Harmonic Analysis on Internally Gelfand Pairs Associated to Groupoids

Let $${\mathcal {G}}$$ be a topological locally compact Hausdorff and second countable groupoid with a Haar system and $${\mathcal {K}}$$ a proper subgroupoid of $${\mathcal {G}}$$ with a Haar system too. $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair if for any u in the unit space, the pair of isotropy groups $$({\mathcal {G}}(u), {\mathcal {K}}(u))$$ is a Gelfand pair. In this work, we prove, when $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair that any representation of $${\mathcal {C}}^{*} ({\mathcal {G}}, {\mathcal {K}},\lambda , \alpha )$$ induced from a positive definite spherical function on an isotropy group is irreducible. We introduce also the notion of spherical bundle and after identifying it with the spectrum of some $$C^*$$ -algebra, we obtain some properties of its topology.

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable sigma-C*-algebras. We also relate equivariant vector bundles to these sigma-C*-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. Mostly we work in the generality of locally compact groupoids with Haar system.

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems.We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter—Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.

ABSTRACTThis article presents the results of measuring the concentrations of ground-level ozone (O3) in ambient air and meteorological parameters (atmospheric pressure, temperature, wind speed, and relative humidity during the year 2008. The measurements were performed at measuring station located near Campo Grande, away from local sources of O3. Fourier analysis, Principal Component Analysis (PCA) and Multiple Linear Regression (MLR) were used as tools for mathematical analysis of the indicated occurrence. The results of Fourier analysis indicate that not only the daily cycles but also the weekly and monthly cycles are visible in all periods as well as the 12-h cycles. The statistical analysis of the obtained data, based on PCA have shown that approximately 60% of the variance in the measured values can be described with two factors. The attempt of O3 distribution modeling, using MLRA, resulted in a linear model whose coefficient of determination was equal to R2 = 0.98. The results of the study indicate that all these methods are useful for understanding the impact of meteorological parameters on variations of ozone concentration over time.

The combined use of order tracking techniques for enhanced Fourier analysis of order components

Introduction A basic result of standard harmonic analysis on ℝ is that every 2π-periodic function can be expanded into an orthonormal basis of trigonometric polynomials of the type {exp( ikx ): k = 0, ±1, ±2,…}. This fundamental fact is exploited in several branches of applied mathematics: in particular – as we shall mention more extensively in the subsequent chapters – it is a main staple in the analysis of stationary stochastic processes. A proof can be provided by elementary techniques, requiring little more than standard calculus – see for instance Rudin [174, p. 88]. It is also well-known (see e.g. [63]) that one can deduce a similar result for fields defined on the sphere S 2 , where the appropriate orthonormal basis is given by the class of spherical harmonics . The aim of this introductory chapter is to put these two results into the broader framework of abstract harmonic analysis on compact groups. At first sight, this choice may seem unduly complicated, but it is indeed necessary for the purposes we are aiming at. On one hand, our standpoint allows us to develop quite straightforwardly Fourier analysis on the sphere. On the other hand, group representation ideas are necessary to understand most of the material presented in the subsequent parts of the book, namely: the characterizations of isotropy, the explicit computation of higher order moments and angular polyspectra, the derivation of high-frequency asymptotic properties, and the discussion of spin ideas at the very end of the book.

In this work, we investigate the subcritical behaviour of a coaxial jet subject to small-amplitude perturbations at the inflow. We use the results of optimal harmonic analysis and dynamic-mode decomposition (DMD) of the flow fields at a Reynolds number, based on the diameter and maximum velocity of the inner inlet pipe, of $Re=200$, to show that, for a sufficiently low value of the Reynolds number, the coherent structures appearing in the perturbed dynamics of the nonlinear system can be effectively described in terms of the harmonic response of the flow. We also show that, for larger subcritical values of the Reynolds number, $Re=400$, a huge amplification of disturbances quickly makes nonlinear effects relevant. Large-scale, near-field coherent dynamics can be still interpreted as an evidence of the preferred response of the system, using DMD of the flow to describe the noise-driven transition to turbulence downstream. The influence of the axial velocity ratio and the rotational motion of the outer stream are assessed as well. Harmonic analysis successfully predicts the prevalence of rotating helical structures observed in the columnar flow for moderate swirl of the outer jet. Finally, we compare the receptivity of the nonlinear system to the optimal linear perturbations with its response to stochastic forcing. Optimal forcing is still more effective than white noise in driving the system to a turbulent state, where nonlinear dynamics prevails. We still conclude that linear optimal forcing may be relevant in investigating the transition to turbulence in coaxial jets, even if more about the transition process could be learnt from a more expensive nonlinear analysis.

In this paper, we consider numerical and trigonometric series with a very general monotonicity condition. First, a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting. In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.

