Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2019
Volume
Vol. 39
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 39 (2019)
Projects
Pages
Articles
Applications of PDEs inpainting to magnetic particle imaging and corneal topography
(Wydawnictwa AGH, 2019) Andrisani, Andrea; Mininni, Rosa Maria; Mazzia, Francesca; Settanni, Giuseppina; Iurino, Alessandro; Tangaro, Sabina; Tateo, Andrea; Bellotti, Roberto
In this work we propose a novel application of Partial Differential Equations (PDEs) inpainting techniques to two medical contexts. The first one concerning recovering of concentration maps for superparamagnetic nanoparticles, used as tracers in the framework of Magnetic Particle Imaging. The analysis is carried out by two set of simulations, with and without adding a source of noise, to show that the inpainted images preserve the main properties of the original ones. The second medical application is related to recovering data of corneal elevation maps in ophthalmology. A new procedure consisting in applying the PDEs inpainting techniques to the radial curvature image is proposed. The images of the anterior corneal surface are properly recovered to obtain an approximation error of the required precision. We compare inpainting methods based on second, third and fourth-order PDEs with standard approximation and interpolation techniques.
Oscillatory results for second-order noncanonical delay differential equations
(Wydawnictwa AGH, 2019) Džurina, Jozef; Jadlovská, Irena; Stavroulakis, Ioannis P.
The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation $\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,$ under the condition $\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.$ Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given.
Decomposition of Gaussian processes, and factorization of positive definite kernels
(Wydawnictwa AGH, 2019) Jørgensen, Palle E.T.; Tian, Feng
We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
On unitary equivalence of bilateral operator valued weighted shifts
(Wydawnictwa AGH, 2019) Kośmider, Jakub
We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We provide an example of unitary equivalence between shifts with weights defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper also contains some remarks regarding unitary operators that can give unitary equivalence of bilateral operator valued weighted shifts.
Description of the scattering data for Sturm-Liouville operators on the half-line
(Wydawnictwa AGH, 2019) Mikitûk, Âroslav Volodimirovič; Suŝik, Natalìâ Stepanìvna
We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to $L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)$, where $\rho:\mathbb{R}_+\to\mathbb{R}_+$ is a monotonically nondecreasing function from some family $\mathscr{R}$. In particular, R includes the functions $\rho(x)=(1+x)^{\alpha}$ with $\alpha \geq 1$.