Point Spectrum Research Articles

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document}, for t∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1)$$\\end{document}, are investigated when they act on the disc algebra A(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A({\\mathbb {D}})$$\\end{document} and on the Hardy spaces Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document}, for 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}. We study the continuity, compactness, spectrum and point spectrum of Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} as well as their linear dynamics and mean ergodicity on these spaces.

Absence of bound states for quantum walks and CMV matrices via reflections

We give a criterion based on reflection symmetries in the spirit of Jitomirskaya–Simon to show absence of point spectrum for (split-step) quantum walks and Cantero–Moral–Velázquez (CMV) matrices. To accomplish this, we use some ideas from a recent paper by the authors and collaborators to implement suitable reflection symmetries for such operators. We give several applications. For instance, we deduce arithmetic delocalization in the phase for the unitary almost-Mathieu operator and singular continuous spectrum for generic CMV matrices generated by the Thue–Morse subshift.

Anderson localization for Schrödinger operators with monotone potentials over circle homeomorphisms

In this paper, we prove pure point spectrum for a large class of Schrödinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough.

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

Visible, near-infrared, and shortwave-infrared spectra as an input variable for digital mapping of soil organic carbon

This study proposes a novel methodology to employ discrete point spectra as input variable for digital mapping of soil organic carbon (SOC). Accordingly, two SOC modeling approaches were used in three agricultural sites in Czech Republic: i) machine learning (ML) including partial least square regression (PLSR), cubist, random forest (RF), and support vector regression (SVR), and ii) regression kriging (RK) by the combination of ordinary kriging (OK) and PLSR (PLSR-K), cubist (cubist-K), RF (RF-K), and SVR (SVR-K). Models were developed on environmental predictor covariates (EPCs) and thirty genetic algorithms (GA)-selected visible, near-infrared, and shortwave-infrared (VNIR–SWIR) wavelengths spectra, individually and combined. Thirty rasters were then created using interpolation of the selected spectra and served as the input variables – with and without EPCs – to test and compare the developed models and SOC predictive maps with each other and with those retrieved from the third approach: iii) kriging using OK of the measured and ML-predicted SOC. The impact of employing selected wavelengths’ spectra and EPCs on models' performance was investigated using independent test samples and the uncertainty associated with the produced maps. Using interpolated spectra as the only input variable yielded a relatively acceptable accuracy (Nová Ves: RMSE = 0.19%, Údrnice: RMSE = 0.12%, Klučov: RMSE = 0.13%). In comparison, the interpolated spectra coupled with EPCs enhanced the results. Regarding the uncertainty, however, the ML-based SOC maps were more reliable, than RK-based ones. Furthermore, maps produced using both spectra and EPCs showed less uncertainty than those constructed on the individual datasets.

Distribution of unit mass on one fractal self-similar web-type curve

In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable$\tau=\sum\nolimits_{n=1}^{\infty}\frac{2\varepsilon_{\tau}}{3^n}\equiv\Delta^g_{\tau_1...\tau_n...}$, where $(\tau_n)$ is a~sequence of independent random variables taking the values $0,1,\cdots,6$ with the probabilities $p_{0n}$, $p_{1n},\cdots,p_{6n}$; $\varepsilon_{6}=0$, $\varepsilon_0$, $\varepsilon_1,\cdots,\varepsilon_5$ are 6th roots of unity. We prove that the set of values of random variable $\tau$ is self-similar six petal snowflake which is a fractal curve $G$ of spider web type with dimension $\log_37$. Its outline is the Koch snowflake. We establish that $\tau$ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable $\tau$, which is the tail set of the seven-symbol representation of the points of the curve $G$. In the case of identical distribution of the random variables $\tau_n$ (namely: $p_{kn}=p_k$) we establish that the spectrum of distribution $\tau$ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension $\alpha_0(E)=\frac{\ln {p_0^{p_0}\cdots p_6^{p_6}}}{-\ln 7}$ with respect to the Hausdorff-Billingsley $\alpha$-measure. The measure is a probabilistic generalization of the Hausdorff $\alpha$-measure. In this case, the random variables $\tau=\Delta^g_{\tau_1\cdots\tau_n\cdots}$ and $\tau'=\Delta^g_{\tau_1'...\tau_n'...}$ defined by different probability vectors $(p_0,\cdots,p_6)$ and $(p'_0,\cdots,p'_6)$ have mutually orthogonal distributions.

