Results in Applied Mathematics最新文献

英文中文

A general law of the iterated logarithm for non-additive probabilities 非加法概率的迭代对数一般规律

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-07-13 DOI: 10.1016/j.rinam.2024.100475

Zhaojun Zong , Miaomiao Gao , Feng Hu

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of $V_{n} ≔ \frac{\sum_{i = 1}^{n} X_{i}}{\sqrt{n} ϕ (n)}$ under some reasonable assumptions, where ${X_{i}}_{i = 1}^{\infty}$ is a sequence of random variables and $ϕ$ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).

受数理经济学、量子力学和金融学中一些有趣问题的启发，非相加概率被用来描述一般非相加的现象。本文在已有成果的基础上，进一步研究了非加概率的迭代对数定律（LIL）。在彭晓峰提出的亚线性期望框架下，我们给出了 Vn≔∑i=1nXinj(n)在一些合理假设下的两个收敛结果，其中 {Xi}i=1∞ 是一个随机变量序列，j 是一个正的非递减函数。由此，对于负相关和非同分布的随机变量，证明了非相加概率的一般 LIL。事实证明，我们的结果是柯尔莫哥洛夫 LIL 和哈特曼-温特纳 LIL 的自然扩展。本文的定理 1 和定理 2 可以看作是 Chen 和 Hu（2014）中定理 1 的扩展。

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引用次数: 0

Oscillatory properties for Emden–Fowler type difference equations with oscillating coefficients 具有振荡系数的埃姆登-福勒型差分方程的振荡特性

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-07-05 DOI: 10.1016/j.rinam.2024.100472

Yaşar Bolat , Murat Gevgeşoğlu , George E. Chatzarakis

In this paper, we give new criteria on the oscillation of the fourth-order Emden–Fowler type delay difference equation with oscillating coefficients of the form $Δ W_{n} + r_{n} y_{n - τ}^{β} = 0, n \geq n_{0},$ where $W_{n} = p_{n} {(Δ^{3} v_{n})}^{α}$ and $v_{n} = y_{n} + q_{n} y_{n - σ}$ . For this we use the Riccati transformation method and the comparison method. Also we give some examples to illustrate our results.

本文给出了四阶埃姆登-福勒延迟差分方程振荡系数的新判据，其振荡系数形式为 ΔWn+rnyn-τβ=0,n≥n0，其中 Wn=pnΔ3vnα 和 vn=yn+qnyn-σ。为此，我们使用了里卡提变换法和比较法。此外，我们还举了一些例子来说明我们的结果。

引用次数: 0

Boundedness and asymptotic behavior in a quasilinear chemotaxis system with nonlinear diffusion and singular sensitivity for alopecia areata 具有非线性扩散和奇异敏感性的准线性趋化系统的有界性和渐近行为，用于治疗脱发症

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-07-02 DOI: 10.1016/j.rinam.2024.100473

Luxu Zhou, Fugeng Zeng, Lei Huang

<div><p>This paper is concerned with a three-component quasilinear chemotaxis system with nonlinear diffusion and singular sensitivity for alopecia areata <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mfenced><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi></mrow></mfenced><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∇</mo><mfenced><mrow><mfrac><mrow><mi>u</mi></mrow><mrow><mi>w</mi></mrow></mfrac><mo>∇</mo><mi>w</mi></mrow></mfenced><mo>+</mo><mi>w</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mfenced><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∇</mo><mi>v</mi></mrow></mfenced><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∇</mo><mfenced><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mi>w</mi></mrow></mfrac><mo>∇</mo><mi>w</mi></mrow></mfenced><mo>+</mo><mi>w</mi><mo>+</mo><mi>r</mi><mi>u</mi><mi>v</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>v</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo>−</mo><mi>w</mi><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>u</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>v</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>w</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><

本文研究的是一个三分量准线性趋化系统，该系统具有非线性扩散和奇异敏感性，适用于脱发症ut=∇D1(u)∇u-χ1∇uw∇w+w-μ1uγ1、(x,t)∈Ω×(0,∞),vt=∇D2(v)∇v-χ2∇vw∇w+w+ruv-μ2vγ2、(x,t)∈Ω×(0,∞),wt=Δw+u+v-w,(x,t)∈Ω×(0,∞),∂u∂ν=∂v∂ν=∂w∂ν=0,(x,t)∈∂Ω×(0,∞),u(x,0)=u0(x),v(x、0)=v0(x),w(x,0)=w0(x),x∈Ω，与凸光滑有界域 Ω⊂R2 中的均相 Neumann 边界条件相关。i=1,2时，参数χi,μi,r为正且γi≥2。对于所有 s≥0，非线性扩散函数 Di(s)∈C2 满足 Di(s)⩾(s+1)αi。我们将深入分析上述系统在特定条件下经典解的全局存在性和有界性。此外，在 γi=2 的情况下，我们建立了一个 Lyapunov 函数，并仔细研究了它的时间演化，以确定共存状态的渐进稳定性。

