scholarly journals The maximal function and conditional square function control the variation: An elementary proof

2016 ◽  
Vol 144 (8) ◽  
pp. 3583-3588 ◽  
Author(s):  
Kevin Hughes ◽  
Ben Krause ◽  
Bartosz Trojan
Keyword(s):  
Maximal Function ◽  
Elementary Proof ◽  
Square Function

Multi-parameter Carnot-Carath´eodory Geometry

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Singular Integral ◽  
Maximal Function ◽  
Integral Operators ◽  
Product Type ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Square Function ◽  
Frobenius Theorem ◽  
Parameter Theory

This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the results are simple variants of results in the single-parameter theory. When the balls are not of product type, these ideas become more difficult. What saves the day is the quantitative Frobenius theorem given in Chapter 2. This can be used to estimate certain integrals, as well as develop an appropriate maximal function and an appropriate Littlewood–Paley square function, all of which are essential to our study of singular integral operators.

scholarly journals Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-13
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Elliptic Equation ◽  
Harmonic Function ◽  
Maximal Function ◽  
Lipschitz Domain ◽  
Elliptic Equations ◽  
Borel Measure ◽  
Square Function ◽  
Square Functions ◽  
Nondivergence Elliptic Equation ◽  
Nontangential Maximal Function

We prove distributional inequalities that imply the comparability of theLpnorms of the multiplicative square function ofuand the nontangential maximal function oflogu, whereuis a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case whenuis a harmonic function is considered.

AN ELEMENTARY PROOF OF A THEOREM OF LEE

1991 ◽  
Vol 11 (3) ◽  
pp. 356-360 ◽  
Author(s):  
Jia'an Yan
Keyword(s):  
Elementary Proof

Error-aware Design Procedure to Implement Energy-efficient Approximate Squaring Hardware

2020 ◽  
Vol 10 (4) ◽  
pp. 471-477
Author(s):  
Merin Loukrakpam ◽  
Ch. Lison Singh ◽  
Madhuchhanda Choudhury
Keyword(s):  
Energy Saving ◽  
Relative Error ◽  
Energy Efficient ◽  
Piecewise Linear ◽  
Arithmetic Operation ◽  
Design Procedure ◽  
Digital Signal ◽  
Maximum Relative Error ◽  
Square Function ◽  
Efficient Design

Background:: In recent years, there has been a high demand for executing digital signal processing and machine learning applications on energy-constrained devices. Squaring is a vital arithmetic operation used in such applications. Hence, improving the energy efficiency of squaring is crucial. Objective:: In this paper, a novel approximation method based on piecewise linear segmentation of the square function is proposed. Methods: Two-segment, four-segment and eight-segment accurate and energy-efficient 32-bit approximate designs for squaring were implemented using this method. The proposed 2-segment approximate squaring hardware showed 12.5% maximum relative error and delivered up to 55.6% energy saving when compared with state-of-the-art approximate multipliers used for squaring. Results: The proposed 4-segment hardware achieved a maximum relative error of 3.13% with up to 46.5% energy saving. Conclusion:: The proposed 8-segment design emerged as the most accurate squaring hardware with a maximum relative error of 0.78%. The comparison also revealed that the 8-segment design is the most efficient design in terms of error-area-delay-power product.

scholarly journals A new elementary proof of a theorem of Minkowski

1926 ◽  
Vol 2 (3) ◽  
pp. 97-99
Author(s):  
Matsusaburô Fujiwara
Keyword(s):  
Elementary Proof
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