The spectrum and stability of travelling pulses in a coupled FitzHugh-Nagumo equation

For a coupled slow-fast FitzHugh-Nagumo(FHN)equation derived from a reaction-diffusionmechanics(RDM)model,Holzer et al.(2013)studied the existence and stability of the travelling pulse,which consists of two fast orbit arcs and two slow ones,where one fast segment passes the unique fold point with algebraic decreasing and two slow ones follow normally hyperbolic critical curve segments.Shen and Zhang(2021)obtained the existence of the travelling pulse,whose two fast orbit arcs both exponentially decrease,and one of the slow orbit arcs could be normally hyperbolic or not at the origin.Here,we characterize both the nonlinear and spectral stability of this travelling pulse.

Time-inconsistent stochastic linear-quadratic control problem with indefinite control weight costs

A time-inconsistent linear-quadratic optimal control problem for stochastic differential equations is studied.We introduce conditions where the control cost weighting matrix is possibly singular.Under such conditions,we obtain a family of closed-loop equilibrium strategies via multi-person differential games.This result extends Yong’s work(2017) in the case of stochastic differential equations,where a unique closed-loop equilibrium strategy can be derived under standard conditions(namely,the control cost weighting matrix is uniformly positive definite,and the other weighting coefficients are positive semidefinite).

The stable Auslander-Reiten components of certain monomorphism categories

Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.

On the Chern conjecture for isoparametric hypersurfaces

For a closed hypersurface Mn?Sn+1(1)with constant mean curvature and constant non-negative scalar curvature,we show that if tr(Ak)are constants for k=3,...,n-1 and the shape operator A,then M is isoparametric.The result generalizes the theorem of de Almeida and Brito(1990)for n=3 to any dimension n,strongly supporting the Chern conjecture.

Optimal orthogonalization processes

Two optimal orthogonalization processes are devised toorthogonalize,possibly approximately,the columns of a very large and possiblysparse matrix A∈C^(n×k).Algorithmically the aim is,at each step,to optimallydecrease nonorthogonality of all the columns of A.One process relies on using translated small rank corrections.Another is a polynomial orthogonalization process forperforming the Löwdin orthogonalization.The steps rely on using iterative methods combined,preferably,with preconditioning which can have a dramatic effect on how fast thenonorthogonality decreases.The speed of orthogonalization depends on howbunched the singular values of A are,modulo the number of steps taken.These methods put the steps of the Gram-Schmidt orthogonalizationprocess into perspective regardingtheir(lack of)optimality.The constructions are entirely operatortheoretic and can be extended to infinite dimensional Hilbert spaces.

Nielsen coincidence theory on infra-solvmanifolds of Sol

We derive averaging formulas for the Lefschetz coincidence numbers,the Nielsen coincidence numbers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected solvable Lie group of type(R).As an application,we compare our formula for the Nielsen coincidence numbers with a result of Jezierski(1992)for pairs of maps on some infra-solvmanifolds of Sol.For all the pairs of self-maps of a nonorientable infra-solvmanifold of Sol,we determine the sets of all the possible values of the Nielsen coincidence numbers and the Reidemeister coincidence numbers.

An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

On U(n)-invariant strongly convex complex Finsler metrics

In this paper,we obtain a necessary and sufficient condition for a U(n)-invariant complex Finsler metric F on domains in C^(n) to be strongly convex,which also makes it possible to investigate the relationship between real and complex Finsler geometries via concrete and computable examples.We prove a rigid theorem which states that a U(n)-invariant strongly convex complex Finsler metric F is a real Berwald metric if and only if F comes from a U(n)-invariant Hermitian metric.We give a characterization of U(n)-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of the U(n)-invariant strongly pseudoconvex complex Finsler metric.Finally,we prove that the real geodesics of some U(n)-invariant complex Finsler metric restricted on the unit sphere S^(2n-1)■C^(n) share a specific property as that of the complex Wrona metric on C^(n).

Global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity

In this paper,we consider an initial-boundary value problem for the 2D incompressible magnetomicropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions.We establish the global well-posedness of strong solutions around the equilibrium(0,e1,0).

Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments.By combining compensated split-step methods and balanced methods,a class of compensated split-step balanced(CSSB)methods are suggested for solving the equations.Based on the one-sided Lipschitz condition and local Lipschitz condition,a strong convergence criterion of CSSB methods is derived.It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions.Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods.Moreover,in order to show the computational advantage of CSSB methods,we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

Riesz potentials of Hardy-Hausdorff spaces and Q-type spaces

This study investigates the restriction problem for the Riesz potentials of Hardy-Hausdorff spaces HH^1-γ(R^n)and Q-type spaces Qγ(R^n).By exploiting a geometric-measure theory generated by the indicatorlike functions of compact sets,it is proved that the Riesz operator Iαcontinuously maps HH^1-γ(R^n)into the weak Morrey spaces L^q,λ/μ,*induced by a Radon measureμ,which obeys a geometric condition.

A function model for the Teichmüller space of a closed hyperbolic Riemann surface

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface.Then we introduce a new metric on the Teichmüller space by using the maximum norm on the function space.We prove that the identity map from the Teichmüller space equipped with the Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Moreover, we prove that the inverse of the identity, i.e., the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the Teichmüller metric, is continuous(but not uniformly). Therefore, the topology induced by the new metric is the same as the topology induced by the Teichmüller metric on the Teichmüller space.Finally, we give a remark about the pressure metric on the function model and the Weil-Petersson metric on the Teichmüller space.

Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds

In this paper, we prove the C^(1,1)-regularity of the plurisubharmonic envelope of a C^(1,1) function on a compact Hermitian manifold. We also present the examples to show this regularity is sharp.

Balanced augmented jackknife empirical likelihood for two sample U-statistics

In this paper, we investigate the two sample U-statistics by jackknife empirical likelihood(JEL),a versatile nonparametric approach. More precisely, we propose the method of balanced augmented jackknife empirical likelihood(BAJEL) by adding two artificial points to the original pseudo-value dataset, and we prove that the log likelihood ratio based on the expanded dataset tends to the χ~2 distribution.

Some results on the regularization of LSQR for large-scale discrete ill-posed problems

LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems(CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition(TSVD) method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.

Two families of Niho sequences having four-valued cross correlation with m-sequences Dedicated to Professor Yingbo Zhang on the Occasion of His 70th Birthday

For two odd integers m and s with 1≤s < m and gcd(m, s) = 1, let h satisfy h(2~s-1) ≡1(mod 2~m+ 1) and d =(h + 1)(2~m-1) + 1. The cross correlation function between a binary m-sequence of period 22~m-1 and its d-decimation sequence is proved to take four values, and the correlation distribution is completely determined. Let n be an even integer and k be an integer with 1≤k≤n/2. For an odd prime p and a p-ary m-sequence {s(t)} of period p^n-1, define u(t) =∑(p^k-1)/2 i=0 s(d_it), where d_i = ip^(n/2) + p^k-i and i = 0, 1,...,(p^k-1)/2. It is proved that the cross correlation function between {u(t)} and {s(t)} is three-valued or four-valued depending on whether k is equal to n/2 or not, and the distribution is also determined.

Multi-recurrence and van der Waerden systems

We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

A weighted Wilcoxon estimate for the covariate-specific ROC curve

The covariate-specific receiver operating characteristic(ROC) curve is an important tool for evaluating the classification accuracy of a diagnostic test when it is associated with certain covariates. In this paper,a weighted Wilcoxon estimator is constructed for estimating this curve under the framework of location-scale model for the test result. The asymptotic normality is established, both for the regression parameter estimator and the estimator for the covariate-specific ROC curve at a fixed false positive point. Simulation results show that the Wilcoxon estimator compares favorably to its main competitors in terms of the standard error, especially when outliers exist in the covariates. As an illustration, the new procedure is applied to the dementia data from the national Alzheimer's coordinating center.

Correlation functions of gauged linear σ-model

This is the second paper in a series following Tian and Xu(2015), on the construction of a mathematical theory of the gauged linear σ-model(GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.

Modulation space estimates for damped fractional wave equation

Instead of the L^p estimates,we study the modulation space estimates for the solution to the damped wave equation.Decay properties for both the linear and semilinear equations are obtained.The estimates in modulation space differ in many aspects from those in L^p space.