Strict LpSolutions for Nonautonomous Fractional Evolution Equations

MSC 2010: 26A33, 34A08, 34K37

Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12

Exact Solutions of Nonlocal BVPs for the Multidimensional Heat Equations

MSC 2010: 44A35, 44A45, 44A40, 35K20, 35K05

Conflict-Controlled Processes Involving Fractional Differential Equations with Impulses

MSC 2010: 34A08, 34A37, 49N70

Oscillation Criteria of Second-Order Quasi-Linear Neutral Delay Difference Equations

2000 Mathematics Subject Classification: 39A10.

Dunkl-Schrödinger Equations with and without Quadratic Potentials

2000 Mathematics Subject Classification: Primary 42A38. Secondary 42B10.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

Computation of rare transitions in the barotropic quasi-geostrophic equations

We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier–Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager–Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherWe adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems.

Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations

We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations. Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction. We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics. Cases with or without detailed balance are studied. We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point. In situations where a first order phase transition is observed, the dynamics are bistable. Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics. For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors.

Lokális energiamódszer kicsi rendben gerjesztett Liénard-egyenletekre (Local energy method for little order forced equations)

Az x''+f(x) x'+g(x) = 0 alakú Liénard-típusú differenciálegyenlet központi szerepet játszik az üzleti ciklusok Káldor-Kalecki-féle [3,4] és Goodwin-féle [2] modelljeiben, sőt egy a munkanélküliség és vállalkozás-ösztönzések ciklikus változásait leíró újabb modellben [1] is. De ugyanez a nemlineáris egyenlettípus a gerjesztett ingák és elektromos rezgőkörök elméletét is felöleli [5]. Az ezzel kapcsolatos irodalom nagyrészt a határciklusok létezését vizsgálja (pl. [5]), pedig az alapvető stabilitási kérdések jóval áttekinthetőbb módon kezelhetők, s a kapott eredmények közvetve a határciklusok létezésének feltételeit is sokkal jobban be tudják határolni. Jelen dolgozatban az egyváltozós analízis hatékony nyelvezetével olyan egyszerűen megfogalmazható eredményekhez jutunk, amelyek képesek kitágítani az üzleti és más közgazdasági ciklusok modelljeinek kereteit, illetve pl. az [1]-beli modellhez újabb szemléltető speciális eseteket is nyerünk. ____ The Liénard type differential equation of the form x00 + f(x) ¢ x0 + g(x) = 0 has a central role in business cycle models by Káldor [3], Kalecki [4] and Goodwin [2], moreover in a new model describing the cyclical behavior of unemployment and entrepreneurship [1]. The same type of nonlinear equation explains the features of forced pendulums and electric circuits [5]. The related literature discusses mainly the existence of limit cycles, although the fundamental stability questions of this topic can be managed much more easily. The achieved results also outline the conditions for the existence of limit cycles. In this work, by the effective language of real valued analysis, we obtain easy-formulated results which may broaden the frames of economic and business cycle models, moreover we may gain new illustrative particular cases for e.g., [1].

An investigation into equations for estimating water requirements and the development of new equations for predicting total water intake

The primary purpose of this study was to investigate agreement among five equations by which clinicians estimate water requirements (EWR) and to determine how well these equations predict total water intake (TWI). The Institute of Medicine has used TWI as a measure of water requirements. A secondary goal of this study was to develop practical equations to predict TWI. These equations could then be considered accurate predictors of an individual’s water requirement. ^ Regressions were performed to determine agreement between the five equations and between the five equations and TWI using NHANES 1999–2004. The criteria for agreement was (1) strong correlation coefficients between all comparisons and (2) regression line that was not significantly different when compared to the line of equality (x=y) i.e., the 95% CI of the slope and intercept must include one and zero, respectively. Correlations were performed to determine association between fat-free mass (FFM) and TWI. Clinically significant variables were selected to build equations for predicting TWI. All analyses were performed with SAS software and were weighted to account for the complex survey design and for oversampling. ^ Results showed that the five EWR equations were strongly correlated but did not agree with each other. Further, the EWR equations were all weakly associated to TWI and lacked agreement with TWI. The strongest agreement between the NRC equation and TWI explained only 8.1% of the variability of TWI. Fat-free mass was positively correlated to TWI. Two models were created to predict TWI. Both models included the variables, race/ethnicity, kcals, age, and height, but one model also included FFM and gender. The other model included BMI and osmolality. Neither model accounted for more than 28% of the variability of TWI. These results provide evidence that estimates of water requirements would vary depending upon which EWR equation was selected by the clinician. None of the existing EWR equations predicted TWI, nor could a prediction equation be created which explained a satisfactory amount of variance in TWI. A good estimate of water requirements may not be predicted by TWI. Future research should focus on using more valid measures to predict water requirements.^

Valutazione dosimetrica e aspetti tecnici di trattamenti VMAT (Volumetric Modulated Arc Therapy) in radioterapia oncologica

