Real Analysis: Measures, Integrals and Applications

Makarov, B. M.; Podkorytov, Anatolii

doi:10.1007/978-1-4471-5122-7

Cited by 40 publications

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“…Thus, the uniform integrability of η ( v k ) is proven. This fact together with () and the Vitali convergence theorem (see, e.g., Makarov and Podkorytov 15, Sec. 4.8.7 ) enable us to conclude that η ( v ) ∈ L 1 (Ω) and η ( v k ) → η ( v ) in L 1 (Ω) as k → ∞ .…”

Section: Auxiliary Resultsmentioning

confidence: 80%

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Boundary value problem for a global‐in‐time parabolic equation

Starovoitov

2020

Math Methods in App Sciences

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The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. In fact, one needs to know the "future" in order to determine the coefficient in this term, that is, the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.

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Section: Auxiliary Resultsmentioning

confidence: 80%

“…Since v k → v in H 1 0 (Ω) as k → ∞, the sequence {v k } converges to v in measure and, as is a continuous function, the sequence { (v k )} converges in measure to (v). Lemma 1 states that the sequence { (v k )} is bounded in L 2 (Ω); therefore, due to the Vitali convergence theorem (see, e.g., Makarov and Podkorytov 15,Sec. 4.8.7…”

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confidence: 99%

Boundary value problem for a global‐in‐time parabolic equation

Starovoitov

2020

Math Methods in App Sciences

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“…for any x ∈ R. Hence, combining (61) and (73), for any x ∈ R, we obtain (59). This concludes the step (a).…”

Section: Appendix G: Proof Of Lemma F1 P 23mentioning

confidence: 79%

On a mathematical theory of coded exposure

Tendero

Osher

2016

Res Math Sci

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This paper proposes a mathematical model and formalism to study coded exposure (flutter shutter) cameras. The model includes the Poisson photon (shot) noise as well as any additive (readout) noise of finite variance. This is an improvement compared to our previous work that only considered the Poisson noise. Closed formulae for the mean square error and signal to noise ratio of the coded exposure method are given. These formulae take into account for the whole imaging chain, i.e., the Poisson photon (shot) noise, any additive (readout) noise of finite variance as well as the deconvolution and are valid for any exposure code. Our formalism allows us to provide a curve that gives an absolute upper bound for the gain of any coded exposure camera in function of the temporal sampling of the code. The gain is to be understood in terms of mean square error (or equivalently in terms of signal to noise ratio), with respect to a snapshot (a standard camera). Keywords: Coded exposure, Computational photography, Flutter shutter, Motion blur, Mean square error (MSE), Signal to noise ratio (SNR) Mathematics Subject Classification: Primary 68U10; Secondary 42A38, 60G99 BackgroundSince the seminal papers [1][2][3][4][5][6] of Agrawal and Raskar coded exposure (flutter shutter) method has received a lot of follow-ups . In a nutshell, the authors proposed to open and close the camera shutter, according to a sequence called "code," during the exposure time. By this clever exposure technique, the coded exposure method permits one to arbitrarily increase the exposure time when photographing (flat) scenes moving at a constant velocity. Note that with a coded exposure method only one picture is stored/transmitted. A rich body of empirical results suggest that the coded exposure method allows for a gain in terms of Mean Square Error (MSE) or Signal to Noise Ratio (SNR) compared to a classic camera, i.e., a snapshot. Therefore, the coded exposure method seems to be a magic tool that should equip all cameras.We now briefly expose the different applications, variants, and studies that surround the coded exposure method. An application of the coded exposure method to bar codes is given in [16,35], to fluorescent cell image in [27], to periodic events in [25,34,36], to multispectral imaging in [10], and to iris in [21]. Application to motion estimation/deblurring are presented in [9,19,20,26,31,33,37,38]. An extension for space-dependent blur is investigated in [28]. Methods to find better or optimal sequences as investigated in [12-© 2016 Tendero and Osher. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,-: vol Tendero and Osher Res Math Sci (2016) 3:4Page 2 of 39 14,22,23,39] o...

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“…, the sequence {v k } converges to v in measure and, as ϕ is a continuous function, the sequence {ϕ(v k )} converges in measure to ϕ(v). Lemma 3.1 states that the sequence {ϕ(v k )} is bounded in L 2 (Ω), therefore, due to the Vitali convergence theorem (see, e.g., [12,Sec. 4…”

Section: Elliptic Problemmentioning

confidence: 99%

Boundary value problem for a global in time parabolic equation

Starovoitov

2020

Preprint

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The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. In fact, one needs to know the "future" in order to determine the coefficient in this term, i.e., the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven.The interaction potential satisfies fairly general conditions and can have an arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.

show abstract

Real Analysis: Measures, Integrals and Applications

Cited by 40 publications

References 0 publications

Boundary value problem for a global‐in‐time parabolic equation

Boundary value problem for a global‐in‐time parabolic equation

On a mathematical theory of coded exposure

Boundary value problem for a global in time parabolic equation

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