Abstract:
We construct a countable inductive limit of weighted Banach spaces of holomorphic functions, which is not a topological subspace of the corresponding weighted inductive limit of spaces of continuous functions. The main step of our construction, using a special sequence of outer holomorphic functions, shows that a certain sequence space is isomorphic to a complemented subspace of a weighted space of holomorphic functions in two complex variables. This example solves in the negative a well-known open problem raised by Bierstedt, Meise and Summers.

Abstract:
We have used normal metal-insulator-superconductor tunnel junctions as thermometers at sub-Kelvin temperatures to study the electron-phonon (e-p) interaction in thin Aluminum films doped with Manganese, as a function of Manganese concentration. Mn in Al is known to be a Kondo impurity with extremely high Kondo temperature $T_K \sim$ 500 K, thus our results probe the e-p coupling in the fully spin compensated, unitary limit. The temperature dependence of the e-p interaction is consistent with the existing theory for disordered metals, however full theory including the Kondo effect has not been worked out yet. The strength of the interaction decreases with increasing Manganese concentration, providing a means to improve sensitivity of detectors and efficiency of solid state coolers.

Abstract:
We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet-Bloch-Gelfand transform.

Abstract:
We have used symmetric normal metal-insulator-superconductor (NIS) tunnel junction pairs, known as SINIS structures, for ultrasensitive thermometry in the temperature range 50 - 700 mK. By Joule heating the electron gas and measuring the electron temperature, we show that the electron-phonon (e-p) scattering rate in the simplest noble metal disordered thin films (Cu,Au) follows a $T^4$ temperature dependence, leading to a stronger decoupling of the electron gas from the lattice at the lowest temperatures. This power law is indicative e-p coupling mediated by vibrating disorder, in contrast to the previously observed $T^3$ and $T^2$ laws.

Abstract:
We have studied thermal gradients in thin Cu and AlMn wires, both experimentally and theoretically. In the experiments, the wires were Joule heated non-uniformly at sub-Kelvin temperatures, and the resulting temperature gradients were measured using normal metal-insulator-superconducting tunnel junctions. The data clearly shows that even in reasonably well conducting thin wires with a short ($\sim 10 \mu$m) non-heated portion, significant temperature differences can form. In most cases, the measurements agree well with a model which includes electron-phonon interaction and electronic thermal conductivity by the Wiedemann-Franz law.

Abstract:
We have studied the electron-phonon (e-p) interaction in thin Cu and Au films at sub-Kelvin temperatures with the help of the hot electron effect, using symmetric normal metal-insulator-superconductor tunnel junction pairs as thermometers. By Joule heating the electron gas and measuring the electron and the lattice temperatures simultaneously, we show that the electron-phonon scattering rate follows a $T^{4}$ temperature dependence in both metals. The result is in accordance with the theory of e-p scattering in disordered films with vibrating boudaries and impurities, in contrast to the $T^{3}$-law expected for pure samples, and $T^{2}$-law for static disorder.

Abstract:
We have investigated one and two dimensional (1D and 2D) arrays of tunnel junctions in partial Coulomb blockade regime. The absolute accuracy of the Coulomb blockade thermometer is influenced by the external impedance of the array, which is not the same in the different topologies of 1D and 2D arrays. We demonstrate, both by experiment and by theoretical calculations in simple geometries, that the 1D structures are better in this respect. Yet in both 1D and 2D, the influence of the environment can be made arbitrarily small by making the array sufficiently large.

Abstract:
We show that the sufficient condition of the above mentioned paper is also necessary for the boundedness of Bergman type projections on a class of regulated domains.

Abstract:
We show that a continuous bilinear mapping P: C(I) \times C(I) \to C(I) can be presented in the form P(f,g) = B((Af)(Ag)), where A and B are bounded linear operators on C(I) and multiplication is defined pointwise, if and only if for all t in I the bilinear form (f,g) -> P(f,g)(t) is integral on C(I) times C(I) and depends in a sense continuously on t. To this end we construct a continuous surjection phi : I \to I^2 admitting a regular averaging operator in the sense of Pelczynski.

Abstract:
We have used normal metal-insulator-superconductor (NIS) tunnel junction pairs, known as SINIS structures, for ultrasensitive thermometry at sub-Kelvin temperatures. With the help of these thermometers, we have developed an ac-technique to measure the electron-phonon (e-p) scattering rate directly, without any other material or geometry dependent parameters, based on overheating the electron gas. The technique is based on Joule heating the electrons in the frequency range DC-10 MHz, and measuring the electron temperature in DC. Because of the nonlinearity of the electron-phonon coupling with respect to temperature, even the DC response will be affected, when the heating frequency reaches the natural cut-off determined by the e-p scattering rate. Results on thin Cu films show a $T^{4}$ behavior for the scattering rate, in agreement with indirect measurement of similar samples and numerical modeling of the non-linear response.