引用次数: 0

Propagation reversal on trees in the large diffusion regime 大扩散系统中树木上的传播逆转

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-06-26 DOI: 10.1016/j.rinam.2024.100468

Hermen Jan Hupkes , Mia Jukić

In this work we study travelling wave solutions to bistable reaction–diffusion equations on bi-infinite $k$ -ary trees in the continuum regime where the diffusion parameter is large. Adapting the spectral convergence method developed by Bates and his coworkers, we find an asymptotic prediction for the speed of travelling front solutions. In addition, we prove that the associated profiles converge to the solutions of a suitable limiting reaction–diffusion PDE. Finally, for the standard cubic nonlinearity we provide explicit formulas to bound the thin region in parameter space where the propagation direction undergoes a reversal.

在这项研究中，我们研究了在扩散参数很大的连续系统中，双稳态反应-扩散方程在双无限 k-ary 树上的行波解。通过采用贝茨及其同事开发的谱收敛方法，我们找到了游波前沿解速度的渐近预测。此外，我们还证明了相关剖面收敛于合适的极限反应-扩散 PDE 的解。最后，对于标准立方非线性，我们提供了明确的公式来约束参数空间中传播方向发生逆转的薄区域。

引用次数: 0

Basis for high order divergence-free finite element spaces 高阶无发散有限元空间的基础

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-06-24 DOI: 10.1016/j.rinam.2024.100469

A. Alonso Rodríguez , J. Camaño , E. De Los Santos , F. Rapetti

A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is here extended to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms.

在此，我们将一种在低多项式程度中用于构建无发散函数空间有限元基础的经典方法扩展到无拓扑限制的有界域的任意多项式程度。该方法使用与两个微分算子（梯度和发散）相关的图，并使用第一个图的生成树来选择基础。该方法可用于表达有限元微分形式的两个主要自由度系列--权值和矩值。

引用次数: 0

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving ψ-Caputo fractional derivative 涉及ψ-卡普托分数导数的时间分数四阶二维扩散波方程的离散谱方法

IF 2 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-06-18 DOI: 10.1016/j.rinam.2024.100466

M.H. Heydari , M. Razzaghi

In this work, the $ψ$ -Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, $ψ$ , is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the $ψ$ -Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the $ψ$ -Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

ψ-卡普托分数导数是经典卡普托分数导数的广义化，其中充分可微函数的分数导数是相对于另一个严格递增函数ψ定义的，本研究利用ψ-卡普托分数导数定义时间分数四阶二维扩散波方程。为求解该方程，开发了一种切比雪夫-高斯-洛巴托方案。这样，就得出了切比雪夫多项式的 ψ-Riemann-Liouville 分数积分的新运算矩阵。通过有限离散切比雪夫级数，并利用所表达的运算矩阵，确定从近似ψ-卡普托分数导数项中提取的代数系统的解，从而获得方程的解。通过求解两个示例检验了所建立方法的有效性。

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引用次数: 0

Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity 洛德-舒尔曼热压电混合时间积分方案的稳定性

IF 2 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-06-14 DOI: 10.1016/j.rinam.2024.100467

Vitalii Stelmashchuk, Heorhiy Shynkarenko

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

根据 Lord-Shulman 热压电现有的初始边界值和变分问题，对压电材料的行为进行了瞬态数值分析。对于变分问题的空间离散化，采用了有限元方法；对于时间离散化，则在双曲方程的纽马克方案和抛物方程的广义梯形法则基础上，构建了混合时间积分方案。通过利用能量平衡定律来处理所获得的时间离散化变分问题，证明了所开发的时间积分方案在某些特定的方案参数值下的无条件稳定性。最后，通过与文献中的数值实验结果进行比较，证明了所构建的数值方案的适用性。

引用次数: 0

Through-the-wall object reconstruction via reinforcement learning 通过强化学习重建穿墙物体

IF 2 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-06-13 DOI: 10.1016/j.rinam.2024.100465

