Mean value bounded variation concept in real sense: an application with new techniques to weighted integrability

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sumfor some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.

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A sharp trace inequality for functions of bounded variation in the ball

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511000758 ◽

2012 ◽

Vol 142 (6) ◽

pp. 1179-1191 ◽

Cited By ~ 8

Author(s):

Andrea Cianchi

Keyword(s):

Bounded Variation ◽

Mean Value ◽

Functions Of Bounded Variation ◽

Trace Inequality ◽

Best Constant ◽

Isoperimetric Constant ◽

Special Case

The best constant in a mean-value trace inequality for functions of bounded variation on admissible domains Ω ⊂ ℝn is shown to agree with an isoperimetric constant associated with Ω. The existence and form of extremals is also discussed. This result is exploited to compute the best constant in the relevant trace inequality when Ω is a ball. The existence and the form of extremals in this special case turn out to depend on the dimension n. In particular, the best constant is not achieved when Ω is a disc in ℝ2.

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On weighted integrability of functions defined by trigonometric series with p-bounded variation coefficients

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2225-1 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Bogdan Szal ◽

Maciej Kubiak

Keyword(s):

Trigonometric Series ◽

Bounded Variation ◽

Necessary Condition ◽

Integrability Of Functions ◽

Sufficient And Necessary Condition ◽

Weighted Integrability

Abstract In this paper we introduce new classes of p-bounded variation sequences and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients belonging to these classes. This is a generalization of the results obtained by the first author [J. Inequal. Appl. 2010:1–19, 2010] and Dyachenko and Tikhonov [Stud. Math. 193(3):285–306, 2009].

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On L1-Convergence of Fourier Series under the MVBV Condition

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-061-6 ◽

2009 ◽

Vol 52 (4) ◽

pp. 627-636 ◽

Cited By ~ 4

Author(s):

Dan Sheng Yu ◽

Ping Zhou ◽

Song Ping Zhou

Keyword(s):

Fourier Series ◽

Function Space ◽

Classical Result ◽

Complex Space ◽

Mean Value ◽

Image Position ◽

Special Case ◽

Nonnegative Sequence ◽

Approximation Of A Function ◽

Mean Value Bounded Variation

AbstractLet f ∈ L2π be a real-valued even function with its Fourier series , and let Sn( f , x), n ≥ 1, be the n-th partial sum of the Fourier series. It is well known that if the nonnegative sequence ﹛an﹜ is decreasing and limn→∞an = 0, thenWe weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the L1-convergence of a function f ∈ L2π in complex space. We also give results on L1-approximation of a function f ∈ L2π under the MVBV condition.

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Pain Experienced during Various Dental Procedures: Clinical Trial Comparing the Use of Traditional Syringes with the Controlled-Flow Delivery Dentapen® Technique

Medicina ◽

10.3390/medicina57121335 ◽

2021 ◽

Vol 57 (12) ◽

pp. 1335

Author(s):

Erick Rafael Fernández-Castellano ◽

Leticia Alejandra Blanco-Antona ◽

Purificación Vicente-Galindo ◽

Víctor Amor-Esteban ◽

Javier Flores-Fraile

Keyword(s):

