Distributional properties of exponential functionals of Levy processes

We study the distribution of the exponential functional I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) \d \eta_t, where $\xi$ and $\eta$ are independent L\'evy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $\eta$ is a Brownian motion with drift we show that this integral equation leads to an important functional equation for the Mellin transform of $I(\xi,\eta)$, which proves to be a very useful tool for studying the distributional properties of this random variable. For general L\'evy process $\xi$ ($\eta$ being Brownian motion with drift) we prove that the exponential functional has a smooth density on $\r \setminus \{0\}$, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that $\xi$ has some positive exponential moments we establish an asymptotic behaviour of \p(I(\xi,\eta)>x) as $x\to +\infty$, and under similar assumptions on the negative exponential moments of $\xi$ we obtain a precise asympotic expansion of the density of $I(\xi,\eta)$ as $x\to 0$. Under further assumptions on the L\'evy process $\xi$ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when $\xi$ has hyper-exponential jumps.Comment: In this version we added a remark after Theorem 1 about extra conditions required for validity of equation (2.3

A formulation of a (q+1,8)-cage

Let $q\ge 2$ be a prime power. In this note we present a formulation for obtaining the known $(q+1,8)$-cages which has allowed us to construct small $(k,g)$--graphs for $k=q-1, q$ and $g=7,8$. Furthermore, we also obtain smaller $(q,8)$-graphs for even prime power $q$.Comment: 14 pages, 2 figure

A construction of small (q-1)-regular graphs of girth 8

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth $8$ and order $2q(q-1)^2$ for all prime powers $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.Comment: 8 pages, 2 figure

Magnetostratigraphy of the Miocene continental deposits of the Montes de Castejón (central Ebro basin, Spain): geochronological and paleoenvironmental implications

A detailed magnetostratigraphic study has been carried out in the early to middle Miocene distal alluvial and lacustrine sediments of the Montes de Castejón (central Ebro Basin). The study was based on the analysis of 196 magnetostratigraphic sites sampled along a stratigraphic interval of about 240 meters. Local magnetostratigraphy yielded a sequence of 12 magnetozones (6 normal and 6 reverse) which could be correlated with the Geomagnetic Polarity Time Scale (GPTS) interval C5Cr to C5AD (between 17 and 14.3 Ma.). The sampled sedimentary sequences include the boundary between two tectosedimentary units (TSU, T5 and T6) already defined in the Ebro Basin. The magnetostratigraphy of the Montes de Castejón allows to date the T5/T6 TSU boundary at 16.14 Ma, within chron C5Cn.1n. This magnetostratigraphy also allows us to analyse in detail as well as to discuss the variations in sedimentation rates through space and time between different lacustrine environments: Outer carbonate lacustrine fringes and distal alluvial plains (Montes de Castejón sections) show higher sedimentation rates than offshore lacustrine areas (San Caprasio section, 50 km east of Montes de Castejón)

An exploratory canonical analysis approach for multinomial populations based on the ϕ\phi-divergence measure

summary:In this paper we consider an exploratory canonical analysis approach for multinomial population based on the $\phi$-divergence measure. We define the restricted minimum $\phi$-divergence estimator, which is seen to be a generalization of the restricted maximum likelihood estimator. This estimator is then used in $\phi$-divergence goodness-of-fit statistics which is the basis of two new families of statistics for solving the problem of selecting the number of significant correlations as well as the appropriateness of the model

Linear bands, zero-momentum Weyl semimetal, and topological transition in skutterudite-structure pnictides

It was reported earlier [Phys. Rev. Lett. 106, 056401 (2011)] that the skutterudite structure compound CoSb$_3$ displays a unique band structure with a topological transition versus a symmetry-preserving sublattice (Sb) displacement very near the structural ground state. The transition is through a massless Dirac-Weyl semimetal, point Fermi surface phase which is unique in that (1) it appears in a three dimensional crystal, (2) the band critical point occurs at $k$=0, and (3) linear bands are degenerate with conventional (massive) bands at the critical point (before inclusion of spin-orbit coupling). Further interest arises because the critical point separates a conventional (trivial) phase from a topological phase. In the native cubic structure this is a zero-gap topological semimetal; we show how spin-orbit coupling and uniaxial strain converts the system to a topological insulator (TI). We also analyze the origin of the linear band in this class of materials, which is the characteristic that makes them potentially useful in thermoelectric applications or possibly as transparent conductors. We characterize the formal charge as Co$^{+}$ $d^8$, consistent with the gap, with its $\bar{3}$ site symmetry, and with its lack of moment. The Sb states are characterized as $p_x$ (separately, $p_y$) $\sigma$-bonded $Sb_4$ ring states occupied and the corresponding antibonding states empty. The remaining (locally) $p_z$ orbitals form molecular orbitals with definite parity centered on the empty $2a$ site in the skutterudite structure. Eight such orbitals must be occupied; the one giving the linear band is an odd orbital singlet $A_{2u}$ at the zone center. We observe that the provocative linearity of the band within the gap is a consequence of the aforementioned near-degeneracy, which is also responsible for the small band gap.Comment: 10 pages, 7 figure

Electron degeneracy and intrinsic magnetic properties of epitaxial Nb:SrTiO3_3 thin-films controlled by defects

We report thermoelectric power experiments in e-doped thin films of SrTiO$_3$ (STO) which demonstrate that the electronic band degeneracy can be lifted through defect management during growth. We show that even small amounts of cationic vacancies, combined with epitaxial stress, produce a homogeneous tetragonal distortion of the films, resulting in a Kondo-like resistance upturn at low temperature, large anisotropic magnetoresistance, and non-linear Hall effect. Ab-initio calculations confirm a different occupation of each band depending on the degree of tetragonal distortion. The phenomenology reported in this paper for tetragonally distorted e-doped STO thin films, is similarto that observed in LaAlO$_3$/STO interfaces and magnetic STO quantum wells.Comment: 5 pages, 5 figure