Complexity of Linear Operators

\ua9 2025 Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, and Vladimir V. Podolskii.Let A be an n-by-n 0/1-matrix with z zeroes and u ones and let x be an n-dimensional vector of formal variables over a semigroup (S, ◦). How many semigroup operations are required to compute the linear operator Ax? It is easy to compute Ax using O(u) semigroup operations. The main question studied in this paper is: can Ax be computed using O(z) semigroup operations? For the case when the semigroup is commutative, we give a constructive proof of an O(z) upper bound. This implies that in the commutative settings, the complements of sparse matrices can be processed as efficiently as sparse matrices, though the corresponding algorithms are more involved. This covers the cases of Boolean and tropical semirings that have numerous applications, e. g., in graph theory. On the other hand, we prove that in general this is not possible: for faithful non-commutative semigroups there exists an n-by-n 0/1-matrix with exactly two zeroes in every row (hence z = 2n) whose complexity is Θ(nα(n)) where α(n) is the inverse Ackermann function. As a simple application of the linear-size construction presented, we show how to multiply two n 7 n matrices over an arbitrary semiring in O(n2) time if one of these matrices is a 0/1-matrix with O(n) zeroes (i. e., the complement of a sparse matrix)

Use of the big liquid argon spectrometer BARS for neutrino and cosmic ray studies

The design of the fine grained 300 t liquid argon calorimeter BARS is described. The BARS electronics include about 30 K channels of low noise amplifiers and ADCs. The DAQ system makes it possible to select channels with signals above the chosen threshold. 48 scintillation hodoscopes placed inside the liquid argon are used to form the first level trigger. The total number of scintillation counters in liquid argon is 384. Sums of ionization signals are used to produce the second level trigger. Results of the first use of liquid argon calorimetry for the measurements of tagged neutrino interactions, cosmic-ray muon spectra and composition of extensive atmospheric showers are discussed. (C) 1998 Elsevier Science B.V. All rights reserved