Daniel Pomerico, Aihua Wood, Philip Cho

This paper addresses the problem of characterizing and localizing objects via through-the-wall radar imaging. We consider two separate problems. First, we assume a single object is located in a room and we use a convolutional neural network (CNN) to classify the shape of the object. Second, we assume multiple objects are located in a room and use a U-net CNN to determine the location of the object via pixel-by-pixel classification. For both problems, we use numerical methods to simulate the electromagnetic field assuming known room parameters and object location. The simulated data is used to train and evaluate both the CNN and U-net CNN. In the case of single objects, we achieve 90% accuracy in classifying the shape of the object. In the case of multiple objects, we show that the U-Net outputs an image segmentation heat map of the domain space, enabling visual analysis to identify the characteristics of multiple unknown objects. Given sufficient data, the U-net heat map highlights object pixels which provide the location and shape of the unknown objects, with precision and recall accuracy exceeding 80%.

本文探讨了通过穿墙雷达成像对物体进行特征描述和定位的问题。我们考虑了两个不同的问题。首先，我们假设房间里只有一个物体，并使用卷积神经网络（CNN）对物体的形状进行分类。其次，我们假设房间里有多个物体，并使用 U-net CNN 通过逐像素分类来确定物体的位置。对于这两个问题，我们使用数值方法模拟电磁场，假设房间参数和物体位置已知。模拟数据用于训练和评估 CNN 和 U-net CNN。在单个物体的情况下，我们对物体形状分类的准确率达到了 90%。在多个物体的情况下，我们发现 U-Net 可以输出域空间的图像分割热图，从而通过视觉分析识别多个未知物体的特征。在数据充足的情况下，U-Net 热图可突出显示提供未知物体位置和形状的物体像素，精确度和召回精度均超过 80%。

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引用次数: 0

Uncovering a generalised gamma distribution: From shape to interpretation 揭示广义伽马分布：从形状到解释

IF 2 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-05-01 DOI: 10.1016/j.rinam.2024.100461

Matthias Wagener , Andriette Bekker , Mohammad Arashi , Antonio Punzo

In this paper, we introduce the flexible interpretable gamma (FIG) distribution, with origins in Weibulisation, power weighting, and a stochastic representation. The FIG parameters have been verified graphically, mathematically, and through simulation as having separable roles in influencing the left tail, body, and right tail shape. The generalised gamma (GG) distribution has become a standard model for positive data in statistics due to its interpretable parameters and tractable equations. Although there are many generalised forms of the GG that can provide a better fit to data, none of them extend the GG so that the parameters are interpretable. We conduct simulation studies on the maximum likelihood estimates and respective sub-models of the FIG. Finally, we assess the flexibility of the FIG relative to existing models by applying the FIG model to hand grip strength and insurance loss data.

在本文中，我们介绍了灵活可解释伽马分布（FIG），它起源于 Weibulisation、幂加权和随机表示法。经图形、数学和模拟验证，FIG 参数在影响左尾、主体和右尾形状方面具有可分离的作用。广义伽马（GG）分布因其可解释的参数和易于理解的方程，已成为统计学中正向数据的标准模型。尽管有许多广义伽马分布的形式可以更好地拟合数据，但它们都没有扩展伽马分布，使其参数可以解释。最后，我们通过将 FIG 模型应用于手部握力和保险损失数据，评估了 FIG 相对于现有模型的灵活性。

引用次数: 0

Inverting the sum of two singular matrices 反转两个奇异矩阵之和

IF 2 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2024-05-01 DOI: 10.1016/j.rinam.2024.100463

Sofia Eriksson, Jonas Nordqvist

Square matrices of the form $\tilde{A} = A + e D f^{*}$ are considered. An explicit expression for the inverse is given, provided $\tilde{A}$ and $D$ are invertible with $rank (\tilde{A}) = rank (A) + rank (e D f^{*})$ . The inverse is presented in two ways, one that uses singular value decomposition and another that depends directly on the components $A$ , $e$ , $f$ and $D$ . Additionally, a matrix determinant lemma for singular matrices follows from the derivations.

研究了形式为 A˜=A+eDf∗ 的正方形矩阵。只要 A˜ 和 D 是可逆的，秩（A˜）=秩（A）+秩（eDf∗），就能给出逆的明确表达式。求逆的方法有两种，一种是使用奇异值分解，另一种是直接取决于 A、e、f 和 D 的分量。

引用次数: 0

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