Given a complex idempotent matrix A, we derive simple, sufficient and necessary conditions for X being nontrivial solution of the Yang-Baxter-like equation AXA = XAX, discriminating commuting solutions from non-commuting ones. On this basis, construct all nonlinear equation.

Let e,8 be *-automorphlsms of avon Neumann algebra M satisfying the operator equation -I -I + =+8 In this paper we prove a general decomposition theorem in the non-commuting situation as compared to the usual commuting case (see references) and prove that there exists a central projection p in M such that 2 82 on Mp and 2 8-2 on M(l-p). KEYS WORDS AND PHRASES. Automorphlsms, central projections...

We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we intr...

Active commuting provides routine-based regular physical activity which can reduce the risk of chronic diseases. Using public transportation involves some walking or cycling to a transit stop, transfers and a walk to the end location and users of public transportation have been found to accumulate more moderate physical activity than non-users. Understanding how public transportation characteri...

The conformal non-compact SL(2, R)/U(1) coset model in two dimensions has been recently shown to embody a nonlinear Ŵ∞ current algebra, consisting of currents of spin ≥ 2 including the energy-momentum tensor. In this letter we explicitly construct an infinite set of commuting quantum Ŵ∞ charges in the model with k = 1. These commuting quantum charges generate a set of infinitely many compatible...

The light-cone approach is reviewed. This method allows to find the underlying quantum field theory for any integrable lattice model in its gapless regime. The relativistic spectrum and S-matrix follows straightforwardly in this way through the Bethe Ansatz. We show here how to derive the infinite number of local commuting and non-local and non-commuting conserved charges in integrable QFT, tak...

We study the variant of the k-local hamiltonian problem which is a natural generalization of k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamiltonian H = ∑i Hi over n qubits, where each Hi acts non-trivially on O(log n) qubits and all the terms Hi commute, and show the following 1. We show that a specific case of O(log n) local commuting hamiltonians over...

We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefini...

Let G be a non-abelian group. The non-commuting graph AG of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with ω(AG) ≤ 57...