Lower bounds for tau coefficients and operator norms using composite matrix norms

Lower bounds for the tau coefficients and operator norms are derived by using composite matrix norms. For a special class of matrices B, our bounds on ‖B‖p (the operator norm of B induced by the ℓp norm) improve upon a general class of Maitre (1967) bounds for p ≥ 2.

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Orthogonal Matrices with Zero Diagonal. II

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-091-x ◽

1971 ◽

Vol 23 (5) ◽

pp. 816-832 ◽

Cited By ~ 53

Author(s):

P. Delsarte ◽

J. M. Goethals ◽

J. J. Seidel

Keyword(s):

General Class ◽

Special Class ◽

Hadamard Matrices ◽

Orthogonal Matrices ◽

Adjacency Matrices ◽

Class Of Matrices

C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v ≡ 2 (mod 4), and the skew C-matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [24] and by Reid and Brown [21].

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Finding Cut-Offs in Leaderless Rendez-Vous Protocols is Easy

Lecture Notes in Computer Science - Foundations of Software Science and Computation Structures ◽

10.1007/978-3-030-71995-1_3 ◽

2021 ◽

pp. 42-61

Author(s):

A. R. Balasubramanian ◽

Javier Esparza ◽

Mikhail Raskin

Keyword(s):

Lower Bounds ◽

Petri Net ◽

Special Class ◽

Upper Bounds ◽

Initial State ◽

Reachability Problem ◽

Final State ◽

Final Configuration ◽

Finite State ◽

State Agents

AbstractIn rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number B such that all initial configurations of the protocol with at least B agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper [17], Horn and Sangnier prove that the cut-off problem is equivalent to the Petri net reachability problem for protocols with a leader, and in "Image missing" for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to "Image missing" and "Image missing" , respectively. The problem of lowering these upper bounds or finding matching lower bounds is left open. We show that the cut-off problem is "Image missing" -complete for leaderless protocols, "Image missing" -complete for symmetric protocols with a leader, and in "Image missing" for leaderless symmetric protocols, thereby solving all the problems left open in [17].

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