A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simpl...

The idea for nding a basis relates to the proof of why a Jordan canonical form exists. What we seek to do is nd a largest possible set of chains (or cycles) of the form {x, (T −λkI)(x), . . . , (T −λkI)(x)} which are linearly independent. By the proof of Jordan canonical form, the number and lengths of these chains can be found from the numbers d0, . . . , d`k . Indeed, let λ = λk be a xed eige...

We establish the bounds 43 6 bν 6 bξ 6 √ 2, where bν and bξ are the NordhausGaddum sum upper bound multipliers, i.e., ν(G)+ν(G) 6 bν |G| and ξ(G)+ξ(G) 6 bξ|G| for all graphs G, and ν and ξ are Colin de Verdière type graph parameters. The Nordhaus-Gaddum sum lower bound for ν and ξ is conjectured to be |G| − 2, and if these parameters are replaced by the maximum nullity M(G), this bound is calle...

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

This is a brief summary of the recent works joint with Dr. Sang Youl Lee, on the Seifert matrices and the (modi ed) Goeritz matrices of knots and links and their invariants: the Alexander polynomial, the Minkowski unit, the signature, the nullity, and the determinant of a knot and a link. We introduce the relationship between the modi ed Goeritz matrices of 2-peroidc links and those of their fa...

a) What are the possible values for the rank of A? Why? Solution By the Rank-Nullity Theorem 0 ≤ rank (A) ≤ min{6, 4} = 4. b) What are the possible values for the dimension of the kernel of A? Why? Solution Since dim(image(A)) ≤ 4, by the Rank-Nullity Theorem 2 ≤ dim ker(A) ≤ 6. c) Suppose the rank of A is as large as possible. What is the dimension of ker(A)? Explain. Solution Since then dim(i...

Despite being a classical topic in robotics, the research on dexterous robotic hands still stirs a lively research activity. The current interest is especially attracted by underactuated robotic hands where a high number of degrees of freedom (DoFs), and a relatively low number of degrees of actuation (DoAs) coexist. The correlation between the DoFs obtained through a wise distribution of actua...

We present a thorough study of the theoretical properties and devise efficient algorithms for persistent Laplacian, an extension standard combinatorial Laplacian to setting pairs (or, in more generality, sequences) simplicial complexes $K \hookrightarrow L$, which was recently introduced by Wang, Nguyen, Wei. In particular, analogy with nonpersistent case, we first prove that nullity $q$th $\De...