In this paper, we establish new sufficient conditions for the polynomial f to be SOS in terms of the Newton polyhedron of f (Theorems 2.6 and 2.12). These new sufficient conditions include results which were proved earlier by Lasserre [13, Theorem 3], Fidalgo and Kovacec [6, Theorem 4.3], Ghasemi and Marshall [7, Theorems 2.1 and 2.3], and Ghasemi and Marshall [8, Theorem 2.3].

This paper deals with the existence and uniqueness of time periodic solutions for the general periodic parabolic equation boundary problem with nonlocal delay. We apply operator semigroup theory and monotone iterative technique of lower and upper solutions to obtain the existence and uniqueness of ω-periodic mild solutions of some abstract evolution equation under some quasimonotone conditions. In the end, applying our abstract results to parabolic equation with nonlocal delay, we get the existence and uniqueness of ω-periodic solution, which generalize the recent conclusions on this issue.

Let M be a 3-manifold obtained by a Dehn-surgery along the figure-eight knot. We give a formula of the Reidemeisiter torsion of M for any SL(2; C)-irreducible representation. It has a rational expression of the trace of the image of the meridian.

In this paper, we focus on relations between holomorphic 1-forms and holomorphic 1-cochains on a closed Riemann surface. Holomorphic 1-cochains are defined by Wilson in 2008 using a combinatorial Hodge theory. A holomorphic 1-form is characterized by its periods. So is a holomorphic 1-cochain. We consider relations between a holomorphic 1-form and a holomorphic 1-cochain which have the same periods and show that holomorphic 1-cochains provide an approximation of holomorphic 1-forms.

We establish some unified lower bounds for the first Neumann and closed eigenvalues of the Finsler-Laplacian on compact Finsler manifolds under the weighted Ricci curvature conditions, which extend some recent theorems on the first eigenvalue of the Riemannian-Laplacian. Moreover, the Lichnerowicz type lower bound for the first Dirichlet eigenvalue of the Finsler-Laplacian is also obtained.

Odaka [16] and Wang [19] proved the intersection formula for the Donaldson-Futaki invariant. In this paper, we generalize this result for the higher Futaki invariants, which are obstructions to asymptotic Chow semistability.

Let M be a weighted Riemannian manifold with non-negative Bakry-Émery-Ricci curvature and N be a complete Riemannian manifold of non-positive sectional curvature. In this paper, the p-harmonic map u: M → N is studied, and a theorem of Liouville type is obtained.

In this paper, we introduce a new class $\mathscr{S}_{H} (k, γ; \phi)$ of harmonic quasiconformal mappings, where $k \in [0,1), γ \in [0,π)$ and $\phi$ is an analytic function. Sufficient conditions for the linear combinations of mappings in such classes to be in a similar class, and convex in a given direction, are established. In particular, we prove that the images of linear combinations in this class, for special choices of $γ$ and $\phi$, are convex.

In this paper, we consider a unified constructions of lattices in splittable solvable Lie groups.

We study convexity of simple closed frontals of Legendre curves in the Euclidean plane by using the curvature of Legendre curves. We show that for a Legendre curve, the simple closed frontal under conditions is convex if and only if the sign of both functions of the curvature of the Legendre curve does not change. We also give some examples of convex simple closed frontals.

We study some class of Dunkl multiplier operators T_k,m; and we give for them an application of the theory of reproducing kernels to the Tikhonov regularization, which gives the best approximation of the operators T_k,m on the Dunkl-type Paley-Wiener spaces H_h.

It is shown that the complex dilatation of the Douady-Earle extension of a strongly symmetric homeomorphism induces a vanishing Carleson measure on the unit disk D. As application, it is proved that the VMO-Teichmüller space is a subgroup of the universal Teichmüller space.

Let f be a class P-homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P-homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never a class P and even more it is not absolutely continuous function when the total product of f-jumps in all the break points is non-trivial.

By use of the Zassenhaus neighborhood of Sp(n,1), we obtain an explicit lower bound for the radius of the largest inscribed ball in quaternionic hyperbolic n-manifold $\mathscr{M} = \mathbf H_\mathbf H ^n/Γ$. As an application, we obtain a lower bound for the volumes of quaternionic hyperbolic n-manifolds.

In this article we give a sufficient and necessary condition for a Kouchnirenko non-degenerate holomorphic function to have an isolated singularity at 0 in terms of its support. As a corollary we give some useful sufficient conditions for singularity to be isolated.