Revisiting Riesz transforms on Heisenberg groups

We characterise higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to multiple Hermite and Laguerre expansions.

Compressive sensing for music signals: comparison of transforms with coherent dictionaries

Compressive Sensing (CS) is a new sensing paradigm which permits sampling of a signal at its intrinsic information rate which could be much lower than Nyquist rate, while guaranteeing good quality reconstruction for signals sparse in a linear transform domain. We explore the application of CS formulation to music signals. Since music signals comprise of both tonal and transient nature, we examine several transforms such as discrete cosine transform (DCT), discrete wavelet transform (DWT), Fourier basis and also non-orthogonal warped transforms to explore the effectiveness of CS theory and the reconstruction algorithms. We show that for a given sparsity level, DCT, overcomplete, and warped Fourier dictionaries result in better reconstruction, and warped Fourier dictionary gives perceptually better reconstruction. “MUSHRA” test results show that a moderate quality reconstruction is possible with about half the Nyquist sampling.

Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = -Delta - |x|(2 similar to) (t) (2) are bounded on L (p) (a''e (n+1)). We also establish an analogue of the Hormander-Mihlin Multiplier Theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator-valued Fourier multiplier theorem due to L. Weis.

Influence of operational and geometrical parameters on the performance of twin thermoacoustic prime mover

Thermoacoustics is the interaction between heat and sound, which are useful in designing heat engines and heat pumps. Research in the field of thermoacoustics focuses on the demand to improve the performance which is achieved by altering operational, geometrical and fluid parameters. The present study deals with improving the performance of twin thermoacoustic prime mover, which has gained the significant importance in the recent years for the production of high amplitude sound waves. The performance of twin thermoacoustic prime mover is evaluated in terms of onset temperature difference, resonance frequency and pressure amplitude of the acoustic waves by varying the resonator length and charge pressures of fluid medium nitrogen. DeltaEC, the free simulation software developed by LANL, USA is employed in the present study to simulate the performance of twin thermoacoustic prime mover. Experimental and simulated results are compared and the deviation is found to be within 10%.

MRT Letter: Experimental Verification of Vectorial Theory to Determine Field at the Geometrical Focus of a Cylindrical Lens

We provide experimental evidence supporting the vectorial theory for determining electric field at and near the geometrical focus of a cylindrical lens. This theory provides precise distribution of field and its polarization effects. Experimental results show a close match (approximate to 95% using (2)-test) with the simulation results (obtained using vectorial theory). Light-sheet generated both at low and high NA cylindrical lens shows the importance of vectorial theory for further development of light-sheet techniques. Potential applications are in planar imaging systems (such as, SPIM, IML-SPIM, imaging cytometry) and spectroscopy. Microsc. Res. Tech. 77:105-109, 2014. (c) 2014 Wiley Periodicals, Inc.

L-p-Integrability, Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in with if its Fourier transform is supported by a set of finite packing -measure where . It is shown that the assertion fails for . The result is applied to prove L-p Wiener Tauberian theorems for R-n and M(2).

DIMENSION FREE BOUNDEDNESS OF RIESZ TRANSFORMS FOR THE GRUSHIN OPERATOR

Let G = -Delta(xi) - vertical bar xi vertical bar(2) partial derivative(2)/partial derivative eta(2) be the Grushin operator on R-n x R. We prove that the Riesz transforms associated to this operator are bounded on L-p(Rn+1), 1 < p < infinity, and their norms are independent of dimension n.

A novel method for studying the buckling of nanotubes considering geometrical imperfections

Buckling of nanotubes has been studied using many methods such as molecular dynamics (MD), molecular mechanics, and continuum-based shell theories. In MD, motion of the individual atoms is tracked under applied temperature and pressure, ensuring a reliable estimate of the material response. The response thus simulated varies for individual nanotubes and is only as accurate as the force field used to model the atomic interactions. On the other hand, there exists a rich literature on the understanding of continuum mechanics-based shell theories. Based on the observations on the behavior of nanotubes, there have been a number of shell theory-based approaches to study the buckling of nanotubes. Although some of these methods yield a reasonable estimate of the buckling stress, investigation and comparison of buckled mode shapes obtained from continuum analysis and MD are sparse. Previous studies show that the direct application of shell theories to study nanotube buckling often leads to erroneous results. The present study reveals that a major source of this error can be attributed to the departure of the shape of the nanotube from a perfect cylindrical shell. Analogous to the shell buckling in the macro-scale, in this work, the nanotube is modeled as a thin-shell with initial imperfection. Then, a nonlinear buckling analysis is carried out using the Riks method. It is observed that this proposed approach yields significantly improved estimate of the buckling stress and mode shapes. It is also shown that the present method can account for the variation of buckling stress as a function of the temperature considered. Hence, this can prove to be a robust method for a continuum analysis of nanosystems taking in the effect of variation of temperature as well.

