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This paper presents a full classification of the short-time behavior of the solution and the interfaces in the Cauchy problem for the nonlinear second order singular parabolic PDE u t - Δ u m + b u β = 0 , x ∈ R N , 0 < t < T with nonnegative initial function u 0 such that s u p p u 0 = { | x | < R } , u 0 ∼ C ( R - | x | ) α , as | x | → R - 0 , where 0 < m < 1 , b , β , C , α > 0 . Depending on the relative strength of the fast diffusion and absorption terms the problem may have infinite ( β ≥ m ) or finite ( β < m ) speed of propagation. In the latter case, the interface surface t = η ( x ) may shrink, expand or remain stationary depending on the relative strength of the fast diffusion and strong absorption terms near the boundary of support, expressed in terms of the parameters m , β , α , and C . In all cases we prove the existence or non-existence of the interfaces, explicit formula for the interface asymptotics, and local solution near the interface or at infinity.

In this paper, we study the well-posedness of the Forward–Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components Y and Z. We establish various sufficient conditions for the well-posedness of an ODE that dominates the characteristic BSDE, which leads to the existence of the desired regular decoupling random field, whence the solvability of the original FBSDE. A synthetic analysis of the solvability is given, as a \"User's Guide,\" for a large class of FBSDEs that are not covered by the existing methods. Some of them have important implications in applications.

In this paper, by analogy with the definition of the nonnegative splitting of a matrix, we introduce the definition of the nonnegative splitting of a tensor. Considering the case that nonnegative splitting of a strong 𝓜-tensor is not necessarily convergent, we establish a new convergence theorem. Since comparison theorems involving the spectral radius of iterative tensors are useful tools in the analysis of convergence rate of tensor splitting iterative (TSI) methods, we derive several comparison theorems for nonnegative splittings of tensors in this paper. These results generalize the previous ones.

In this paper stability analysis of fractional-order nonlinear systems is studied. An extension of Lyapunov direct method for fractional-order systems using Bihari’s and Bellman–Gronwall’s inequality and a proof of comparison theorem for fractional-order systems are proposed.

We prove an analogue of Scholze’s Primitive Comparison Theorem for proper rigid spaces over an algebraically closed non-archimedean field K of characteristic p . This implies a v-topological version of the Primitive Comparison Theorem for proper finite type morphisms f : X → Y of analytic adic spaces over Z p . We deduce new cases of the Proper Base Change Theorem for p -torsion coefficients and the Künneth formula in this setting.

The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on an ad hoc basis. Recently, Gaussian approximation has been used to justify bootstrap adjustments in large scale simultaneous inference in some general settings when n ≫ (log p)⁷, where p is the multiplicity of the inference problem and n is the sample size. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high dimension. In this paper, we reduce the sample size requirement to n ≫ (log p)⁵ for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov–Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anticoncentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable or strong enough to produce desired results.

This paper investigates dynamical behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte to the growth of an immunogenic tumor. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs), and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

The goal of this paper was to study the oscillations of a class of fourth-order nonlinear delay differential equations with a middle term. Novel oscillation theorems built on a proper Riccati-type transformation, the comparison approach, and integral-averaging conditions were developed, and several symmetric properties of the solutions are presented. For the validation of these theorems, several examples are given to highlight the core results.

We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with -range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie–Yeroshkin and Kuwae–Li.