Search Results (42)

We develop a non-Archimedean framework for the representation and analysis of genomic sequences based on the arithmetic and geometric structure of the ring of 5-adic integers. The proposed approach associates RNA sequences with points in a compact ultrametric space through an injective symbolic-to-arithmetic embedding that transforms genomic information into a hierarchical geometric object. We prove that the embedding is a global isometry between a natural symbolic prefix metric and the induced 5-adic metric, and we show that its image forms a compact Cantor-type subset of ${\mathbb{Z}}_5$. Building upon this representation, we formulate a continuous-time evolutionary model governed by a Vladimirov pseudo-differential operator. The resulting non-Archimedean diffusion equation provides a mathematically rigorous mechanism for describing evolutionary transitions across hierarchical genomic scales and admits an explicit fundamental solution obtained through 5-adic Fourier analysis. We further introduce a finite-resolution projection onto quotient rings of ${\mathbb{Z}}_5$ and develop an alignment-free phylogenetic inference framework based directly on the 5-adic valuation. The induced distance function is ultrametric and naturally encodes hierarchical relationships through shared symbolic prefixes. The proposed construction establishes a bridge between p-adic analysis, ultrametric geometry, pseudo-differential operators, and computational phylogenetics. As an illustration, we discuss its application to Hantavirus genomic sequences, demonstrating how hierarchical evolutionary organization can be represented within a unified non-Archimedean mathematical framework. Full article

We study the phase-space dynamics of quantum systems with $\mathrm{SU}(1,1)$ group symmetry using coherent-state representations on the Poincaré disk. The resulting evolution equation combines transport terms with nonlocal contributions generated with the spectral functions of the Casimir operator, which admit a natural interpretation as pseudodifferential operators associated with the hyperbolic Laplace–Beltrami operator. Using this pseudodifferential structure, we classify the phase-space generators according to the type of the underlying PDE: compact quadratic dynamics ($\widehat{H}\propto {\widehat{K}}_0^2$) yield a degenerate hyperbolic operator of the transport type, and noncompact dynamics ($\widehat{H}\propto {\widehat{K}}_2^2$) give rise to a mixed-order differential–pseudodifferential operator. For numerical evaluation, we reformulate the propagator as an oscillatory integral and develop two complementary strategies: a Fourier-series reduction exploiting the periodicity of compact orbits and a Levin-type spectral collocation method for the noncompact case. Both approaches are stable, accurate, and free of the stiffness issues that afflict direct PDE evolution on the Poincaré disk. Full article

We extend the classical fractal uncertainty principle (FUP) framework in time-frequency analysis by exploring several novel directions. First, we generalize the FUP beyond the classical Gaussian window by investigating non-Gaussian windows and the corresponding generalized Fock space techniques. Second, we develop uncertainty estimates in alternative joint representations, including the continuous wavelet transform and directional representations such as shearlets. Third, we study fractal uncertainty on random and anisotropic fractal sets, providing probabilistic and geometric refinements of the FUP. Fourth, we connect these results with semiclassical and microlocal analysis, thereby elucidating the role of fractal geometry in resonance theory and pseudodifferential operators. Finally, we extend the analysis beyond Gaussian Gabor multipliers by considering non-Gaussian generating functions and irregular lattice samplings. Our results yield new operator norm estimates and spectral properties, with potential applications in signal processing, quantum mechanics, and numerical analysis. Full article

The Dirac factorization method (DFM) is the key feature of the present investigation. It is addressed to the relevant use in diverse fields of research, regarding, e.g., the handling of pseudo-operators arising in quantum mechanics and fractional calculus. We explore the role that the factorization plays in a classical context too, including the study of d’Alembert and Laplace equations. Its strong entanglement with the Cauchy–Riemann conditions and with complex analysis is discussed. We complete our study with the extension of DFM to second-order ordinary differential equations, to classical analytical mechanics, and to higher-order partial differential equations. Full article

