scholarly journals On a hierarchy of Boolean functions hard to compute in constant depth

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Anna Bernasconi
Keyword(s):  
Harmonic Analysis ◽  
Complexity Theory ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Combinatorial Property ◽  
Constant Depth ◽  
Models Of Computation ◽  
Mathematical Properties ◽  
International Audience ◽  
Mathematical Techniques

International audience Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals The Epistemological Implications of Critical Complexity Thinking for Operational Research

Systems ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 5 ◽  
Author(s):  
Hennie Kruger ◽  
Anné Verhoef ◽  
Rika Preiser
Keyword(s):  
Complexity Theory ◽  
Real World ◽  
Operational Research ◽  
Initial Conditions ◽  
Complex Nature ◽  
The Real ◽  
Complexity Thinking ◽  
The World ◽  
Mathematical Techniques ◽  
Complex Phenomena

Classical operational research (OR) is mainly concerned with the use of mathematical techniques and models used in decision-making situations. The basic assumptions of OR presuppose that the structure of the world is one of order and predictability. Although this positivistic approach produces significant results when levels of certainty and initial conditions are stable, it is limited when faced with an acknowledgement of the complex nature of the real world. This paper aims to highlight that by drawing on a general understanding of complexity theory, classical OR approaches can be enriched and broadened by adopting an epistemology based on the assumption that the underlying mechanisms governing the world are complex. It is argued that complexity theory (as interpreted by the philosopher Paul Cilliers) acknowledges the complex nature of the real world and helps to identify the characteristics of complex phenomena. By aligning OR epistemologies with the acknowledgment of complexity, new modelling methods could be developed. In addition, the implications for knowledge generating processes through boundary setting, as well as the provisional nature of such knowledge and what (ethical) responsibilities accompany the study of complex phenomena, will be discussed. Examples are presented to highlight the epistemological implications of complexity thinking for OR.

scholarly journals Lower Bounds and Separations for Constant Depth Multilinear Circuits

2009 ◽  
Vol 18 (2) ◽  
pp. 171-207 ◽  
Author(s):  
Ran Raz ◽  
Amir Yehudayoff
Keyword(s):  
Lower Bounds ◽  
Constant Depth

scholarly journals Lower bounds to the complexity of symmetric Boolean functions

1990 ◽  
Vol 74 (3) ◽  
pp. 313-323 ◽  
Author(s):  
L. Babai ◽  
P. Pudlák ◽  
V. Rödl ◽  
E. Szemeredi
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Symmetric Boolean Functions

On the lower bounds of the second order nonlinearities of some Boolean functions

2010 ◽  
Vol 180 (2) ◽  
pp. 266-273 ◽  
Author(s):  
Sugata Gangopadhyay ◽  
Sumanta Sarkar ◽  
Ruchi Telang
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order

scholarly journals Three-Dimensional Texture Analysis After Bunge and Roe: Correspondence Between the Respective Mathematical Techniques

1982 ◽  
Vol 5 (2) ◽  
pp. 95-125 ◽  
Author(s):  
C. Esling ◽  
E. Bechler-Ferry ◽  
H. J. Bunge
Keyword(s):  
Harmonic Analysis ◽  
Texture Analysis ◽  
Series Expansion ◽  
Three Dimensional ◽  
Analysis Methods ◽  
Mathematical Techniques

Bunge's and Roe's three-dimensional texture analysis methods, although both founded on harmonic analysis, show some differences between the various mathematical techniques used.This paper establishes the correspondence relation between the respective mathematical techniques allowing one to compare works done in either variant. Taking the latest developments in three dimensional texture analysis into account, the correspondence relations hold for the odd degrees l as well as for the even ones.Finally numerical tables give the extension of the symmetry coefficients B:l4mμ (after Bunge) and R4nμl (after Roe) to all the degrees l of the series expansion, even and odd, including l = 34.

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