Regularity issues for the null-controllability of the linear 1-d heat equation

The fact that the heat equation is controllable to zero in any bounded domain of the Euclidean space, any time T>0 and from any open subset of the boundary is well known. On the other hand, numerical experiments show the ill-posedness of the problem. In this paper we develop a rigorous analysis of the 1-d problem which provides a sharp description of this ill-posedness. To be more precise, each initial data y0∈L2(0,1) of the 1-d linear heat equation has a boundary control of the minimal L 2(0,T)-norm which drives the state to zero in time T>0. This control is given by a solution of the homogeneous adjoint equation with some initial data φ0, minimizing a suitable quadratic cost. Our aim is to study the relationship between the regularity of y0 and that of φ0. We show that there are regular data y0 for which the corresponding φ0 are highly irregular, not belonging to any negative exponent Sobolev space. Moreover, the class of such initial data y 0 is dense in L2(0,1). This explains the severe ill-posedness of the numerical algorithms developed for the approximation of the minimal L2(0,T)-norm control of y0 based on the computation of φ0. The lack of polynomial convergence rates for Tychonoff regularization processes is a consequence of this phenomenon too

Noncommutative Quantum Hall Effect and Aharonov-Bohm Effect

We study a system of electrons moving on a noncommutative plane in the presence of an external magnetic field which is perpendicular to this plane. For generality we assume that the coordinates and the momenta are both noncommutative. We make a transformation from the noncommutative coordinates to a set of commuting coordinates and then we write the Hamiltonian for this system. The energy spectrum and the expectation value of the current can then be calculated and the Hall conductivity can be extracted. We use the same method to calculate the phase shift for the Aharonov-Bohm effect. Precession measurements could allow strong upper limits to be imposed on the noncommutativity coordinate and momentum parameters $\Theta$ and $\Xi$.Comment: 9 pages, RevTeX4, references added, small changes in the tex

The decay constants of pseudoscalar mesons in a relativistic quark model

The decay constants of pseudoscalar mesons are calculated in a relativistic quark model which assumes that mesons are made of a valence quark antiquark pair and of an effective vacuum like component. The results are given in terms of quark masses and of some free parameters entering the expression of the internal wave functions of the mesons. By using the pion and kaon decay constants $F_{\pi^+}=130.7~MeV,~F_{K^+}=159.8~MeV$ to fix the parameters of the model one gets $60~MeV\leq F_{D^+}\leq 185~MeV,~95~MeV\leq F_{D_s}\leq230~MeV,~80~MeV\leq F_{B^+}\leq205~MeV$ for the light quark masses $m_u=5.1~MeV,~m_d=9.3~MeV,~m_s=175~MeV$ and the heavy quark masses in the range: $1.~GeV\leq m_c\leq1.6~GeV,~4.1~GeV\leq m_b\leq4.5~GeV$. In the case of light neutral mesons one obtains with the same set of parameters $F_{\pi^0}\approx 138~MeV,~F_\eta\approx~130~MeV,F_{\eta'} \approx~78~MeV$. The values are in agreement with the experimental data and other theoretical results.Comment: 11 pages, LaTe

Coupling constants and transition potentials for hadronic decay modes of a meson

Within the independent-harmonic-oscillator model for quarks inside a hadron, a rigorous method is presented for the calculation of coupling constants and transition potentials for hadronic decay, as needed in a multi-channel description of mesons.Comment: 19 pages, 4 figure

Inter-observer reliability of ultrasound detection of tendon abnormalities at the wrist and ankle in patients with rheumatoid arthritis

OBJECTIVE: To assess inter-observer reliability in US detection of tendon inflammatory and structural changes at wrists and ankles in RA patients. METHODS: Fourteen consecutive RA patients underwent bilateral US assessment of the extensor carpi ulnaris (ECUT) and tibialis posterior tendons (TPTs) by two blinded rheumatologists, with different level of experience in musculoskeletal (MS) US. Grey scale and power Doppler (PD) US assessment was focused on detection of tenosynovitis, tenosynovial and intra-tendon PD signal and structural lesions (i.e. tendinosis, tendon erosion, partial or total rupture). RESULTS: The frequency of US findings detected by Investigator 1 was 28.6% for inflammatory changes and 51.8% for structural damage changes while Investigator 2 detected 34 and 53.6% for the corresponding abnormalities. A high overall agreement (82.7%) was found for inflammatory pathology and 89.7% for structural lesions in all tendons. Mean kappa (κ) values for all tendons and pathology was moderate (κ = 0.42), with fair level of agreement for the wrist region (0.27-0.34) and moderate to good values for the ankle region (κ = 0.47-0.62). Subclinical abnormalities were detected in 37.5% of the tendons by Investigator 1 and 28.6% of the tendons by Investigator 2. CONCLUSIONS: MSUS showed high overall agreement and fair to moderate inter-observer κ-values between investigators with different levels of experience in detection of tendon pathology at the wrist and ankle in RA patients. Further standardization of scanning method and pathology definitions may improve MSUS reproducibility