Mária Hutníková

An alternative description of gabor spaces and gabor-toeplitz operators

Abstract We show a representation of the spaces of Gabor transforms using the general decomposition method of Vasilevski. This provides an interesting characterization of a class of Gabor-Toeplitz operators and enables to obtain their representation formulae, Wick calculus and the commutativity of the algebra of such operators particularly simply.

Affine coherent states and Toeplitz operators

We study a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderon reproducing formula (= resolution of unity on ) and the specific admissible wavelets (= affine coherent states in ) related to Laguerre functions. Symbols of such Calderon–Toeplitz operators as individual coordinates of the affine group (= upper half-plane with the hyperbolic geometry) are considered. In this case, a certain class of pseudo-differential operators, their properties and their operator algebras are investigated. As a result of this study, the Fredholm symbol algebras of the Calderon–Toeplitz operator algebras for these particular cases of symbols are described.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

On Toeplitz localization operators

We present a unified approach to study properties of Toeplitz localization operators based on the Calderón and Gabor reproducing formula. We show that these operators with functional symbols on a plane domain may be viewed as certain pseudo-differential operators (with symbols on a line, or certain compound symbols).

Continuous Stockwell transform: Coherent states and localization operators

There are several methods of interest in signal analysis and image processing. The most widely used are the wavelet transform, the short-time Fourier transform, the shearlet transform, and the Stockwell transform. Putting a filter in the corresponding reproducing formula, one gets the well-known Calderon-Toeplitz and Gabor-Toeplitz localization operators widely studied in the context of time scale, time-frequency as well as shearlet analysis. Using Vasilevski’s technique of Hilbert-space decomposition (applied for the space of continuous Stockwell transforms of L2(ℝ)-functions), we get the structural results of the transform space and we study Toeplitz operators in this context with many desirable properties of localization operators. We find their unitary equivalent images for the case of separable generating symbols and show that the Wick symbols for these operators are associated with a well-defined calculus. Also, certain algebras generated by these operators are described in detail. Thus, the results...

Characterization of native and plasticized starch using solid state NMR

The crystallinity of native starches influences numerous properties of starch-based materials and for this reason, its assessment is of great importance. It is known that the phase composition of native starch can be obtained by the decomposition of its 13C NMR spectrum into the sub-spectra of the ordered (crystalline and interfacial) and amorphous phases. The percentage of amorphous phase can be obtained from the intensity of C4 carbon resonance. The deconvolution of the C1 carbon resonance in the sub-spectrum of the ordered phase gives information on the ratio of crystalline and interfacial phases and the starch crystalline type, which is inferred from the presence of triplet (A-type) or doublet (B-type) of even intensities. Four native starches - soltex (wheat, A-type), amylex (potato, B-type) meritena and maisita (both corn A-type) were studied and as their amorphous counterparts the starches plasticized with urea were used. The 13C NMR spectra of the studied samples were recorded using the cross polarization (CP) and the magic angle spinning (MAS) techniques and using the decomposition of the spectra the phase compositions of native starches were obtained. In order to evaluate the mobility of different phases in native as well as plasticized samples the measurements of carbon spin-lattice relaxation times T1(13C) in laboratory frame were carried out. The T1(13C) values obtained fitting the bi-exponential function to the C1 carbon intensity decay and taking into account the calculated composition of the samples (ordered and disordered phases) show that A-type starches have close T1(13C) values which are shorter (16.6 – 20.8 s) for disordered phase and longer (53.9 – 80.1 s) for ordered phase proving different mobility of these phases. The B-type sample has lower T1(13C) values (1.7 s, 37.0 s) which are in line with less spatial constraints in its structure. The mono-exponential functions fitted to the data obtained for plasticized samples provided T1(13C) values in the interval (43.9–79.9 s) which could be explained by different molecular structure of native samples.The crystallinity of native starches influences numerous properties of starch-based materials and for this reason, its assessment is of great importance. It is known that the phase composition of native starch can be obtained by the decomposition of its 13C NMR spectrum into the sub-spectra of the ordered (crystalline and interfacial) and amorphous phases. The percentage of amorphous phase can be obtained from the intensity of C4 carbon resonance. The deconvolution of the C1 carbon resonance in the sub-spectrum of the ordered phase gives information on the ratio of crystalline and interfacial phases and the starch crystalline type, which is inferred from the presence of triplet (A-type) or doublet (B-type) of even intensities. Four native starches - soltex (wheat, A-type), amylex (potato, B-type) meritena and maisita (both corn A-type) were studied and as their amorphous counterparts the starches plasticized with urea were used. The 13C NMR spectra of the studied samples were recorded using the cross pola...

Laboratory experiment for the study of friction forces using rotating apparatus

The standard experimental set-up enabling observation of rotational motion of a bar around its centre of mass, which is set into motion due to the external torque generated by the small weight, was extended with an optical gate and position sensor and connected to a computer with software, which made it possible to display measured values of bar half-rotations during accelerated and decelerated motion as well as to process the data immediately. The detailed analysis of experimental data obtained for decelerated rotational motion due to frictional torque only (without small weight) showed that, besides the constant term due to dry friction at an axle, the expression for friction forces in the system has to include terms depending on the first and/or second power of angular speed, which is evidence that viscous forces influence the motion of a bar. The frictional torque due to viscous forces can be evaluated as the difference between the effective frictional torque acting on the system and the frictional torque due to dry friction at an axle. The data obtained in the experiment in which the bar performed damped oscillatory motion provided the values of effective frictional torque and the moment of inertia of rotating bodies. The frictional torque due to dry friction can be obtained as a minimum torque (calculated using minimum mass of weight) needed to start rotational motion. The last two proposed experiments can be included in undergraduate laboratory practicals.

Some special functions identities arising from commuting operators

Commuting is an important property in many cases of investigation of properties of operators as well as in various applications, especially in quantum physics. Using the observation that the generalized weighted differential operator of order k and the weighted Hardy-type operator commute we derive a number of new and interesting identities involving some functions of mathematical physics.