Narrow Results

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator $A$ defined in a Banach space $E$, in which by choosing $E$ and $A$ we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.

We establish a regularity result for stochastic heat equations in probabilistic evolution spaces of Besov type and we use it to prove a global in time existence and uniqueness of solution to a stochastic magnetohydrodynamics equation. The existence result holds with a positive probability which can be made arbitrarily close to one. The work is carried out by blending harmonic analysis tools such as Littlewood–Paley decomposition, Jean–Micheal Bony paradifferential calculus and stochastic calculus. The law of large numbers is a key tool in our investigation. Our global existence result is new in three-dimensional spaces.

The dynamical behavior of nonlinear maps undergoing degenerate period doubling or degenerate Hopf bifurcations is studied via a frequency-domain approach. The technique is based on a discrete-time feedback representation of the system and the application of the well-known engineering tools of harmonic balance to approximate the emerging solutions. More specifically, the results are a higher-order extension of the previous developments obtained by the authors for nondegenerate bifurcations. Two examples are included for illustration.

In this paper, we describe a numerical continuation method that enables harmonic analysis of nonlinear periodic oscillators. This method is formulated as a boundary value problem that can be readily implemented by resorting to a standard continuation package — without modification — such as AUTO, which we used. Our technique works for any kind of oscillator, including electronic, mechanical and biochemical systems. We provide two case studies. The first study concerns itself with the autonomous electronic oscillator known as the Colpitts oscillator, and the second one with a nonlinear damped oscillator, a nonautonomous mechanical oscillator. As shown in the case studies, the proposed technique can aid both the analysis and the design of the oscillators, by following curves for which a certain constraint, related to harmonic analysis, is fulfilled.

This short survey deals with the large-scale geometry of solvable groups. Instead of giving a global overview of this wide subject, we chose to focus on three aspects which illustrate the broad diversity of methods employed in this subject. The first one has probabilistic and analytic flavors, the second is related to cohomological properties of unitary representations, while the third one deals with the Dehn function. To keep the exposition concrete, we discuss lots of examples, mostly among solvable linear groups.

Periodicity is recurrent in nature and society. The problem of the periodicity is faced here with the support of fractal methodology. Some analytical tools for the understanding of the one-dimensional projections of periodic phenomena are proposed. The reconstruction of an unknown periodic sampled variable is approached, assuming a deterministic self-affine nature in its small oscillations. Fractal trigonometric polynomials are defined by means of suitable iterated function systems. These objects are fractal perturbations of the classical circular functions. The coefficients of the system enable the control and modification of the properties of the originals. Additionally, Fourier parameters and approximants for sampled signals are computed and the density of the fractal trigonometric polynomials in the most common spaces of periodic functions is proved.

A study of the quality and accuracy of the methods, based on frequency analysis, for fractal characterization of surfaces is carried out. The study is based on computer simulation of fractal surfaces images and then a discussion of the mathematical algorithms used for computer generation of fractal surfaces is also made. The main conclusion is that the studies of fractal characterization by frequency analysis reported in previous papers as well as the conclusions about the performance of the different methods are doubtful and questionable. We propose new methods for frequency analysis that in some cases yield the more accurate results.

We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.

Starting with a complex Hilbert space, using inductive limits, we build Lie algebras, and find families of representations. They include those often studied in mathematical physics in order to model quantum statistical mechanics or quantum fields. We explore natural actions on infinite tensor algebras T(H) built with a functorial construction, starting with a fixed Hilbert space H.

While our construction applies also when H is infinite-dimensional, the case with N ≔ dim H finite is of special interest as the symmetry group we consider is then a copy of the non-compact Lie group U(N, 1). We give the tensor algebra T(H) the structure of a Hilbert space, i.e. the unrestricted infinite tensor product Fock space . The tensor algebra T(H) is naturally represented as acting by bounded operators on , and U(N, 1) as acting as a unitary representation. From this we built a covariant system, and we explore how the fermion, the boson, and the q on Hilbert spaces are reduced by the representations. In particular we display the decomposition into irreducible representations of the naturally defined U(N, 1) representation.

We study reduction schemes for functions of “many” variables into system of functions in one variable. Our setting includes infinite dimensions. Following Cybenko–Kolmogorov, the outline for our results is as follows: We present explicit reduction schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in “many” variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation.

Partially motivated by a class of nonlinear equations of interest for string theory, cosmology and p-adic analysis, we continue our research on Sobolev spaces on arbitrary locally compact abelian (LCA) groups. In this paper we focus on an analog for LCA groups of the classical Rellich compactness theorem. As an application, we prove the existence of continuous solutions to a generalized Euclidean bosonic string equation posed on a compact abelian group.

