GENERAL COUPLED MEAN-FIELD REFLECTED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations(FBSDEs),whose coefficients not only depend on the solution but also on the law of the solution.The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations(BSDEs)under Lipschitz conditions,and for the one-dimensional case a comparison theorem is studied.With the help of this comparison result,we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions.It should be mentioned that,under appropriate assumptions,we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner.

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

IMPROVED REGULARITY OF HARMONIC DIFFEOMORPHIC EXTENSIONS ON QUASIHYPERBOLIC DOMAINS

Let X be a Jordan domain satisfying certain hyperbolic growth conditions.Assume that φ is a homeomorphism from the boundary ?X of X onto the unit circle.Denote by h the harmonic diffeomorphic extension of φ from X onto the unit disk.We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of h.These generalize the Sobolev regularity of h in [A.Koski,J.Onninen,Sobolev homeomorphic extensions,J.Eur.Math.Soc.23(2021) 4065-4089,Theorem 3.1].

REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES

Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this short note,we introduce a notion of p-energy for u which is slightly different from the original definition of Korevaar and Schoen.We show that each minimizing p-harmonic mapping(p≥2)associated to our notion of p-energy is locally Holder continuous whenever its image lies in a compact subset of X.

A ROBUST COLOR EDGE DETECTION ALGORITHM BASED ON THE QUATERNION HARDY FILTER

This paper presents a robust filter called the quaternion Hardy filter(QHF)for color image edge detection.The QHF can be capable of color edge feature enhancement and noise resistance.QHF can be used flexibly by selecting suitable parameters to handle different levels of noise.In particular,the quaternion analytic signal,which is an effective tool in color image processing,can also be produced by quaternion Hardy filtering with specific parameters.Based on the QHF and the improved Di Zenzo gradient operator,a novel color edge detection algorithm is proposed;importantly,it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique.From the experimental results,we conclude that the minimum PSNR improvement rate is 2.3%and the minimum SSIM improvement rate is 30.2%on the CSEE database.The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.

Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching

A model for the mechanics of woven fabrics is developed in the framework of two-dimensional elastic surface theory. Thickness effects are modeled indirectly in terms of appropriate constitutive equations. The model accounts for the strain of the fabric and additional effects associated with the normal bending, geodesic bending, and twisting of the constituent fibers.

Uniform convergence rates for spot volatility estimation

This study presents the uniform convergence rate for spot volatility estimators based on delta sequences.Kernel and Fourier-based estimators are examples of this type of estimator.We also present the uniform convergence rates for kernel and Fourier-based estimators of spot volatility as applications of the main result.

UNIFORM OPTIMAL-ORDER ESTIMATES FOR FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION EQUATIONS

This article summarizes our recent work on uniform error estimates for various finite elementmethods for time-dependent advection-diffusion equations.

An orthogonally accumulated projection method for symmetric linear system of equations

A direct as well as iterative method(called the orthogonally accumulated projection method, or the OAP for short) for solving linear system of equations with symmetric coefficient matrix is introduced in this paper. With the Lanczos process the OAP creates a sequence of mutually orthogonal vectors, on the basis of which the projections of the unknown vectors are easily obtained, and thus the approximations to the unknown vectors can be simply constructed by a combination of these projections. This method is an application of the accumulated projection technique proposed recently by the authors of this paper, and can be regarded as a match of conjugate gradient method(CG) in its nature since both the CG and the OAP can be regarded as iterative methods, too. Unlike the CG method which can be only used to solve linear systems with symmetric positive definite coefficient matrices, the OAP can be used to handle systems with indefinite symmetric matrices. Unlike classical Krylov subspace methods which usually ignore the issue of loss of orthogonality, OAP uses an effective approach to detect the loss of orthogonality and a restart strategy is used to handle the loss of orthogonality.Numerical experiments are presented to demonstrate the efficiency of the OAP.

RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS

Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.

FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O（△t ＋ △x^2）-order error estimate is derived in the discrete L2-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.

Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms

The main purpose of this paper is to study different types of sampling formulas of quaternionic functions,which are bandlimited under various quaternion Fourier and linear canonical transforms.We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms.In addition,the relationships among different types of sampling formulas under various transforms are discussed.First,if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are identical.If this rectangle is not symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are different from each other.Second,using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform,we derive sampling formulas under various quaternion linear canonical transforms.Third,truncation errors of these sampling formulas are estimated.Finally,some simulations are provided to show how the sampling formulas can be used in applications.

ANURBS-Enhanced Finite Volume Method for Steady Euler Equations with Goal-Oriented h-Adaptivity

In[A NURBS-enhanced finite volume solver for steady Euler equations,X.C.Meng,G.H.Hu,J.Comput.Phys.,Vol.359,pp.77–92],aNURBS-enhanced finite volume method was developed to solve the steady Euler equations,in which the desired high order numerical accuracy was obtained for the equations imposed in the domain with a curved boundary.In this paper,the method is significantly improved in the following ways:(i)a simple and efficient point inversion technique is designed to compute the parameter values of points lying on a NURBS curve,(ii)with this new point inversion technique,the h-adaptive NURBS-enhanced finite volume method is introduced for the steady Euler equations in a complex domain,and(iii)a goal-oriented a posteriori error indicator is designed to further improve the efficiency of the algorithm towards accurately calculating a given quantity of interest.Numerical results obtained from a variety of numerical experiments with different flow configurations successfully show the effectiveness and robustness of the proposed method.

Stochastic maximum principle for systems driven by local martingales with spatial parameters

We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we obtain the stochastic maximum principle as the necessary condition for an optimal control,and we also prove its sufficiency under proper conditions.The stochastic linear quadratic problem in this setting is also discussed.