Comprehensive investigation of two substituted benzimidazoles: Design, synthesis, biological assessment, and characterization

Benzimidazole is a bicyclic molecule that is mostly used in medicinal chemistry. The 2-substituted benzimidazoles that are now on the market include omeprazole, flubendazole, mebendazole, and albendazole. Antioxidants are compounds that impede the oxidation process, which is the chemical process that produces free radicals. A computational technique called "in-silico screening" is used in drug discovery to effectively estimate a molecule's likelihood of interacting with a target of interest. Antihelminthics are a class of antiparasitic drugs that either kill or paralyze internal parasites like parasitic worms, all without endangering the host. A standard procedure was used to create the 2-substituted benzimidazole. PyRx software was used to do the in-silico screening for anthelmintic drugs. The DPPH and ABTS methods were used to measure the antioxidant activity. Melting point, TLC, and IR spectra were used to confirm the structures of the produced compounds. The binding affinity for sulphosalicylic acid in the anthelmentic in silico screening was -7.7. Salicylic acid demonstrated excellent antioxidant activity, as seen by its IC50 values of less than 5 mg/ml for the ABTS method and 37 mg/ml for the DPPH method. Overall, the combination of theoretical and practical yield data, along with melting points, ATR Spectra and Rf values, offers valuable insights into the synthesis and characterization of chemical compounds. The salicylic acid derivative exhibits good antioxidant activity by both the methods. The Sulphosalicylic acid derivative shows prominent In-silico anthelmintic activity result hence the synthesized compounds can be tested for anthelmintic activity in future.

Ghana’s Free Senior High School Policy within the Spectrum of the Focal Points of Curriculum Development

This study performs a meticulous examination of Ghana's Free Senior High School (FSHS) policy, considering the focal points of curriculum development. Ghana has significantly augmented its financial commitment to the educational sector, recognising the pivotal importance of education in sustainable development. Despite this monetary dedication, the government has implemented the FSHS policy to guarantee the extensive availability of high-quality education. The study seeks to thoroughly analyse the policies of curriculum development, assessing its influence on access, fairness, curriculum content, educational excellence, stakeholder engagement, and long-term viability. The research aims to provide detailed insights into the effectiveness of the FSHS policy by thoroughly examining various aspects. It goes beyond just looking at enrollment numbers and evaluates how the policy contributes to a well-rounded educational experience that aligns with both national and global educational goals. The expected results are positioned to offer useful insights for policymakers, educators, and stakeholders, enabling continuous improvements to the Free Senior High School project and the overall educational environment in Ghana.

On the Point Spectrum of a Non-Self-Adjoint Quasiperiodic Operator

On the Point Spectrum of a Non-Self-Adjoint Quasiperiodic Operator

The point spectrum of the three-particle Schrödinger operator for a system comprising two identical bosons and one fermion on Z

The point spectrum of the three-particle Schrödinger operator for a system comprising two identical bosons and one fermion on Z

Identification and characterization of phytochemicals in methanolic extract of roots of Datura fastuosa using various techniques

BackgroundPlant extracts have attracted significant interest among researchers due to their potential bioactivity and crucial contribution to the production of pharmaceutical compounds. In this study, the primary objective was to extract, analyze and characterize the bioactive compounds found in the methanol root extract of Datura fastuosa (D. fastuosa). This was achieved using various analytical techniques such as gas chromatography/mass spectrometry (GC–MS), ultra-violet visible spectrophotometry, Fourier-transform infrared (FT-IR), nuclear magnetic radiation spectrometry (NMR) and DPPH free radical scavenging activity assay.ResultsGC–MS analysis of the methanol root crude extract identified 49 compounds. Three compounds were isolated via column chromatography; one was pure, with a sharp melting point and clean IR spectrum, while the other two showed broad melting points and IR interferences. Comprehensive investigation of the pure extract revealed a UV profile with two distinct bands (300–800 nm) and confirmed functional groups (alcohol, alkanes, alkenes, carbonyl, methylene, and methyl) through FT-IR analysis. The 1HNMR (proton nuclear magnetic resonance spectroscopy) signal confirmed the presence of forty-nine non-equivalent protons, 13CNMR (Carbon-13 nuclear magnetic resonance spectroscopy) signal confirmed the presence of 32 non-equivalent carbons and DEPT-135 (distortionless enhancement by polarization transfer-135) signal confirmed the presence of 24 carbons (17 for odd and 7 for even) which are protons containing carbons in the compound. Combining the above mentioned analyses with data obtained from the GC/MS analysis of National Institute of Standards and Technology (NIST) library, the isolated pure compound exhibited a structural similarity to 1-(7-(3-hydroxyphenyl)-1,1,4a,5,6,9,10a,10b-octamethyl-1,2,3,4,4a,4b,6a,7,8,9,10,10a,10b,11,12,12a-hexadecahydrochrysen-2-yl)propan-1-one, with a chemical formula of C35H50O2.ConclusionsThe presence of various notable compounds, including phenolics, flavonoids, alkaloids, steroids etc., within the methanol root extract of D. fastuosa signifies its pharmacological potential. The methanol crude extract demonstrated antioxidant potential compared to standard ascorbic acid, exhibiting DPPH scavenging activity. Previous research has demonstrated the bioactivity of some of these compounds, further elucidating the plant’s medicinal properties. These findings not only suggest opportunities for developing synthetic drugs but also underscore its direct therapeutic potential in addressing diverse ailments.

