scholarly journals Offdiagonal complexity: A computationally quick complexity measure for graphs and networks

2007 ◽  
Vol 375 (1) ◽  
pp. 365-373 ◽  
Author(s):  
Jens Christian Claussen
Keyword(s):  
Complexity Measure ◽  
Graphs And Networks

Word Complexity Measure

2010 ◽  
Author(s):  
Carol Stoel-Gammon
Keyword(s):  
Complexity Measure

scholarly journals On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

2021 ◽  
Vol 13 (3) ◽  
pp. 1-21
Author(s):  
Suryajith Chillara
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complexity Measure ◽  
Natural Question ◽  
Multilinear Polynomial ◽  
Arithmetic Circuit ◽  
Theory Of Computing ◽  
Small Constant ◽  
Multilinear Polynomials ◽  
The Individual

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

Deciding Clause Classes by Semantic Clash Resolution

1993 ◽  
Vol 18 (2-4) ◽  
pp. 163-182
Author(s):  
Alexander Leitsch
Keyword(s):  
Decision Procedure ◽  
Decision Procedures ◽  
General Concept ◽  
Complexity Measure ◽  
Syntactical Structure

It is investigated, how semantic clash resolution can be used to decide some classes of clause sets. Because semantic clash resolution is complete, the termination of the resolution procedure on a class Γ gives a decision procedure for Γ. Besides generalizing earlier results we investigate the relation between termination and clause complexity. For this purpose we define the general concept of atom complexity measure and show some general results about termination in terms of such measures. Moreover, rather than using fixed resolution refinements we define an algorithmic generator for decision procedures, which constructs appropriate semantic refinements out of the syntactical structure of the clause sets. This method is applied to the Bernays – Schönfinkel class, where it gives an efficient (resolution) decision procedure.

scholarly journals Isoscattering strings of concatenating graphs and networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michał Ławniczak ◽  
Adam Sawicki ◽  
Małgorzata Białous ◽  
Leszek Sirko
Keyword(s):  
Trace Function ◽  
Quantum Graphs ◽  
Mathematical Approach ◽  
Large Graphs ◽  
Graphs And Networks ◽  
Scattering Matrices ◽  
Theoretical Predictions ◽  
Infinite Strings ◽  
Microwave Networks ◽  
Insight Into

AbstractWe identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for $$n \rightarrow \infty $$ n → ∞ . The theoretical predictions are confirmed experimentally using $$n=2$$ n = 2 units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the $$2n \times 2n $$ 2 n × 2 n scattering matrices $${\hat{S}}$$ S ^ of the systems to 2n diagonal elements, while the old measures of isoscattering require all $$(2n)^2$$ ( 2 n ) 2 entries. The studied problem generalizes a famous question of Mark Kac “Can one hear the shape of a drum?”, originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.

Information and complexity measures in molecular reactivity studies

2014 ◽  
Vol 16 (28) ◽  
pp. 14928-14946 ◽  
Author(s):  
Meressa A. Welearegay ◽  
Robert Balawender ◽  
Andrzej Holas
Keyword(s):  
Complexity Measure ◽  
Complexity Measures ◽  
Molecular Reactivity

The usefulness of the information and complexity measure in molecular reactivity studies.

Graphs and networks

1982 ◽  
Vol 10 (1) ◽  
pp. 117-118
Author(s):  
C.J. Pursglove
Keyword(s):  
Graphs And Networks

Using complexity measure factor to predict protein subcellular location

Amino Acids ◽  
2004 ◽  
Vol 28 (1) ◽  
pp. 57-61 ◽  
Author(s):  
X. Xiao ◽  
S. Shao ◽  
Y. Ding ◽  
Z. Huang ◽  
Y. Huang ◽  
...  
Keyword(s):  
Subcellular Location ◽  
Complexity Measure ◽  
Protein Subcellular Location
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