We present a type system for parametric object mutability and reference nullity in an imperative object oriented language. We present a simple but powerful system for generic nullity constraints, and build on previous work to provide safe initialization of objects which is not bound to constructors. The system is expressive enough to handle initialization of cyclic immutable data structures, an...

Given a finite and connected two-dimensional CW complex K with fundamental group $$\Pi $$ second integer cohomology $$H^2(K;\mathbb {Z})$$ of odd order, we prove that: (1) for each local coefficient system $$\alpha :\Pi \rightarrow \textrm{Aut}(\mathbb over K, the corresponding twisted $$H^2(K;_{\alpha }\!\mathbb is say order $$\mathfrak {c}^{*}(\alpha )$$ , there exists natural function—which ...

We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on $T$ count for fault-tolerant computation. First, we stabilizer nullity multi-qubit unitary, which is based subgroup quotient Pauli group associated with unitary. This unitary extends state-stabilizer by Beverland et al. a dynamic version. In particular, show this measure has d...

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...

A banded invertible matrix has a remarkable inverse. All “upper” and “lower” submatrices of have low rank (depending on the bandwidth in ). The exact rank condition is known, and it allows fast multiplication by full matrices that arise in the boundary element method. We look for the “right” proof of this property of . Ultimately it reduces to a fact that deserves to be better known: Complement...

A graph G is singular of nullity (> 0), if its adjacency matrix A is singular, with the eigenvalue zero of multiplicity . A singular graph having a 0-eigenvector, x, with no zero entries, is called a core graph.We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the en...

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is not isomorphic to nK1. In this paper, we characterize the extremal graphs attaining the upper bound n− 2 and the second upper bound n− 3. The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is al...