.Hopf bifurcation including its direction and stability for the model is also studied.We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system.It is also observed that for a fixed strength of fear,an increase in the Allee parameter makes the system unstable,whereas an increase in prey refuge drives the system toward stability.However,higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction.Further,we explore the variation of densities of the populations in different bi-parameter spaces,where the coexistence equilibrium point remains stable.Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software.

Statistical Procedures for Estimating and Detecting Climate Changes

This paper provides a concise description of the philosophy, mathematics, and algorithms for estimating, detecting, and attributing climate changes. The estimation follows the spectral method by using empirical orthogonal functions, also called the method of reduced space optimal averaging. The detection follows the linear regression method, which can be found in most textbooks about multivariate statistical techniques. The detection algorithms are described by using the space-time approach to avoid the non-stationarity problem. The paper includes （1） the optimal averaging method for minimizing the uncertainties of the global change estimate, （2） the weighted least square detection of both single and multiple signals, （3） numerical examples, and （4） the limitations of the linear optimal averaging and detection methods.

LONG-TIME ASYMPTOTIC OF STABLE DAWSON-WATANABE PROCESSES IN SUPERCRITICAL REGIMES

Let W =(W_t)_(t≥0) be a supercritical a-stable Dawson-Watanabe process(withα∈(0,2]) and f be a test function in the domain of-(-△)^(α/2) satisfying some integrability condition. Assuming the initial measure W_0 has a finite positive moment, we determine the long-time asymptotic of arbitrary order of W_t(f). In particular, it is shown that the local behavior of Wt in long-time is completely determined by the asymptotic of the total mass W_t(1), a global characteristic.

Using deep neural networks coupled with principal component analysis for ore production forecasting at open-pit mines

Ore production is usually affected by multiple influencing inputs at open-pit mines.Nevertheless,the complex nonlinear relationships between these inputs and ore production remain unclear.This becomes even more challenging when training data(e.g.truck haulage information and weather conditions)are massive.In machine learning(ML)algorithms,deep neural network(DNN)is a superior method for processing nonlinear and massive data by adjusting the amount of neurons and hidden layers.This study adopted DNN to forecast ore production using truck haulage information and weather conditions at open-pit mines as training data.Before the prediction models were built,principal component analysis(PCA)was employed to reduce the data dimensionality and eliminate the multicollinearity among highly correlated input variables.To verify the superiority of DNN,three ANNs containing only one hidden layer and six traditional ML models were established as benchmark models.The DNN model with multiple hidden layers performed better than the ANN models with a single hidden layer.The DNN model outperformed the extensively applied benchmark models in predicting ore production.This can provide engineers and researchers with an accurate method to forecast ore production,which helps make sound budgetary decisions and mine planning at open-pit mines.

A HYBRID VISCOSITY APPROXIMATION METHOD FOR A COMMON SOLUTION OF A GENERAL SYSTEM OF VARIATIONAL INEQUALITIES,AN EQUILIBRIUM PROBLEM,AND FIXED POINT PROBLEMS

In this paper,we introduce a new iterative method based on the hybrid viscosity approximation method for finding a common element of the set of solutions of a general system of variational inequalities,an equilibrium problem,and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space.We prove a strong convergence theorem of the proposed iterative scheme under some suitable conditions on the parameters.Furthermore,we apply our main result for W-mappings.Finally,we give two numerical results to show the consistency and accuracy of the scheme.

Solvability of Parabolic Anderson Equation with Fractional Gaussian Noise

This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model ∂u/∂t=1/2△+u˙W on[0,∞)×R^(d) with d≥1 has a unique randomfield solution,where W(t,x)is a fractional Brownian sheet on[0,∞)×Rd and formally ˙W=∂d+1/∂t+∂x_(1)…∂x_(d)=W(t,x).When the noise W(t,x) is white in time,our condition is both necessary and sufficient when the initial data u(0,x)is bounded between two positive constants.When the noise is fractional in time with Hurst parameter H_(0)>1/2,our sufficient condition,which improves the known results in the literature,is different from the necessary one.

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

Mathematical modeling of the dynamics of COVID-19 variants of concern:Asymptotic and finite-time perspectives

The COVID-19 pandemic has seen multiple waves,in part due to the implementation and relaxation of social distancing measures by the public health authorities around the world,and also caused by the emergence of new variants of concern(VOCs)of the SARS-Cov-2 virus.As the COVID-19 pandemic is expected to transition into an endemic state,how to manage outbreaks caused by newly emerging VOCs has become one of the primary public health issues.Using mathematical modeling tools,we investigated the dynamics of VOCs,both in a general theoretical framework and based on observations from public health data of past COVID-19 waves,with the objective of understanding key factors that determine the dominance and coexistence of VOCs.Our results show that the transmissibility advantage of a new VOC is a main factor for it to become dominant.Additionally,our modeling study indicates that the initial number of people infected with the new VOC plays an important role in determining the size of the epidemic.Our results also support the evidence that public health measures targeting the newly emerging VOC taken in the early phase of its spread can limit the size of the epidemic caused by the new VOC(Wu et al.,2139Wu,Scarabel,Majeed,Bragazzi,&Orbinski,Wu et al.,2021).