Construction and analysis of the quadratic finite volume methods on tetrahedral meshes

We construct and analyze a family of quadratic finite volume method(FVM)schemes over tetrahedral meshes.In order to prove the stability and the error estimate,we propose the minimum V-angle condition on tetrahedral meshes,and the surface and volume orthogonal conditions on dual meshes.Through the technique of element analysis,the local stability is equivalent to a positive definiteness of a 9 × 9 element matrix,which is difficult to analyze directly or even numerically.With the help of the surface orthogonal condition and the congruent transformation,this element matrix is reduced into a block diagonal matrix,and then we carry out the stability result under the minimum V-angle condition.It is worth mentioning that the minimum V-angle condition of the tetrahedral case is very different from a simple extension of the minimum angle condition for triangular meshes,while it is also convenient to use in practice.Based on the stability,we prove the optimal H^(1) and L^(2) error estimates,respectively,where the orthogonal conditions play an important role in ensuring the optimal L^(2) convergence rate.Numerical experiments are presented to illustrate our theoretical results.

Generalized Rectangular Convexity

In this paper we investigate the tr-convexity and the rectangular biconvexity in Euclidean spaces.

An updated estimation of the risk of transmission of the novel coronavirus(2019-nCov)

The basic reproduction number of an infectious agent is the average number of infections one case can generate over the course of the infectious period,in a naïve,uninfected population.It is well-known that the estimation of this number may vary due to several methodological issues,including different assumptions and choice of parameters,utilized models,used datasets and estimation period.With the spreading of the novel coronavirus(2019-nCoV)infection,the reproduction number has been found to vary,reflecting the dynamics of transmission of the coronavirus outbreak as well as the case reporting rate.Due to significant variations in the control strategies,which have been changing over time,and thanks to the introduction of detection technologies that have been rapidly improved,enabling to shorten the time from infection/symptoms onset to diagnosis,leading to faster confirmation of the new coronavirus cases,our previous estimations on the transmission risk of the 2019-nCoV need to be revised.By using time-dependent contact and diagnose rates,we refit our previously proposed dynamics transmission model to the data available until January 29th,2020 and re-estimated the effective daily reproduction ratio that better quantifies the evolution of the interventions.We estimated when the effective daily reproduction ratio has fallen below 1 and when the epidemics will peak.Our updated findings suggest that the best measure is persistent and strict self-isolation.The epidemics will continue to grow,and can peak soon with the peak time depending highly on the public health interventions practically implemented.

Dynamics of Quantum Coherence in Bell-Diagonal States under Markovian Channels

We study the curves of coherence for the Bell-diagonal states including l_1-norm of coherence and relative entropy of coherence under the Markovian channels in the first subsystem once. For a special Bell-diagonal state under bit-phase flip channel, we find frozen coherence under l_1 norm occurs, but relative entropy of coherence decrease. It illustrates that the occurrence of frozen coherence depends on the type of the measure of coherence. Also, we study the coherence evolution of Bell-diagonal states under Markovian channels in the first subsystem n times and find that coherence under depolarizing channel decreases initially then increases for small n and tends to zero for large n. The dynamics of coherence of the Bell-diagonal state under two independent same type local Markovian channels is discussed.

Sparse polynomial interpolation based on diversification

We consider the problem of interpolating a sparse multivariate polynomial over a finite field,represented with a black box.Building on the algorithm of Ben-Or and Tiwari(1988)for interpolating polynomials over fields with the characteristic 0,we develop a new Monte Carlo algorithm over finite fields by doing additional probes.To interpolate a polynomial f∈Fq[x1,...,xn]with a partial degree bound D and a term bound T,our new algorithm costs O~(nTlog^(2)q+nT√Dlogq)bit operations and uses 2(n+1)T probes to the black box.If q O(n T^(2)D),it has a constant success rate to return the correct polynomial.Compared with the previous algorithms over a general finite field,our algorithm has better complexity in the parameters n,T and D,and is the first one to achieve the complexity of the fractional power about D,while keeping linear in n and T.A key technique is a randomization which makes all the coefficients of the unknown polynomial distinguishable,producing a diverse polynomial.This approach,called diversification,was proposed by Giesbrecht and Roche(2011).Our algorithm interpolates each variable independently by using O(T)probes,and then uses the diversification to correlate terms in different images.At last,we get the exponents by solving the discrete logarithms and obtain the coefficients by solving a linear system.We have implemented our algorithm in Maple.Experimental results show that our algorithm can be applied to the sparse polynomials with large degree within a few minutes.We also analyze the success rate of the algorithm.

On abelian 2-ramification torsion modules of quadratic fields

For a number field F and a prime number p, the Z_(p)-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T_(p)(F), is an important subject in the abelian p-ramification theory. In this paper, we study the group T_(2)(F) = T_(2)(m) of the quadratic field F = Q(m_(1/2)). Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk4(T_(2)(-m))= rk2(T_(2)(-m))-rank(R), where R is a certain explicitly described Rédei matrix over F_(2). Furthermore, using this formula, we obtain the 4-rank density formula of T_(2)-groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain the results about the 2-power divisibility of orders of T_(2)(±l) and T_(2)(±2l), both of which are cyclic 2-groups. In particular, we find that #T_(2)(l) ≡ 2#T_(2)(2l) ≡ h_(2)(-2l)(mod 16) if l ≡ 7(mod 8),where h_(2)(-2l) is the 2-class number of Q((-2l)_(1/2)). We then obtain the density results for T_(2)(±l) and T_(2)(±2l)when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about T_(p)(F) when F varies over real or imaginary quadratic fields for any prime p, and about T_(2)(±l)and T_(2)(±2l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T_(2)(l) case is closely connected to Shanks-Sime-Washington’s speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units.