Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The analysis of the fractional version (of order $\nu$) of the Fresnel equation is also performed and, in detail, some specific cases, like $\nu=1/2$, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process $F(t)$, $t>0$ with real sign-varying density is constructed and some of its properties examined. The equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. The composition of F with reflecting Brownian motion $B$ yields the law of biquadratic heat equation while the composition of $F$ with the first passage time $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure

Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: the case of the relativistic harmonic oscillator

We solve the generalized relativistic harmonic oscillator in 1+1 dimensions, i.e., including a linear pseudoscalar potential and quadratic scalar and vector potentials which have equal or opposite signs. We consider positive and negative quadratic potentials and discuss in detail their bound-state solutions for fermions and antifermions. The main features of these bound states are the same as the ones of the generalized three-dimensional relativistic harmonic oscillator bound states. The solutions found for zero pseudoscalar potential are related to the spin and pseudospin symmetry of the Dirac equation in 3+1 dimensions. We show how the charge conjugation and $\gamma^5$ chiral transformations relate the several spectra obtained and find that for massless particles the spin and pseudospin symmetry related problems have the same spectrum, but different spinor solutions. Finally, we establish a relation of the solutions found with single-particle states of nuclei described by relativistic mean-field theories with scalar, vector and isoscalar tensor interactions and discuss the conditions in which one may have both nucleon and antinucleon bound states.Comment: 33 pages, 10 figures, uses revtex macro

Weak Values with Decoherence

The weak value of an observable is experimentally accessible by weak measurements as theoretically analyzed by Aharonov et al. and recently experimentally demonstrated. We introduce a weak operator associated with the weak values and give a general framework of quantum operations to the W operator in parallel with the Kraus representation of the completely positive map for the density operator. The decoherence effect is also investigated in terms of the weak measurement by a shift of a probe wave function of continuous variable. As an application, we demonstrate how the geometric phase is affected by the bit flip noise.Comment: 17 pages, 3 figure

Relativistic quantum mechanics of a Dirac oscillator

The Dirac oscillator is an exactly soluble model recently introduced in the context of many particle models in relativistic quantum mechanics. The model has been also considered as an interaction term for modelling quark confinement in quantum chromodynamics. These considerations should be enough for demonstrating that the Dirac oscillator can be an excellent example in relativistic quantum mechanics. In this paper we offer a solution to the problem and discuss some of its properties. We also discuss a physical picture for the Dirac oscillator's non-standard interaction, showing how it arises on describing the behaviour of a neutral particle carrying an anomalous magnetic moment and moving inside an uniformly charged sphere.Comment: 19 pages, 1 figur

Posterior Consistency via Precision Operators for Bayesian Nonparametric Drift Estimation in SDEs

We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function

JJ-self-adjoint operators with C\mathcal{C}-symmetries: extension theory approach

A well known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac-Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ($J$-self-adjoint) Hamiltonians with complex point-interactions. We demonstrate that the resulting Hamiltonians are bijectively related with so called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices . General properties of the $\cC$ operators for these Hamiltonians are derived. A detailed study of $\cC$-operator parametrizations and Krein type resolvent formulas is provided for $J$-self-adjoint extensions of symmetric operators with deficiency indices . The technique is exemplified on 1D pseudo-Hermitian Schr\"odinger and Dirac Hamiltonians with complex point-interaction potentials

Foundation Pattern, Productivity and Colony Success of the Paper Wasp, Polistes versicolor

Polistes versicolor (Olivier) (Hymenoptera: Vespidae) colonies are easily found in anthropic environments; however there is little information available on biological, ecological and behavioral interactions of this species under these environmental conditions. The objective of this work was to characterize the foundation pattern, the productivity, and the success of colonies of P. versicolor in anthropic environments. From August 2003 to December 2004, several colonies were studied in the municipal district of Juiz de Fora, Southeastern Brazil. It was possible to determine that before the beginning of nest construction the foundress accomplishes recognition flights in the selected area, and later begins the construction of the peduncle and the first cell. As soon as new cells are built, the hexagonal outlines appear and the peduncle is reinforced. Foundation of nests on gypsum plaster was significantly larger (p < 0.0001; χ2 test) in relation to the other types of substrate, revealing the synantropism of the species. On average, the P. versicolor nest presents 244.2 ± 89.5 (100–493) cells and a medium production of 171.67 ± 109.94 (37–660) adults. Cells that produced six individuals were verified. Usually, new colonies were founded by an association of females, responsible for the success of 51.5%. Although these results enlarge knowledge on the foundation pattern of P. versicolor in anthropic environments, other aspects of the foundation process require further investigation

Stochastic thermodynamics of holonomic systems

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