Mixed norm estimates for the Riesz transforms on the Heisenberg group

In this paper we prove weighted mixed norm estimates for Riesz transforms on the Heisenberg group and Riesz transforms associated to the special Hermite operator. From these results vector-valued inequalities for sequences of Riesz transforms associated to generalised Grushin operators and Laguerre operators are deduced.

MIXED NORM ESTIMATES FOR THE RIESZ TRANSFORMS ON SU (2)

In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.

Size Effect and Geometrical Effect of Polycrystals and Thin Film/Substrate System in Micro-indentation Test

Micro-indentation test at scales on the order of sub-micron has shown that the measured hardness increases strongly with decreasing indent depth or indent size, which is frequently referred to as the size effect. Simultaneously, at micron or sub-micron scale, the material microstructure size also has an important influence on the measured hardness. This kind of effect, such as the crystal grain size effect, thin film thickness effect, etc., is called the geometrical effect by here. In the present research, in order to investigate the size effect and the geometrical effect, the micro-indentation experiments are carried out respectively for single crystal copper and aluminum, for polycrystal aluminum, as well as for a thin film/substrate system, Ti/Si3N4. The size effect and geometrical effect are displayed experimentally. Moreover, using strain gradient plasticity theory, the size effect and the geometrical effect are simulated. Through comparing experimental results with simulation results, length-scale parameter appearing in the strain gradient theory for different cases is predicted. Furthermore, the size effect and the geometrical effect are interpreted using the geometrically necessary dislocation concept and the discrete dislocation theory. Member Price: $0; Non-Member Price: $25.00

Size Effect and Geometrical Effect of Solids in Micro-Indentation Test

Micro-indentation tests at scales of the order of sub-micron show that the measured hardness increases strongly with decreasing indent depth or indent size, which is frequently referred to as the size effect. At the same time, at micron or sub-micron scale, another effect, which is referred to as the geometrical size effects such as crystal grain size effect, thin film thickness effect, etc., also influences the measured material hardness. However, the trends are at odds with the size-independence implied by the conventional elastic-plastic theory. In the present research, the strain gradient plasticity theory (Fleck and Hutchinson) is used to model the composition effects (size effect and geometrical effect) for polycrystal material and metal thin film/ceramic substrate systems when materials undergo micro-indenting. The phenomena of the "pile-up" and "sink-in" appeared in the indentation test for the polycrystal materials are also discussed. Meanwhile, the micro-indentation experiments for the polycrystal Al and for the Ti/Si_3N_4 thin film/substrate system are carried out. By comparing the theoretical predictions with experimental measurements, the values and the variation trends of the micro-scale parameter included in the strain gradient plasticity theory are predicted.

Tuning The Geometrical Parameters Of Biomimetic Fibrillar Structures To Enhance Adhesion

Fibrillar structures are common features on the feet of many animals, such as geckos, spiders and flies. Theoretical analyses often use periodical array to simulate the assembly, and each fibril is assumed to be of equal load sharing (ELS). On the other hand, studies on a single fibril show that the adhesive interface is flaw insensitive when the size of the fibril is not larger than a critical one. In this paper, the Dugdale Barenblatt model has been used to study the conditions of ELS and how to enhance adhesion by tuning the geometrical parameters in fibrillar structures. Different configurations in an array of fibres are considered, such as line array, square and hexagonal patterns. It is found that in order to satisfy flaw-insensitivity and ELS conditions, the number of fibrils and the pull-off force of the fibrillar interface depend significantly on the fibre separation, the interface interacting energy, the effective range of cohesive interaction and the radius of fibrils. Proper tuning of the geometrical parameters will enhance the pull-off force of the fibrillar structures. This study may suggest possible methods to design strong adhesion devices for engineering applications.