In this work we extend the first part of the previous paper [F4] to higher dimensional local fields. We introduce a nontrivial translation invariant measure on the additive group of higher dimensional local fields, and then develop elements of integration and harmonic analysis. We also discuss its relation with several other measure theories. For an n-dimensional local field F a translation invariant measure μ is defined on a certain ring A of measurable sets and takes values in R ((X1)) . . . ((Xn−1)) (which itself is an n-dimensional local field whose last residue field is the archimedean field R). The algebra A in the case of higher dimensional fields with finite last residue field is the algebra generated by characteristic functions of shifts of fractional ideals, i.e. a + bO with a, b ∈ F and O the ring of integers of F with respect to the n-dimensional structure. The measure is countably additive in a refined sense (see sections 7 and 8). Elements of integration theory are introduced in sections 9–13. The additive group of a higher dimensional field has a certain topology on it with respect to which it is not locally compact for n > 1. This topology plays a key role in higher class field theory [F1–F3]. The additive group of F is self dual, which together with the measure and integration leads to analogs of many classical results in Fourier analysis (section 14). In particular, for functions in certain space we introduce their transform

Fourier–Stieltjes Algebras of Locally Compact Groupoids

We theoretically investigate a new type of piezoelectric actuator with tubular interdigitated electrodes. A partial model is adopted due to the axial symmetry of the actuator, which is divided into non-overlapping opposing electrode pairs on the center plane of the branch electrode. We present a finite element model of actuator and analyze its static-dynamic properties by using the commercially available finite element software ABAQUS. The length-to-diameter ratio of the tubular actuator can be as larger as necessary, so we are more concerned with the axial mechanical properties of the actuator. Thus we studied the influence of the electrode’s shape and structure size on the clamping stress and free strain of the partial models. We also analyzed an ordinary piezoelectric model of the same size. We demonstrated that the partial model displayed significantly improved axial performance over the ordinary one. The axial stress of the partial model can reach 2.75 times that of the ordinary one, and the axial free strain can reach 1.55 times that of the ordinary one. According to the results of the static analysis, appropriate electrodes and structural parameters were selected to establish a tubular actuator model, with which modal and harmonic response analysis were performed. A modal analysis was used to obtain the frequencies and the shapes of the axial vibration. Each pair of electrodes is shunted to a sinusoidal varying voltage. Using the results of the harmonic response analysis, we obtained the actuator’s axial resonance curve.

Topological string in harmonic space and correlation functions in [formula omitted] stringy cosmology

It is well known that the vibrational behavior of a mistuned bladed disk differs strongly from that of a tuned bladed disk. A large number of publications dealing with the dynamics of mistuned bladed disks are available in the literature. The vibrational phenomena analyzed in these publications are either forced vibrations or self-excited flutter vibrations. Nearly, all published literature on the forced vibrations of mistuned blades disks considers harmonic, i.e., steady-state, vibrations, whereas the self-excited flutter vibrations are analyzed by the evaluation of the margin against instabilities by means of a modal, or rather than eigenvalue, analysis. The transient forced response of mistuned bladed disk is not analyzed in detail so far. In this paper, a computationally efficient mechanical model of a mistuned bladed disk to compute the transient forced response is presented. This model is based on the well-known fundamental model of mistuning (FMM). With this model, the statistics of the transient forced response of a mistuned bladed disk is analyzed and compared to the results of harmonic forced response analysis.

Based on the characters that the piston rod of labyrinth reciprocating compressors is supported by a guide bearing system in the direction of movement, simplified mechanical model of piston rod is established. To determine the effect of relevant parameters on the vibration and stability of piston rod, modal analysis, harmonic analysis and buckling analysis are performed with ANSYS software. The Analysis result is used for the optimization of the guide support system of piston rod, and good result is obtained.

Carbon monoxide (CO) total columns over European Russia (ER) and western Siberia (WS) have been analyzed using MOPITT (V5, TIR/NIR, L3) IR-radiometer data obtained in 2000–2014. High CO contents are revealed over large urban and industrial agglomerations and over regions of oil-and-gas production. A stable local CO maximum is observed over the Moscow agglomeration. Statistical characteristics of CO total columns observed in the atmosphere over ER and WS in 2000–2014 are presented. An analysis of long-term changes in CO content reveals nonlinear changes in the CO total column over northern Eurasia in 2000–2014. Results of a comparative analysis of annual variations in atmospheric CO contents over ER and WS are given. Based on Fourier analysis, empirical models of annual variations in total CO contents over ER and WS are proposed. Relations between regional CO contents and fire characteristics and between spatial CO distributions and features of large-scale atmospheric dynamics under conditions of weather and climate anomalies in the summers of 2010 in ER and 2012 in WS are analyzed. Data on total CO contents measured with a MOPITT satellite radiometer and a ground-based spectrometer operating at the Zvenigorod Scientific Station of the Obukhov Institute of Atmospheric Physics are compared.