Given a simple graph $G$ with $m$ edges, we are looking for a bijection $f$ from $E(G)$ to the integer set $\{ k+1,k+2,\ldots,k+m \}$ such that the vertex sum of each vertex $v$, $\phi(v)$, defined as the sum of $f(e)$ over all edges $e$ incident to $v$ is unique. If such a bijection $f$ exists, we say $G$ is $k$-shifted antimagic. This is a generalization of the antimagic graphs proposed by Hartsfield and Ringel [7]. In this paper, we proved that every tree of diameter four or five, except for two previous known examples, is $k$-shifted antimagic for every integer $k$.

Let $R$ be the Ehrhart ring of the stable set polytope of a cycle graph which is not Gorenstein. We describe the non-Gorenstein locus of $\operatorname{Spec}R$. Further, we show that $R$ is almost Gorenstein. Moreover, we show that the conjecture of Hibi and Tsuchiya is true.

We present an iterative algorithm based on the concept of the forward and backward observers to reconstruct the initial state of the linear system with polynomial coefficients. Its discussions of the convergence in the continuous space, semi-discrete space and full-discrete space are considered. Finally numerical examples are given to illustrate the effectiveness of our algorithm.

This paper deals with the well-posedness and asymptotic behavior for a pseudo-parabolic equation involving $p$-biharmonic operator and logarithmic nonlinearity under Navior boundary condition. By combining Galerkin approximation, the method of potential well, the technique of differential inequality and improved logarithmic Sobolev inequality, we establish the local and global solvability, infinite and finite time blow-up phenomena of weak solutions in different energy levels. Moreover, we obtain the growth rate of weak solutions, life span in different energy cases and also give a result of extinction phenomenon.

In this work, we study the weighted Kirchhoff problem \[ \begin{cases} g\big( \int_{B} (w(x) |\Delta u|^{2}) \, dx \big) [\Delta (w(x) \Delta u)] = f(x,u) &\textrm{in $B$}, \\ u > 0 &\textrm{in $B$}, \\ u = \frac{\partial u}{\partial n} = 0 &\textrm{on $\partial B$}, \end{cases} \] where $B$ is the unit ball of $\mathbb{R}^{4}$, $w(x) = \big( \log \frac{e}{|x|} \big)^{\beta}$, the singular logarithm weight in Adam's embedding, $g$ is a continuous positive function on $\mathbb{R}^{+}$. The nonlinearities are critical growth in view of Adam's inequalities. We prove the existence of a positive ground state solution using mountain pass method combined with a concentration compactness result. The associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.

In this paper, we investigate several two-parameter weak Orlicz–Hardy martingale spaces generated by the $p$-convex and $q$-concave functions, and establish their atomic decomposition theorems. Using the atomic decomposition, we obtain a sufficient condition for the boundedness of a sublinear operator defined on the two-parameter weak Orlicz–Hardy martingale spaces. Furthermore, the dual spaces of the two-parameter weak Orlicz–Hardy martingale spaces are considered.

In this paper, we consider the Cauchy problem for the Laplace operator. We construct approximate solutions by using the iterative method proposed by Bastay, Kozlov and Turesson. In the iterative method, we solve the corresponding boundary value problems repeatedly. Then, we show that we construct them more stably when we choose the smaller domain where we consider the boundary value problems. Furthermore, since the iterative method is applicable to the case where we know only approximations to the exact data with error, we also deal with this case.

Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.