Symplectic analysis of time-frequency spaces

We present a different symplectic point of view in the definition of weighted modulation spaces and weighted Wiener amalgam spaces. All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the τ-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions \mu(A)(f\otimes\bar g), where \mu(A) is the metaplectic operator and A is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications

Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gr\"ochenig and Rzeszotnik in [24] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators [11], which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schr\"odinger equation with bounded perturbations, cf. [7].Comment: 26 page

Metaplectic Gabor Frames and Symplectic Analysis of Time-Frequency Spaces

We introduce new frames, called \textit{metaplectic Gabor frames}, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions. Namely, we develop the theory of metaplectic atoms in a full-general setting and prove an inversion formula for metaplectic Wigner distributions on $\mathbb{R}^d$. Its discretization provides metaplectic Gabor frames. Next, we deepen the understanding of the so-called shift-invertible metaplectic Wigner distributions, showing that they can be represented, up to chirps, as rescaled short-time Fourier transforms. As an application, we derive a new characterization of modulation and Wiener amalgam spaces. Thus, these metaplectic distributions (and related frames) provide meaningful definitions of local frequencies and can be used to measure effectively the local frequency content of signals

Wigner Representation of Schr\"odinger Propagators

We perform a Wigner analysis of Fourier integral operators (FIOs), whose main examples are Schr\"odinger propagators arising from quadratic Hamiltonians with bounded perturbations. The perturbation is given by a pseudodifferential operator $\sigma(x,D)$ with symbol in the H\"ormander class $S^0_{0,0}(\mathbb{R}^{2d})$. We compute and study the Wigner kernel of these operators. They are special instances of a more general class of FIOs named $FIO(S)$, with $S$ the symplectic matrix representing the classical symplectic map. We shall show the algebra and the Wiener's property of this class. The algebra will be the fundamental tool to represent the Wigner kernel of the Schr\"odinger propagator for every $t\in\mathbb{R}^d$, also in the caustic points. This outcome underlines the validity of the Wigner analysis for the study of Schr\"odinger equations.Comment: A more general definition of Wigner kernel is given, the readability of some proofs has been improved, and misprints have been correcte

Wigner Analysis of Fourier Integral Operators with symbols in the Shubin classes

We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes $\Gamma^m(\mathbb{R^{2d}})$, with negative order $m$. The phases considered are the so-called tame ones, which appear in the Schr\"odinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the H\"ormander's class $S^{0}_{0,0}(\mathbb{R^{2d}})$. Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order $m$.Comment: 21 page

Biochemical Alterations in Semen of Varicocele Patients: A Review of the Literature

Oxidative stress is a mechanism underlying different kinds of infertility in human males. However, different results can be observed in relation to the method used for its evaluation. Varicocele patients show a number of biochemical abnormalities, including an altered distribution of coenzyme Q between seminal plasma and sperm cells and also an apparent defect in the utilization of antioxidants. Moreover, an influence of systemic hormones on seminal antioxidant system was observed too. Finally, the effects of surgical treatment on oxidativestress indexes and the possible usefulness of some medical therapies, like coenzyme Q supplementation, are discussed. In conclusion, published data show a role of oxidative stress in varicocele-related male infertility, but at present we do not know the precise molecular mechanisms underlying these phenomena

Beyond ‘BRICS’: ten theses on South–South cooperation in the twenty-first century

Grounded in a review of past and present academic South–South cooperation literatures, this article advances ten theses that problematise empirical, theoretical, conceptual and methodological issues essential to discussions of South–South cooperation in the 21st century. This endeavour is motivated by the perceived undermining, especially in the contemporary Anglophone academic South–South cooperation literature, of the emancipatory potential historically associated with South–South cooperation. By drawing on the interventionist South–South cooperation agendas of ‘left’-leaning Latin America-Caribbean governments, the article seeks to establish a dialogue between social science theories and less ‘visible’ analyses from academic (semi)peripheries. The ten theses culminate in an exploration of the potential of South–South cooperation to promote ‘alternative’ development

Excursus on modulation spaces via metaplectic operators and related time-frequency representations

We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the sub- manifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces