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...

Bounds for the ranks of upper-right submatrices of a generalized inverse of a strictly lower k-banded matrix are obtained. It is shown that such ranks can be exactly predicted under some conditions. The proof uses the Nullity Theorem and bordering technique for generalized inverse.

unenforceability (inopposabilité) is a double sanction. because this sanction combines in the same time “validity” and “nullity”, validity between the parties and nullity to third parties. our law obtains this sanction from french law. thus for an exact study, referring to original law is inevitable. in iranian law there are sporadic instances of unenforceability. but sometimes unenforceability...

Let G = (V, E) be a graph with V = {1, 2, . . . , n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A = [ai,j ] with ai,j = 0 if i = j and i and j are non-adjacent, ai,j = 0 if i = j and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-defi...

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero o...

We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants. 2010 Mathematics Subject Classification: primary: 57M25, secondary: 57M27, 57N70