Mean Value ◽

Ease Of Use ◽

P Value ◽

New Techniques ◽

Dental Procedures ◽

Adult Participants ◽

Before And After ◽

Single Centre Study ◽

Controlled Flow ◽

Randomised Controlled

Background and Objectives: Currently, one of the most discouraging aspects for many patients undergoing dental procedures is the administration of anaesthesia. Consequently, there is a constant search for new techniques to avoid the invasive and painful nature of the injection. A new motorised syringe system (Dentapen®) has recently been developed, standing out for its convenience and ease of use. Material and Methods: Randomised, controlled, single-blind, and single-centre study including 178 voluntary adult participants aged between 18 and 90 years. Individuals were randomly assigned using a randomised table. Patients were asked to rate the level of pain experienced during the injections, using a 10-point visual analogue scale (VAS). The following data were recorded: pain index, heart rate, blood pressure, and saturation, both before and after anaesthesia. Results: Of the total 178 participants, 87 participants (48.9%) were men and 91 (51.1%) were women. The first variable to be assessed was the pain experienced by patients when anaesthetised with a syringe, obtaining a mean value of 2.63 ± 1.86 on the VAS with the conventional syringe and 1.06 ± 1.28 with the Dentapen® syringe, showing statistically significant differences (p-value < 0.01). When stratifying, based on the procedure that was undertaken, differences were also significant for all treatments (p-value < 0.01) except for endodontics, where differences were likely to be significant (p-value = 0.02). Conclusions: In conclusion, from a clinical standpoint, the Dentapen® syringe is a valid alternative to traditional infiltration syringes, causing minimum pain with the injection.

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Comparison of Different Final Impression Techniques for Management of Resorbed Mandibular Ridge: A Case Report

Case Reports in Dentistry ◽

10.1155/2014/253731 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Bhupender Yadav ◽

Manisha Jayna ◽

Harish Yadav ◽

Shrey Suri ◽

Shefali Phogat ◽

...

Keyword(s):

Case Report ◽

Clinical Examination ◽

Mean Value ◽

Complete Denture ◽

Complete Dentures ◽

New Techniques ◽

Impression Materials ◽

Significant Difference ◽

History Of ◽

Green Stick

The history of complete denture impression procedures has been influenced largely by the development of impression materials from which new techniques and ideas arose. The purpose of this study was to compare the retention of complete dentures made by using different impression techniques like conventional, admixed, all green, and functional techniques. The results showed that there was significant difference in retention between the six techniques where functional technique showed the highest mean value of retention followed by elastomeric, all green, and admixed, while cocktail and green stick compound showed the lowest mean value. However, on clinical examination, the retention produced by the six techniques was satisfactory.

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Some Remarks on Results Related to ∇-Convex Function

Journal of Mathematical and Fundamental Sciences ◽

10.5614/j.fund.math.sci.2021.53.1.5 ◽

2021 ◽

Vol 53 (1) ◽

pp. 67-85

Author(s):

Asif Raza Khan ◽

Faraz Mehmood ◽

Faisal Nawaz ◽

Aamna Nazir

Keyword(s):

Convex Function ◽

Present Article ◽

Divided Difference ◽

Mean Value ◽

Higher Order ◽

Two Dimensions ◽

Mean Value Theorem ◽

New Techniques ◽

Differentiable Functions ◽

The Mean

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.

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Balls minimize trace constants in BV

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0098 ◽

2017 ◽

Vol 2017 (725) ◽

Cited By ~ 1

Author(s):

Andrea Cianchi ◽

Vincenzo Ferone ◽

Carlo Nitsch ◽

Cristina Trombetti

Keyword(s):

Bounded Variation ◽

Mean Value ◽

Functions Of Bounded Variation ◽

Optimal Constant ◽

Boundary Trace ◽

Type Boundary ◽

Trace Inequalities ◽

Euclidean Domains

AbstractBalls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincaré type boundary trace inequalities for functions of bounded variation with vanishing median or mean value.

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Some Remarks on Results Related to ∇-Convex Function

Journal of Mathematical and Fundamental Sciences ◽

10.5614/j.math.fund.sci.2021.53.1.5 ◽

2021 ◽

Vol 53 (1) ◽

pp. 67-85

Author(s):

Asif Raza Khan ◽

Faraz Mehmood ◽

Faisal Nawaz ◽

Aamna Nazir

Keyword(s):

Convex Function ◽

Present Article ◽

Divided Difference ◽

Mean Value ◽

Higher Order ◽

Two Dimensions ◽

Mean Value Theorem ◽

New Techniques ◽

Differentiable Functions ◽

The Mean

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.

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