Supremum distribution of Bessel process of drifting Brownian motion

Let (B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t + \mu t) be a three-dimensional Brownian motion with drift \mu, starting at the origin. Then X_t = ||(B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t +\mu t)||, its distance from the starting point, is a diffusion with many applications. We investigate the distribution of the supremum of (X_t), give an infinite-series formula for its density and an exact estimate by elementary functions

Inversion, duality and Doob hh-transforms for self-similar Markov processes

We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $\alpha>0$ can be represented as a path transformation of some Markov additive process (MAP) $(\theta,\xi)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(\theta,-\xi)$ through this extended Lamperti transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$, with respect to the measure $\pi(x/\|x\|)\|x\|^{\alpha-d}dx$, if and only if $(\theta,\xi)$ is reversible with respect to the measure $\pi(ds)dx$, where $\pi(ds)$ is some $\sigma$-finite measure on $S_{d-1}$ and $dx$ is the Lebesgue measure on $\mathbb{R}$. Besides, the dual process $\widehat{X}$ has the same law as the inversion $(X_{\gamma_t}/\|X_{\gamma_t}\|^2,t\ge0)$ of $X$, where $\gamma_t$ is the inverse of $t\mapsto\int_0^t\|X\|_s^{-2\alpha}\,ds$. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable L\'evy processes

The application of the multiple criteria decision making/aiding methodology to evaluation and redesign of logistics systems

The paper presents the methodological background of Multiple Criteria Decision Making/Aiding (MCDM/A) and its practical application in logistics systems. It explains why MCDM/A methodology is important while dealing with different categories of decision problems that arise in those systems. Major features and basic notions of MCDM/A methodology are presented. Different categories of MCDM/A methods are characterized and classified. Two case studies demonstrate possible applications of MCDM/A methodology in logistics. In the first case study multiple objective optimization of the distribution system is carried out and compared with the single objective optimization. The decision problem is formulated as multiple criteria mathematical programming problem and solved by an extended version of MS Excel Solver – Premium Solver Plus. The second case study focuses on the multiple criteria evaluation and ranking of the logistics infrastructure objects, i.e. a set of warehouses – distribution centers. The decision problem is formulated as a multiple criteria ranking problem and solved with an application of ELECTRE III/IV method

Quantum Computing with Electron Spins in Quantum Dots

Several topics on the implementation of spin qubits in quantum dots are reviewed. We first provide an introduction to the standard model of quantum computing and the basic criteria for its realization. Other alternative formulations such as measurement-based and adiabatic quantum computing are briefly discussed. We then focus on spin qubits in single and double GaAs electron quantum dots and review recent experimental achievements with respect to initialization, coherent manipulation and readout of the spin states. We extensively discuss the problem of decoherence in this system, with particular emphasis on its theoretical treatment and possible ways to overcome it.Comment: Lecture notes for Course CLXXI "Quantum Coherence in Solid State Systems" Int. School of Physics "Enrico Fermi", Varenna, July 2008, 61 pages, 20 figure

A few remarks on the theory of non-nilpotent graphs

We prove a few results about non-nilpotent graphs of symmetric groups $S_n$ -- namely that they have a Hamiltonian cycle and they satisfy a conjecture of Nongsiang and Saikia. The latter is likewise proven for alternating groups $A_n$. We also show that the class of non-nilpotent graphs does not have any ''local'' properties, ie. for every simple graph $X$ there is a group $G$, such that its non-nilpotent graph has $X$ as an induced subgraph