A sharp square function estimate for the cone in
                    ℝ<sup>3</sup>

Let X be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of X-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents p ≥ 2 if and only if the space X is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

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Weighted and vector-valued variational estimates for ergodic averages

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.27 ◽

2016 ◽

Vol 38 (1) ◽

pp. 244-256 ◽

Cited By ~ 2

Author(s):

BEN KRAUSE ◽

PAVEL ZORIN-KRANICH

Keyword(s):

Hölder Inequality ◽

Square Function ◽

Reverse Hölder Inequality ◽

Ergodic Averages ◽

Function Estimate ◽

Reverse Hölder ◽

Vector Valued

We prove weighted and vector-valued variational estimates for ergodic averages on$\mathbb{R}^{d}$. The weighted square function estimate relating ergodic averages to the dyadic martingale is obtained using an$\ell ^{r}$version of a reverse Hölder inequality for variation seminorms.

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

Nanoscience &Nanotechnology-Asia ◽

10.2174/2210681209666190807143557 ◽

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.

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VAT Efficiency—A Discussion on the VAT System in the European Union

Sustainability ◽

10.3390/su13094768 ◽

2021 ◽

Vol 13 (9) ◽

pp. 4768

Author(s):

Anna Kowal ◽

Grzegorz Przekota

Keyword(s):

European Union ◽

Tax Evasion ◽

Value Added ◽

Square Function ◽

The European Union ◽

Tax System ◽

Aggregated Data ◽

Tax Gap ◽

Standard Rate ◽

Tax Fraud

The effectiveness of the tax system can be analysed in various ways. According to the authors one of manifestations of such effectiveness is resistance to tax evasion. This phenomenon is influenced by multiple factors, with few being the level of VAT rates and the number of rates in force in the country concerned. The aim of the considerations is therefore to analyse how the standard VAT rate as well as the number of rates affect the effectiveness of this tax. The research was based on a literature query in the field of value added tax in the European Union. In addition, the problem of tax evasion was indicated and the aggregated data on the size of the tax gap in the Member States were presented. Then, there are the results of the research for 27 European Union countries for 2011–2019. The efficiency of VAT collection was modelled using square function, determining the significance of the parameters of this function, as well as the value of abscissa, which made it possible to group the countries based on how they maintained the efficiency of VAT collection over the analysed period of time. The final part of the study concentrates on the relationship between the efficiency of tax collection and the amount of both the basic rate and the number of rates. The conclusions of the research are as follows: a tax system with a small number of reduced rates, and preferably with one relatively low standard rate, is the system least susceptible to tax fraud. The research also shows a positive correlation between the value of the basic VAT rate along with the number of preferential rates and the scale of the tax gap, i.e., in countries with a higher standard VAT rate and a greater number of preferential rates, the tax gap is greater. The study will enable further investigation into the strategy of determining the optimal VAT rate and the process of its unification. Proposed changes may contribute to increasing the efficiency of VAT administration in EU countries, reducing the shadow economy, tax fraud and positively influencing economic growth.

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