Lie-algebraic Poisson structures, related to the superalgebra of super-pseudodifferential operators on the circle over the even component of the ${\mathbb{Z}}_2$-graded Grassmann algebra, have been studied in detail; the corresponding coadjoint orbits, generated by the Casimir invariants, regarding the different superalgebra splittings into the subalgebras, are analyzed. The related Lax-type completely integrable Hamiltonian flows are constructed on suitably defined functional manifolds with respect to the canonical super-Lie–Poisson structures on them. An approach was proposed allowing the extension of the related coadjoint flows by means of the respectively constructed super-evolution flows on the adjoint super-subalgebras, specially deformed by means of super-pseudodifferential operator elements, depending on the generalized eigenfunctions of the corresponding super-linear Lax-type spectral problem. As a consequence, it is stated that all constructed new coadjoint superflows generate on the suitably extended supermanifolds Lax-type integrable Hamiltonian systems. The centrally extended super-Lie-algebraic structures have been analyzed and the related coadjoint orbits described, generated by the corresponding Casimir invariants and coinciding with integrable Hamiltonian systems on suitably defined supermanifolds. Full article

On a compact n-dimensional Riemannian manifold without boundary $(M,g)$, it is well-known that the $L^2$-normalized Laplace eigenfunctions with semiclassical parameter h satisfy the universal $L^{\infty }$ growth bound of $O(h^{{\displaystyle \frac{1-n}{2}}})\mathrm{as}h\to 0^{+}.$ In the context of a quantum completely integrable system on M, which consists of n commuting self-adjoint pseudodifferential operators $P_1\left(h\right),\dots ,P_n\left(h\right)$, where $P_1\left(h\right)=-h^2{\mathrm{\Delta }}_g+V\left(x\right)$, Galkowski-Toth showed polynomial improvements over the standard $O(h^{{\displaystyle \frac{1-n}{2}}})$ bounds for typical points. Specifically, in the two-dimensional case, such an improved upper bound is $O(h^{-1/4})$. In this study, we aim to further enhance this bound to $O\left(\right|lnh{|}^{1/2})$ at the points where a strictly monotonic condition is satisfied. Full article

In the present paper we define and study a new wavelet transformation associated to the linear canonical Dunkl transform (LCDT), which has been widely used in signal processing and other related fields. Then we define and study a class of pseudo-differential operators known as time-frequency (or localization) operators and we give criteria for its boundedness and Schatten class properties. Full article

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called “naive” approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis. Full article

This paper presents a low-power 60 GHz low-noise amplifier (LNA) designed for Gbit/s applications using 28 nm CMOS technology. The LNA exploits a single-stage pseudo-differential architecture with integrated input transformer for both electrostatic discharge (ESD) protection and simultaneous noise/impedance matching. An effective power-constrained design strategy is adopted to pursue the lowest current consumption at the minimum noise figure (NF), with the best tradeoff between gain and frequency bandwidth. The LNA, which has been designed to drive an on–off keying (OOK) demodulator, is operated at a supply voltage as low as 0.9 V and achieves a voltage gain of about 21 dB with a 3 dB bandwidth of 2 GHz around 60 GHz. Thanks to the proper impedance transformation at the 60 GHz input, the amplifier exhibits an NF of 6.3 dB, also including the input transformer loss with a very low power consumption of about 5 mW. The adoption of a single-stage topology also allows an excellent input 1 dB compression point (IP_1dB) of −4.7 dBm. The input transformer guarantees up to 2 kV human body model (HBM) ESD protection. Full article

In this paper, a rail-to-rail transconductance stable and enhanced ultra-low-voltage operational transconductance amplifier (OTA) is proposed for electrocardiogram (ECG) signal processing. The variation regularity of the bulk transconductance of pMOS and nMOS transistors and the cancellation mechanism of two types of transconductance variations are revealed. On this basis, a transconductance stabilization and enhancement technique is proposed. By using the “current-reused and transconductance-boosted complementary bulk-driven pseudo-differential pairs” structure, the bulk-driven pseudo-differential pair during the input common-mode range (ICMR) is stabilized and enhanced. The proposed OTA based on this technology is simulated using the TSMC 0.18 μm process in a Cadence environment. The proposed OTA consumes a power below 30 nW at a 0.4 V voltage supply with a DC gain of 54.9 dB and a gain-bandwidth product (GBW) of 14.4 kHz under a 15 pF capacitance load. The OTA has a high small signal figure-of-merit (FoM) of 7410 and excellent common-mode voltage (VCM) stability, with a transconductance variation of about 1.35%. Based on a current-scaling version of the proposed OTA, an OTA-C low-pass filter (LPF) for ECG signal processing with VCM stability is built and simulated. With a −3 dB bandwidth of 250 Hz and a power consumption of 20.23 nW, the filter achieves a FoM of 3.41 × 10⁻¹³, demonstrating good performance. Full article

