We show that any rotationally symmetric Riemannian manifold has the L¹-Liouville property for harmonic functions, i.e., any integrable harmonic function on it must be identically constant. We also give a characterization of a manifold which carries a non-constant L¹ nonnegative subharmonic function.

We study two initial value problems of the linear diffusion equation u_t = u_xx and the nonlinear diffusion equation u_t = (1 + u_x²)⁻¹u_xx, when Cauchy data u(x,0) = u₀(x) are bounded and oscillate mildly. The latter nonlinear heat equation is the equation of the curvature flow, when the moving curves are represented by graphs. In the case of lim_|x|→+∞|xu′₀(x)|= 0, by using an elementary scaling technique, we show

lim_t→+∞|u($\sqrt{t}$x,t) − (F(−x)u₀(− $\sqrt{t}$) + F(+ x)u₀(+ $\sqrt{t}$))| = 0

for the linear heat equation u_t = u_xx, where x $in$ R and F(z): = $\frac{1}{2\sqrt \pi}\int_{-\infty}^z e^{-\frac{y^2}{4}} dy$. Further, by combining with a theorem of Nara and Taniguchi, we have the same result for the curvature equation u_t = (1 + u_x²)⁻¹u_xx. In the case of lim_|x|→+0|xu′₀(x)| = 0 and in the case of sup_x$in$R|xu′₀(x)| < +∞, respectively, we also give a similar remark for the linear heat equation u_t = u_xx.

Real hypersurfaces M²ⁿ⁻¹ of type (B) in CHⁿ(c), n ≥ 2 are known as interesting examples of Hopf hypersurfaces with constant principal curvatures. They are homogeneous in this ambient space. Moreover, the numbers of distinct principal curvatures of all real hypersurfaces of type (B) with radius r ≠ (1/$\sqrt{|c|}$) log_e(2 + $\sqrt{3}$) are 3. When r = (1/$\sqrt{|c|}$) log_e(2 + $\sqrt{3}$), the real hypersurface of type (B) has two distinct principal curvatures. The purpose of this paper is to characterize this Hopf hypersurface having two distinct constant principal curvatures.

Let (X,L) be a polarized manifold with dim X = n ≥ 3 and Bs|L|= ∅. In this paper, we classify (X,L) with g(X,L)= q(X) + m and h⁰(L) = n + m − 1.

We construct a complete bounded immersed null holomorphic curve in C³, which is a recovery of the previous paper of the last three authors on this subject.

This paper considers some properties of meromorphic solutions of the nonlinear difference equation

(f(z + 1) + f(z))(f(z) + f(z − 1)) = $\frac{P(z,f(z))}{Q(z,f(z))}$,

where P(z, f(z)) and Q(z, f(z)) are polynomials in f having rational coefficients and no common roots.

Let Xⁿ(2) be a subset of C^{n + 1} $\backslash$ {0} any two elements of which are not propotional. We estimate the number of exceptional points in Xⁿ(2) for several holomorphic curves and we consider the defect relation for holomorphic curves. We shall give an example for which the defect relation is extremal and then give some holomorphic curves for which the defect relation is not extremal over Xⁿ(2). Another defect relation is also considered.

This paper is concerned with a class of Nicholson's blowflies model with a nonlinear density-dependent mortality term. Under appropriate conditions, we establish some criteria to ensure that the solutions of this model converge globally exponentially to a positive periodic solution. Moreover, we give an example and its numerical simulation to illustrate our main results.

For a surface in the Euclidean 4-space, we prove a reduction theorem for the codimension of a surface all whose points are inflection points.

In this paper, we prove that a complete manifold whose m-Bakry-Émery curvature satisfies

Ric_f,m(x) ≥ −(m − 1) $\frac{K_0}{(1+r(x))^2}$

for some constant K₀ < $-\frac{1}{4}$ should be compact. We also get an upper bound estimate for the diameter.

The starting point of Fontaine theory is the possibility of translating the study of a p-adic representation of the absolute Galois group of a finite extension K of Q_p into the investigation of a (φ, Γ)-module. This is done by decomposing the Galois group along a totally ramified extension of K, via the theory of the field of norms: the extension used is obtained by means of the cyclotomic tower which, in turn, is associated to the multiplicative Lubin-Tate group. It is known that one can insert different Lubin-Tate groups into the "Fontaine theory" machine to obtain equivalences with new categories of (φ, Γ)-modules (here φ may be iterated). This article uses only (φ, Γ)-theoretical terms to compare the different (φ, Γ) modules arising from various Lubin-Tate groups.

The aim of this paper is to establish the semilocal convergence of a parameter based continuation method combining the Chebyshev's and the Super-Halley's methods for solving nonlinear equations in Banach spaces. The parameter α $in$ [0,1] be such that for α = 0 it reduces to the Chebyshev's method and for α = 1 to the Super-Halley's method. This convergence is established using recurrence relations under the assumption that the second order Fréchet derivative satisfies the ω-continuity condition. This condition is milder than the Lipschitz and the Hölder continuity conditions used for this purpose. A numerical example is given to show that the second order Fréchet derivative satisfies the ω-continuity condition even when it fails to satisfy the Lipschitz and the Hölder continuity conditions. A number of recurrence relations are derived based on two parameters. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α $in$ [0, 1] for the solution x* is given. Two numerical examples are worked out to demonstrate the efficacy of the method. It is observed that our method gives better existence and uniqueness regions of the solution for both the examples when compared with the results obtained in [4] for both the Chebyshev's method (α = 0) and the Super-Halley's method (α = 1).

As is well-known, a homogeneous Reinhardt domain in C* coinsides with C*. In this paper, generalizing this fact, we show that a pseudoconvex homogeneous Reinhardt domain in (C*)ⁿ coinsides with (C*)ⁿ itself.