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Qi Ding

Publications and source records attributed to Qi Ding.

50 records · Page 3Linked to original sources
arXiv2017

On distance matrices of graphs

Distance well-defined graphs consist of connected undirected graphs, strongly connected directed graphs and strongly connected mixed graphs. Let $G$ be a distance well-defined graph, and let ${\sf D}(G)$ be the distance matrix of $G$. Graham, Hoffman and Hosoya [3] showed a very attractive theorem, expressing the determinant of ${\sf D}(G)$ explicitly as a function of blocks of $G$. In this paper, we study the inverse of ${\sf D}(G)$ and get an analogous theory, expressing the inverse of ${\sf D}(G)$ through the inverses of distance matrices of blocks of $G$ (see Theorem 3.3) by the theory of Laplacian expressible matrices which was first defined by the first author [9]. A weighted cactoid digraph is a strongly connected directed graph whose blocks are weighted directed cycles. As an application of above theory, we give the determinant and the inverse of the distance matrix of a weighted cactoid digraph, which imply Graham and Pollak's formula and the inverse of the distance matrix of a tree.

math.CO↗
arXiv2016

Matching preclusion for $n$-grid graphs

A matching preclusion set of a graph is an edge set whose deletion results in a graph without perfect matching or almost perfect matching. The Cartesian product of $n$ paths is called an $n$-grid graph. In this paper, we study the matching preclusion problems for $n$-grid graphs and obtain the following results. If an $n$-grid graph has an even order, then it has the matching preclusion number $n$, and every optimal matching preclusion set is trivial. If the $n$-grid graph has an odd order, then it has the matching preclusion number $n+1$, and all the optimal matching preclusion sets are characterized.

math.CO↗
arXiv2016

Subsequent singularities of mean convex mean curvature flows in smooth manifolds

For any $n$-dimensional smooth manifold $Σ$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $Σ$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_{\infty}$ (maybe empty) at infinity. Previously this was shown (i) for $n\leq 7$, and (ii) for arbitrary $n$ up to the first singular time without the smooth condition for $M_{\infty}$.

math.DG↗
arXiv2013

Minimal graphic functions on manifolds of non-negative Ricci curvature

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Si$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Si$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

math.DG↗
arXiv2013

Volume growth, eigenvalue and compactness for self-shrinkers

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\ir{3}$ obtained by Colding-Minicozzi under weaker conditions.

math.DG↗
arXiv2012

A Compressed PCA Subspace Method for Anomaly Detection in High-Dimensional Data

Random projection is widely used as a method of dimension reduction. In recent years, its combination with standard techniques of regression and classification has been explored. Here we examine its use with principal component analysis (PCA) and subspace detection methods. Specifically, we show that, under appropriate conditions, with high probability the magnitude of the residuals of a PCA analysis of randomly projected data behaves comparably to that of the residuals of a similar PCA analysis of the original data. Our results indicate the feasibility of applying subspace-based anomaly detection algorithms to randomly projected data, when the data are high-dimensional but have a covariance of an appropriately compressed nature. We illustrate in the context of computer network traffic anomaly detection.

stat.ME↗
arXiv2012

Entire spacelike translating solitons in Minkowski space

In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.

math.DG↗
arXiv2012

The Rigidity Theorems of Self Shrinkers via Gauss maps

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1 and Theorem 3.2 not only improve the previous results, but also are optimal. In higher codimensional case, using geometric properties of the Grassmanian manifolds (the target manifolds of the Gauss map) we give a rigidity theorem for self-shrinking graphs.

math.DG↗
arXiv2012

Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrödinger equation with time-dependent nonlinearity

In this paper, we consider a general form of nonlinear Schrödinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schrödinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

nlin.SI↗
arXiv2011

The rigidity theorems for Lagrangian self shrinkers

By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in \cite{HW} and \cite{CCY} by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in \cite{CCY}.

math.DG↗
arXiv2011

The rigidity theorems of self shrinkers

By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.

math.DG↗
arXiv2010

Some results on Chern's problem

For a compact minimal hypersurface $M$ in $S^{n+1}$ with the squared length of the second fundamental form $S$ we confirm that there exists a positive constant $\de(n)$ depending only on $n,$ such that if $n\leq S\leq n +δ(n)$, then $S\equiv n$, i.e., $M$ is a Clifford minimal hypersurface, in particular, when $n\ge 6,$ the pinching constant $\de(n)=\f{n}{23}.$

math.DG↗