A weighted curvature flow for shape deformation

In this paper we shall discuss a weighted curvature flow for a regular curve in the 2D Euclidean space. The weighted curvature flow for planar curves is a generalization of the well-known curvature flow discussed by Gage, Hamilton and Grayson. Under a suitable weighted curvature flow, convex curves will remain convex in the deformation process. However, the curve may not converge to a round point for general weights. Indeed, for a nonnegative weight function ω(u) with k isolated zeros, a curve will converge to a limiting k-polygon. The weighted curvature flow will have many useful properties which have applications to image processing. We shall also present some numerical simulations to illustrate how curves deform under the weighted curvature flow with different weight functions ω(u). Moreover, our algorithm is very effective and stable. The approximation of higher derivatives in our new algorithm only involve in the neighboring points.     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

Nonnegative sectional curvature hypersurfaces in a real space form

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form M ⁿ⁺¹(c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M ⁿ⁺¹(c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S ^k (a) × S ^n-k (√1 ? a ²), 1 ≤ k ≤ n ? 1, in S ⁿ⁺¹(1) and the Riemannian product H ^k (tanh² r ? 1) × S ^n-k (coth² r ? 1), 1 ≤ k ≤ n ? 1, in H ⁿ⁺¹(?1).     

On the generalized mean curvature

We study some properties of graphs whose mean curvature (in distributional sense) is a vector Radon measure. In particular, we prove that the distributional mean curvature of the graph of a Lipschitz continuous function u is a measure if and only if the distributional divergence of T u is a measure. This equivalence fails to be true if Lipschitz continuity is relaxed, as it is shown in a couple of examples. Finally, we prove a theorem of approximation in W ^(1,1) and in the sense of mean curvature of C ² graphs by polyhedral graphs. A number of examples illustrating different situations which can occur complete the work.     

A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature

We prove a pointwise gradient bound for bounded solutions of Δu+F^′(u)=0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature.We also obtain some rigidity results when equality in the bound holds at some point.     

An extremal case of the equation of prescribed Weingarten curvature

In this paper we investigate the existence and regularity of solutions to a Dirichlet problem for a Hessian quotient equation on the sphere. The equation in question arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cylinder tangentially and whose k-th Weingarten curvature is a given function. This is a generalization of a Gaussian curvature problem treated in [13]. Essentially given ${\Omega \subset \mathbb{R}^n}$ we seek a convex function u such that graph(u) has a prescribed k-th curvature ψ and |Du(x)| → ∞ as x → ?Ω. Under certain regularity assumptions on ψ and Ω we are able to demonstrate the existence of a solution whose graph is C ^3,α provided that ${\psi^{-\frac{1}{k}} = \psi^{-\frac{1}{k}}(x, \nu)}$ is convex in x and a certain compatibility condition between ψ| _?Ω and Ω is satisfied.     

Compactness and gradient bounds for solutions of the mean curvature system in two independent variables

We establish here a priori estimates for the gradient of solutions of the minimal surface system in two independent variables and for the curvature of their graphs. With the intent of extending these results to graphs with nonzero mean curvature vectors, we then analyze the compactness properties of smooth (C ²) solutions of the mean curvature system. Using a geometric measure theory approach we are able to classify the possible behaviors of a sequence {u _?(x)} of such solutions, under the assumption that a uniform bound on the area of the graphs holds and suitable hypotheses on the length of the mean curvature vectorH(x). In particular, this implies the existence of an a priori gradient bound depending on the oscillation of the solutionu(x).     

A note on the Riccati differential equation

In this note, we obtain some results for the Riccati differential equations u′=A(z)+u² with nonentire meromorphic functions A(z). Some examples are given to illustrate our some results are sharp.     

The inverse of a matrix polynomial

An explicit representation is obtained for P(z)^?1 when P(z) is a complex n×n matrix polynomial in z whose coefficient of the highest power of z is the identity matrix. The representation is a sum of terms involving negative powers of z?λ for each λ such that P(λ) is singular. The coefficients of these terms are generated by sequences u_k, v_k of 1×n and n×1 vectors, respectively, which satisfy u₁≠0, v₁≠0, ∑^k?1_h=0(1?h!)u_k?hP^(h)(λ)=0, ∑^k?1_h=0(1?h!)P^(h)(λ)v_k?h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at λ but not necessarily a polynomial, the terms in the representation involving negative powers of z?λ provide the principal part of the Laurent expansion for P(z)^?1 in a punctured neighborhood of z=λ.     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

偶数阶自共轭微分算子谱的离散性条件

In this paper, we consider differential operators of 2nd-order
a[u]=(-1)^k(a_k(x)u^(k)(x))(k), x∈(0, ∞)
whose coefficients a_k(x) are restricted by powers of e^x, and give conditions on the coefficients sufficient to ensure that the spectrum is discrete; next we formulate necessary and sufficient conditions for the discreteness of the spectrum of differential operators whose coefficients a_k(x) may increase as e^αkx as x→∞.     

The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w²+R(z)²(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w²+P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w²+(z−z₀)P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w²+A(z−z₀)²(w^′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab²=1, a²=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B²=C and q^′(z)=±P(z).     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

Isolated singularities for some types of curvature equations

We consider the removability of isolated singularities for the curvature equations of the form H_k[u]=0, which is determined by the kth elementary symmetric function, in an n-dimensional domain. We prove that, for 1?k?n−1, isolated singularities of any viscosity solutions to the curvature equations are always removable, provided the solution can be extended continuously at the singularities. We also consider the class of “generalized solutions” and prove the removability of isolated singularities.     

Property (X) for Second-Order Nonlinear Differential Equations of Generalized Euler Type

In this paper the generalized nonlinear Euler differential equation t²k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X⁺), which is very important for the existence of periodic solutions and oscillation theory.     

The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc

We consider the complex differential equations of the form

Ak(z)f(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=F(z),     

Regularity of subelliptic Monge-Ampère equations

In dimension n?3, for k≈|x|^2m that can be written as a sum of squares of smooth functions, we prove that a C² convex solution u to a subelliptic Monge-Ampère equation detD²u=k(x,u,Du) is itself smooth if the elementary (n−1)st symmetric curvature kn−1 of u is positive (the case m?2 uses an additional nondegeneracy condition on the sum of squares). Our proof uses the partial Legendre transform, Calabi's identity for ∑uijσij where σ is the square of the third order derivatives of u, the Campanato method Xu and Zuily use to obtain regularity for systems of sums of squares of Hörmander vector fields, and our earlier work using Guan's subelliptic methods.     

On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).     

一类高阶整函数系数微分方程解的增长级、下级与超级

该文研究了一类高阶整函数系数微分方程解的增长性,对方程f~(k)+A_(k-1)(z)e~(ak-1z).f~(k-1)+…+A_0(z)e~(a0z)f=0与方程f~(k)+(A_(k-1)(z)e~(ak-1z)+D_(k-1)(z))f~(k-1)+…+(A_0(z)e~(a0z)+D_0(z))f=0中a_j(0≤j≤k-1)幅角主值不全相等的情形,得到了解的增长级、下级与超级的精确估计.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.