Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes

The inversion of a Levy measure was first introduced (under a different name) in Sato 2007. We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Levy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Levy process as time approaches infinity. This allows us to get self contained conditions for a Levy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Levy processes and for the important class of tempered stable Levy processes. For this latter class, we give detailed results in terms of their Rosinski measures

Convolution-type derivatives, hitting-times of subordinators and time-changed C0C_0-semigroups

In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.Comment: Final version, Potential analysis, 201

Numerical approximations for the tempered fractional Laplacian: Error analysis and applications

In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of $O(h^\epsilon)$, for $u \in C^{0, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{1, \alpha-1+\epsilon} (\bar{\Omega})$ if $\alpha \ge 1$) with $\epsilon > 0$, suggesting the minimum consistency conditions. The accuracy can be improved to $O(h^2)$, for $u \in C^{2, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{3, \alpha - 1 + \epsilon} (\bar{\Omega})$ if $\alpha \ge 1$). Numerical experiments confirm our analytical results and provide insights in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our method only requires the solution $u \in C^{1,1}(\bar{\Omega})$ for any $0<\alpha<2$. Moreover, if the solution of tempered fractional Poisson problems satisfies $u \in C^{p, s}(\bar{\Omega})$ for $p = 0, 1$ and $0<s \le 1$, our method has the accuracy of $O(h^{p+s})$. Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen-Cahn equation and Gray-Scott equations.Comment: 21 pages, 11 figures, 3 table

U-FAST technology in Al2O3 transparent ceramic fabrication

Przeświecalna ceramika Al2O3 jest pożądanym materiałem z uwagi na swoje właściwości porównywalne z szafirem. Charakteryzuje się wysoką odpornością na zużycie, wysoką stabilnością chemiczną i doskonałą odpornością na korozję. Głównymi obszarami jej zastosowań są: wysokotemperaturowe wizjery, drobne elementy opancerzenia, detektory podczerwieni oraz koperty emiterów podczerwieni. Do wytwarzania transparentnej ceramiki stosowane są techniki HP, SPS i HIP. W niniejszej pracy materiały otrzymano z zastosowaniem technologii U-FAST (ang. Upgraded-Field Assisted Sintering Technology), jednego z najnowocześniejszych rozwiązań na rynku urządzeń do spiekania. Jest to efektywna i energooszczędna technologia przeznaczona do konsolidacji materiałów proszkowych, w tym nanokrystalicznych. Procesy spiekania przeprowadzono z wykorzystaniem handlowego proszku tlenku glinu, bez użycia dodatków. Zbadano wpływ parametrów procesu na mikrostrukturę i stopień zagęszczenia uzyskanych spieków, jak również na transmisję w zakresie światła widzialnego.Transparent Al2O3 ceramics arouses of great interest due to its properties comparable to sapphire. Alumina is characterized by high wear-resistance, high chemical stability, excellent corrosion resistance. The main areas of its applications are high-temperature light transmitting materials, small armor protection, infrared detectors, IR emitters envelopes. For the production of transparent ceramics, HP and SPS and HIP techniques are used. In this paper, the materials were made using the U-FAST technology (Upgraded-Field Assisted Sintering Technology), one of the most modern solutions on the market of sintering devices. It is an effective and energy-saving technology dedicated to consolidation of powder materials, including nanocrystalline materials. The sintering processes were carried out using commercial alumina powder without the use of additives. The influence of sintering parameters on the microstructure and density of the obtained sinters, as well as on transmittance in visible range were studied

Some Remarks and Calculations Concerning Collisional Fine Structure Mixing in Alkali Metal Atoms

The analysis of all existing theoretical and experimental results concerning collisional fine structure (FS) mixing, suggesting that their agreement depends on FS separation and FS components intensity anomaly, showed that the test of the theoretical model should be made on sodium atoms. Therefore, the mixing cross sections for the nP (n = 6 ÷ 24) states of sodium atoms colliding with noble gas atoms, N2 and sodium atoms in the ground state have been calculated theoretically. Cross sections also for the lowest nP states of Na and Rb atoms have been estimated

The Safety Level Analysis of the SWIM System in Air Traffic Management

Safety level analysis is included in the paper in context of SWIM system in air traffic management. The example is given. It examines the limitations and drawbacks of current ICT systems used for air traffic management. Analyzing the development of communication systems for the management of general air traffic, it can be concluded that the development of the terrestrial segment of the exchange of information between the parties relating to the air traffic will fluctuate towards a solution based on a service-oriented architecture SOA. This architecture will be the basis for the implementation of the concept of an information exchange system SWIM