An assessment of direct and indirect effects of climate change for populations of the Rocky Mountain Apollo butterfly （ Parnassius smintheus Doubleday）

Climate change is occurring and insects are responding. Current challenges for ecologists and managers are predicting how organisms will respond to continuing climate change and determining how to mitigate potential negative effects. In contrast to broad scale predictions for climate change involving the distribution of species, in this article we highlight the many ways in which local populations of the Rocky Mountain Apollo butterfly （Parnassius smintheus Doubleday） are predicted to respond to climate change. Using experimental and observational data collected over the past 15 years, we detail both direct and indirect effects. In addition, we identify limitations in our knowledge restricting the ability to predict how populations will respond to climate change. Some changes, such as warmer winter temperatures, may have beneficial effects; however, most of the effects of climate change will be detrimental. Variability in snow cover during the overwintering period and habitat loss due to forest encroachment have the largest potential negative effects.

Semilinear automorphisms of classical groups and quivers

For a classical group G over a field F together with a finite-order automorphism θ that acts compatibly on F, we describe the fixed point subgroup of θ on G and the eigenspaces of θ on the Lie algebra g in terms of cyclic quivers with involution. More precise classification is given when g is a loop Lie algebra, i.e.,when F = C((t)).

A mathematical modeling study of the HIV epidemics at two rural townships in the Liangshan Prefecture of the Sichuan Province of China

Background:As a response to a severe HIV epidemic in the Liangshan Prefecture,one of the worst in China,population based HIV interventions,including two population-wide HIV screening,have been carried out since 2005 at two townships in a remote mountainous region of Liangshan.The objective of our mathematical modeling study is to assess the temporal dynamics of the HIV epidemic in the two townships based on the data collected in the study area during the period 2005e2010.Methods:A mathematical model was set up to describe the population dynamics of HIV transmission in study area.The model was calibrated by fitting it to the HIV testing and treatment data from 2005 to 2008.Validation of the model was done by comparing its predicted value of HIV prevalence in 2010 to the prevalence data obtained in the 2010 population wide HIV testing.The validated model was used to produce estimation of HIV incidence,prevalence and death.Results:Our model estimations show that population-based HIV interventions have significantly slowed down the rise of the HIV epidemic in the two townships.Over the five-year period from 2005 to 2010,the year-over-year rate of increase in HIV incidence,prevalence,and death has declined by 91.5%,28.7%,and 52.3%,respectively.Conclusion:Mathematical models,when integrated with epidemiological and surveillance data,can be an effective tool for predicting the temporal dynamics of HIV and assessing the impacts of HIV interventions.

Robustness properties of dimensionality reduction with Gaussian random matrices

In this paper, motivated by the results in compressive phase retrieval, we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal restricted isometry property(RIP) constants, which plays a central role in the study of phaseless compressed sensing. As a byproduct of our results, we also establish the robustness property of Gaussian random finite frames under erasure.

SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS

This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.

PREFACE

There are large amount of research papers on stochastic partial differential equations (SPDEs). This volume attempts to collect some recent progresses on some very special topics in this broad field. Our concentration will be the stochastic heat (wave) equations driven by Gaussian noises.

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Due to its highly oscillating solution,the Helmholtz equation is numerically challenging to solve.To obtain a reasonable solution,a mesh size that is much smaller than the reciprocal of the wavenumber is typically required(known as the pollution effect).High-order schemes are desirable,because they are better in mitigating the pollution effect.In this paper,we present a high-order compact finite difference method for 2D Helmholtz equations with singular sources,which can also handle any possible combinations of boundary conditions(Dirichlet,Neumann,and impedance)on a rectangular domain.Our method is sixth-order consistent for a constant wavenumber,and fifth-order consistent for a piecewise constant wavenumber.To reduce the pollution effect,we propose a new pollution minimization strategy that is based on the average truncation error of plane waves.Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes,particularly in the critical pre-asymptotic region where kh is near 1 with k being the wavenumber and h the mesh size.

Rational Schur Superalgebras

We develop and study the generalization of rational Schur algebras to the super setting.Similar to the classical case,this provides a new method for studying rational supermodules of the general linear supergroup GL(m|n).Furthermore,we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple.

Convergence to Diffusion Waves for Nonlinear Evolution Equations with Ellipticity and Damping, and with Different End States

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects： {ψt=-（1-α）ψ-θx＋αψxx, θt=-（1-α）θ＋νψx＋（ψθ）x＋αθxx（E） with initial data （ψ,θ）（x,0）=（ψ0（x）,θ0（x））→（ψ±,θ±）as x→±∞ where α and ν are positive constants such that α 〈 1, ν 〈 4α（1 - α）. Under the assumption that ｜ψ＋ - ψ-｜ ＋ ｜θ＋ - θ-｜ is sufficiently small, we show the global existence of the solutions to Cauchy problem （E） and （I） if the initial data is a small perturbation. And the decay rates of the solutions with exponential rates also are obtained. The analysis is based on the energy method.

Homogeneous wavelets and framelets with the refinable structure

Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.