A low-noise amplifier (LNA) design with the operation of concurrent dual-band for Global Navigation Satellite System (GNSS) receivers with single channel is presented in this work. This LNA structure has an inductively degenerated cascode architecture and is pseudo-differential, operating at two frequencies simultaneously (1.2 GHz and 1.57 GHz). Two noise reduction/cancellation techniques, using load capacitor and feedforward path, respectively, are proposed resulting in an excellent improvement in the noise figure (NF). The input matching circuit uses both series and parallel resonant components to enable concurrency. The adopted pseudo-differential structure results in input balun elimination. Inductively degenerated cascode topology provides both input impedance and optimum noise impedance matching. The soundness of the proposed approach has been demonstrated in a 0.18-µm CMOS technology by TSMC. Simulation results show that at 1.2 GHz and 1.57 GHz the LNA achieves −13 dB and −11 dB of input matching, 24.6 dB and 24.7 dB of gain, 1.47 dB and 1.43 dB of NF, respectively. The input-referred 1-dB compression point (IP_1dB) is around −16 dBm, while the input-referred third-order intercept point (IIP₃) achieves −2.2 dBm at 1.2 GHz and −0.6 dBm at 1.57 GHz. The LNA draws about 13 mA from a 1.8-V supply voltage. Full article

The description of the electron–electron interactions in two-dimensional materials has a dimensional mismatch, where electrons live in (2 + 1)D while photons propagate in (3 + 1)D. In order to define an action in (2 + 1)D, one may perform a dimensional reduction of quantum electrodynamics in (3 + 1)D (QED4) into pseudo-quantum electrodynamics (PQED). The main difference between this model and QED4 is the presence of a pseudo-differential operator in the Maxwell term. However, besides the Coulomb repulsion, electrons in a material are subjected to several microscopic interactions, which are inherent in a many-body system. These are expected to reduce the range of the Coulomb potential, leading to a short-range interaction. Here, we consider the coupling to a scalar field in PQED for explaining such a mechanism, which resembles the spontaneous symmetry breaking (SSB) in Abelian gauge theories. In order to do so, we consider two cases: (i) by coupling the quantum electrodynamics to a Higgs field in (3 + 1)D and, thereafter, performing the dimensional reduction; and (ii) by coupling a Higgs field to the gauge field in PQED and, subsequently, calculating its effective potential. In case (i), we obtain a model describing electrons interacting through the Yukawa potential and, in case (ii), we show that SSB does not occur at one-loop approximation. The relevance of the model for describing electronic interactions in two-dimensional materials is also addressed. Full article

In this paper, I explore a specific class of bi-parameter pseudo-differential operators characterized by symbols $\sigma (x_1,x_2,{\xi }_1,{\xi }_2)$ falling within the product-type Hörmander class ${\mathbf{S}}_{\rho ,\delta }^m$. This classification imposes constraints on the behavior of partial derivatives of $\sigma$ with respect to both spatial and frequency variables. Specifically, I demonstrate that for each multi-index $\alpha ,\beta$, the inequality $|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }\sigma (x_1,x_2,{\xi }_1,{\xi }_2)|\le {\mathcal{C}}_{\alpha ,\beta }{(1+|\xi \left|\right)}^m{\prod }_{i=1}^2(1+|{\xi }_i{\left|\right)}^{-\rho |{\alpha }_i|+\delta |{\beta }_i|}$ is satisfied. My investigation culminates in a rigorous analysis of the $L^p$-boundedness of such pseudo-differential operators, thereby extending the seminal findings of C. Fefferman from 1973 concerning pseudo-differential operators within the Hörmander class. Full article

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation. Full article

The parabolic wave equation describes wave propagation in a preferable direction, which is usually horizontal in underwater acoustics and vertical in seismic applications. For dense receiver arrays (receiver spacing is less than signal wavelength), this equation can be used for propagating the recorded wavefield back into the medium for imaging sources and scattering objects. Similarly, for multiple source and receiver array acquisition, typical for seismic exploration and potentially beneficial for ocean acoustics, one can model data in one run using an extension of the parabolic equation—the pseudo-differential Double Square Root (DSR) equation. This extended equation allows for the modeling and imaging of multi-source data but operates in higher-dimensional space (source, receiver coordinates, and time), which makes its numerical computation time-consuming. In this paper, we apply a faster ray method for solving the DSR equation. We develop algorithms for both kinematic and dynamic ray tracing applicable to either data modeling or true-amplitude recovery. Our results can be used per se or as a basis for the future development of more elaborated asymptotic techniques that provide accurate and computationally feasible results. Full article