This paper concerns the storm surge calculation based on the algorithm of ensemble empirical mode decomposition (EEMD). An accurate storm surge result is key information for coastal disaster warning and prevention. Separation of storm surge magnitude from sea level data has typically been done by specifying tidal input from main tidal harmonics. Obtaining accurate storm surge magnitude with harmonic analysis (HA) requires at least one month. This study discusses possible storm surge separation from short-term sea level time series using EEMD. The current work reveals that EEMD is predominant for short-term sea level data analysis shorter than thirty-five days. Due to different residues obtained from EEMD, this work proposes a method to determine most ideal residue for representing the storm surge.

The paper investigates the dynamic behavior of thin-walled reservoirs containing an ideal liquid taking into account the effects of hydroelastic interaction and sloshing. A mathematical statement of the problem is based on the principle of virtual displacements, which accounts for the pre-stressed non-deformed state of the shell caused by various force factors. The behavior of compressible liquid is described by linearized Euler equations, which are transformed by the Bubnov–Galerkin method. The dynamics of partially filled circular and elliptical cylindrical reservoirs are investigated numerically using a finite element procedure. It has been shown that allowing for sloshing considerably reduces eigenfrequencies of vibrations of the examined systems but has inessential effect on the displacements of the structure under non-stationary loads. Based on the modal analysis we present a classification of eigenmodes of free surface oscillations of a liquid in vertical tanks. It has been found that due to consideration of the sloshing effect, the frequency spectrum of the system can split into two parts in the case when the vibration frequencies of liquid differ from the vibration frequencies of an empty shell. Moreover, under harmonic excitation consideration of liquid sloshing leads to a more complicated amplitude-frequency curve characterized by displacement jumps.

We expand the Bochner technique on the following Lebedev-Stieltjes integrals

The recently proposed empirical wavelet transform was based on a particular type of filter. In this paper, we aim to propose a general framework for the construction of empirical wavelet systems in the continuous case. We define a well-suited formalism and then investigate some general properties of empirical wavelet systems. In particular, we provide some sufficient conditions to the existence of a reconstruction formula. In the second part of the paper, we propose the construction of empirical wavelet systems based on some classic mother wavelets.

We consider the problem of calculating the pseudorotational energy levels associated with the molecular Jahn-Teller effect for electronic triplets. We show that the energy levels are related to the spectra of Laplace-Beltrami operators on line bundles over the leaves of Cartan's isoparametric foliation of S⁴. These spectra are computed explicitly using the Peter-Weyl theorem and harmonic analysis on line bundles.

The following sections are included:

• REMINISCENCES

• RESEARCH
  • High Indices
  • Reflexivity and Summability
  • Harmonic Analysis
  • Change of Variable
  • Fourier Series and Generalized Variation
  • Representation of Functions, Orthogonal Series
  • Real Analysis
  • Summability
  • Survey Papers

• PUBLICATIONS

• DOCTORAL STUDENTS

The crankshaft is the most important moving part of a reciprocating compressor. As such, it plays a key role in the petrochemical industry. Frequently, the crankshaft cracks and the crank bearing scuffs because of torsional vibration. Therefore, further study on crankshaft vibration is needed to promote development of the high-speed reciprocating compressor. The Modal and Harmonic analysis of the crankshaft was done with the Finite Element Method. The first 6 modals of the crankshaft were calculated using the Lanzos method, and the 3rd and 6th modal shapes of the shaft are torsional. The results showed that bending, torsion and combinations of both are modal shapes of the shaft. Torsional resonance is the main form of shafting dynamic characteristics. Through the harmonic analysis, it found that the response frequency of maximum stress and distortion is 140Hz, and that maximum stress occurs at the transition of two cranks or crankpin junctions. These conclusions can provide some guidance for structure design and for solving the default problem for reciprocating compressor.

This paper proposes a bottom-up, probabilistic harmonic assessment technique for residential loads. The modeling, divided into two levels, uses the Markov-Chain Monte Carlo (MCMC) method to build harmonic profiles of 24h. At the top, Monte Carlo controls the household size, the ownership of electric devices and other factors that result in different use patterns of home appliances between families. At the bottom, Markov-Chain is used to determine the starting time of each appliance in individual household. A varying harmonic equivalent circuit based on human behavior representing a residential house, combining user activity model with corresponding electric harmonic model, is thus derived. The total harmonic produced by residential loads is carried out by accumulating multiple residential houses supplied with residential feeders. Field measurements have confirmed the validity of the proposed technique.

Yves Meyer has made numerous contributions to mathematics, several of which will be reviewed here, in particular in number theory, harmonic analysis and partial differential equations.

His work in harmonic analysis led him naturally to take an interest in wavelets, when they emerged in the early 1980s; his synthesis of the advanced theoretical results in singular integral operator theory, established by himself and others, and of the requirements imposed by practical applications, led to enormous progress for wavelet theory and its applications. Wavelets and wavelet packets are now standard, extremely useful tools in many disciplines; their success is due in large measure to the vision, the insight and the enthusiasm of Yves Meyer.