Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ differential operators $L_1$ $L_{2}$, that has implications for spectral properties of $K$. This work complements our explicit characterization commuting pairs $KL=LK$ provides an exhaustive list kernels admitting or sesquicommuting operators.

The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a super linear upper bound $\widetilde{O}(n\sqrt{m})$ are known, which prompts the question of relaxing the problem: either by asking for $1 \pm \varepsilon$ appro...

The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is t...

In this paper, we study three types of rotational surfaces in Galilean 3-spaces. We classify satisfying $$L_1G=F(G+C)$$ for some constant vector $C\in \mathbb{G}^3$ and smooth function $F$, where $L_1$ denotes the Cheng-Yau operator.

Local and global well-posedness of the coagulation-fragmentation equation with size diffusion are investigated. Owing to semilinear structure equation, a semigroup approach is used, building upon generation results previously derived for linear fragmentation-diffusion operator in suitable weighted $L_1$-spaces.

‎In the present paper we investigate the $L_1$-weak ergodicity of‎ ‎nonhomogeneous continuous-time Markov processes with general state‎ ‎spaces‎. ‎We provide a necessary and sufficient condition for such‎ ‎processes to satisfy the $L_1$-weak ergodicity‎. ‎Moreover‎, ‎we apply‎ ‎the obtained results to establish $L_1$-weak ergodicity of quadratic‎ ‎stochastic processes‎.

Techniques based on minimal graph cuts have become a standard tool for solving combinatorial optimization problems arising in image processing and computer vision applications. These techniques can be used to minimize objective functions written as the sum of a set of unary and pairwise terms, provided that the objective function is submodular. This can be interpreted as minimizing the $l_1$-no...