La VMAT (Volumetric Modulated Arc Therapy) è una delle più recenti tecniche radioterapiche, in cui, oltre alla modulazione geometrica della fluenza del campo di radiazione come avviene nell’IMRT (Intensity Modulated Radiotherapy), sono variati durante il trattamento anche la velocità del gantry e il rateo di dose. La radiazione è erogata senza interruzioni lungo uno o più archi continui della testata dell’acceleratore, così da ridurre i tempi di trattamento in modo sostanziale rispetto all’IMRT. Nelle tecniche ad intensità modulata, ed in particolare nella VMAT, il lettino porta paziente modifica la distribuzione di dose durante gli irraggiamenti posteriori, riducendo quella al target e aumentando quella superficiale. Il presente lavoro di tesi, che ha proprio l’obiettivo di valutare questi aspetti dosimetrici in un’ottica pre-clinica, è stato svolto presso il Servizio di Fisica Sanitaria del Policlinico S.Orsola–Malpighi, Azienda Ospedaliero-Universitaria di Bologna. Le misure sono state effettuate presso le U.O. Radioterapia-Morganti e Radioterapia-Frezza f.f. della medesima Azienda Sanitaria, al fine di caratterizzare dal punto di vista dosimetrico il lettino di trattamento iBEAM evo dell’acceleratore lineare Synergy Elekta. L’attenuazione misurata in caso di incidenza perpendicolare del fascio sul lettino, in buon accordo con gli articoli di riferimento, è stata: (2.81±0.06)% per fotoni di energia di 6 MV, (1.81±0.10)% a 10 MV e (1.38±0.05)% a 15 MV. L’attenuazione massima misurata con fotoni di energia di 6 MV si è avvicinata al 4% negli irraggiamenti obliqui. Infine, è stato analizzato il confronto con statistica gamma fra distribuzione di dose pianificata e misurata prima e dopo aver modellizzato le caratteristiche del lettino nel software per l’elaborazione del piano di trattamento. Solo dopo tale operazione, i due casi VMAT analizzati (tumore dell’orofaringe e prostatico) superano i criteri più restrittivi e affidabili utilizzati nella pratica clinica.

Improving minimum rate predictors algorithm for compression of volumetric medical images

Medical imaging technologies are experiencing a growth in terms of usage and image resolution, namely in diagnostics systems that require a large set of images, like CT or MRI. Furthermore, legal restrictions impose that these scans must be archived for several years. These facts led to the increase of storage costs in medical image databases and institutions. Thus, a demand for more efficient compression tools, used for archiving and communication, is arising. Currently, the DICOM standard, that makes recommendations for medical communications and imaging compression, recommends lossless encoders such as JPEG, RLE, JPEG-LS and JPEG2000. However, none of these encoders include inter-slice prediction in their algorithms. This dissertation presents the research work on medical image compression, using the MRP encoder. MRP is one of the most efficient lossless image compression algorithm. Several processing techniques are proposed to adapt the input medical images to the encoder characteristics. Two of these techniques, namely changing the alignment of slices for compression and a pixel-wise difference predictor, increased the compression efficiency of MRP, by up to 27.9%. Inter-slice prediction support was also added to MRP, using uni and bi-directional techniques. Also, the pixel-wise difference predictor was added to the algorithm. Overall, the compression efficiency of MRP was improved by 46.1%. Thus, these techniques allow for compression ratio savings of 57.1%, compared to DICOM encoders, and 33.2%, compared to HEVC RExt Random Access. This makes MRP the most efficient of the encoders under study.

Temporal Coding of Volumetric Imagery

'Image volumes' refer to realizations of images in other dimensions such as time, spectrum, and focus. Recent advances in scientific, medical, and consumer applications demand improvements in image volume capture. Though image volume acquisition continues to advance, it maintains the same sampling mechanisms that have been used for decades; every voxel must be scanned and is presumed independent of its neighbors. Under these conditions, improving performance comes at the cost of increased system complexity, data rates, and power consumption.

This dissertation explores systems and methods capable of efficiently improving sensitivity and performance for image volume cameras, and specifically proposes several sampling strategies that utilize temporal coding to improve imaging system performance and enhance our awareness for a variety of dynamic applications.

Video cameras and camcorders sample the video volume (x,y,t) at fixed intervals to gain understanding of the volume's temporal evolution. Conventionally, one must reduce the spatial resolution to increase the framerate of such cameras. Using temporal coding via physical translation of an optical element known as a coded aperture, the compressive temporal imaging (CACTI) camera emonstrates a method which which to embed the temporal dimension of the video volume into spatial (x,y) measurements, thereby greatly improving temporal resolution with minimal loss of spatial resolution. This technique, which is among a family of compressive sampling strategies developed at Duke University, temporally codes the exposure readout functions at the pixel level.

Since video cameras nominally integrate the remaining image volume dimensions (e.g. spectrum and focus) at capture time, spectral (x,y,t,\lambda) and focal (x,y,t,z) image volumes are traditionally captured via sequential changes to the spectral and focal state of the system, respectively. The CACTI camera's ability to embed video volumes into images leads to exploration of other information within that video; namely, focal and spectral information. The next part of the thesis demonstrates derivative works of CACTI: compressive extended depth of field and compressive spectral-temporal imaging. These works successfully show the technique's extension of temporal coding to improve sensing performance in these other dimensions.

Geometrical optics-related tradeoffs, such as the classic challenges of wide-field-of-view and high resolution photography, have motivated the development of mulitscale camera arrays. The advent of such designs less than a decade ago heralds a new era of research- and engineering-related challenges. One significant challenge is that of managing the focal volume (x,y,z) over wide fields of view and resolutions. The fourth chapter shows advances on focus and image quality assessment for a class of multiscale gigapixel cameras developed at Duke.

Along the same line of work, we have explored methods for dynamic and adaptive addressing of focus via point spread function engineering. We demonstrate another form of temporal coding in the form of physical translation of the image plane from its nominal focal position. We demonstrate this technique's capability to generate arbitrary point spread functions.