Loading conditions for self‐organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability that a particle changes type at each step. In the case where , the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers . It has been proved that, if , and whether or and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of and be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net . This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.

Computing Klein-Gordon Spectra

Abstract We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.

Weyl's Theorem for Algebraically (\(\wp, \wr, \varrho\))-Paranormal and Algebraically (\(\wp, \wr, \varrho\))-*-Paranormal Operators

Present L be an algebraically (\(\wp, \wr, \varrho\))-Paranormal and algebraically (\(\wp, \wr, \varrho\))-*-Paranormal operators on \(L^2\) space. We examine Weyl's theorem, a-Browder's theorem and spectral mapping theorem holds for weyl's spectrum of L and essential approximate point spectrum of L.

Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom

A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.

Study on the spectrum model of light response of Manglietia yuyuanensis

Abstract To study the spectral characteristics of the photosynthetic response of Manglietia yuyuanensis, this paper uses a self-developed monochromatic light generator to generate 22 kinds of monochromatic light, and measures the net photosynthetic rate of Manglietia yuyuanensis under the irradiation of 22 kinds of monochromatic light, then obtains its light response curve under different wavelengths. The right-angle hyperbolic correction model was used to fit the light response curves under each kind of monochromatic light, and the completed spectrum of the light saturation points of Manglietia yuyuanensis was obtained. It makes up for the disadvantage that the self-developed monochromatic light generator cannot detect the light saturation point of the plant under some monochromatic light due to the low radiation intensity. This paper supposes that the coefficients α, β, γ, and Rd in the right-angle hyperbolic correction model are functions related to the wavelength λ, then fits a Gaussian function of α, β, γ, and Rd to wavelength λ. From the fitting results, the fitting degree of α, β, and γ can reach 0.8 or more, which indicates the fitting effect is good. The fitting degree of Rd is about 0.5. According to the fitting function, a new photosynthesis model related to wavelength and light radiation was obtained, and the fitting degree of the data by the new model is good, laying a foundation for future research on photosynthetic response curves of other plants.

A note on the spectral mapping theorem for point spectrum

A note on the spectral mapping theorem for point spectrum

Consistent spectral approximation of Koopman operators using resolvent compactification

Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L 2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.

Synthesis, DFT investigation, ADME-T properties, molecular docking and molecular dynamics simulation of new α-aminophosphonate inhibitor targeting Mpro and RdRp enzymes in SARS-CoV-2

A specific α-aminophosphonate molecule, namely diethyl (phenyl (4-pyridinylamino) methyl) phosphonate (α-DAP), as a potential SARS-CoV-2 inhibitor, has been synthesized via a Kabachnick-Fields reaction, a three-component, dalfampridine, benzenecarbaldehyde and triethoxyphosphine, and its molecular structure was confirmed by spectroscopic and physicochemical methods such FT-IR, UV–vis, melting point, 1H NMR, 13C NMR and 31P NMR spectra. The Density Functional Theory (DFT) at the B3LYP/6–31 G (d, p) level was used to calculate the optimized structure, reactivity, stability of the target molecule. In this context, several characteristics have been calculated and the obtained theoretical results are in good agreement with the experimental ones. Also, predicted ADME-T were performed and the obtained parameters indicated that the compound under investigation should have good oral bioavailability. In addition, the molecular docking has been used, further, the compound showed high potential docking with SARS-CoV-2 major protease (Mpro) and RNA dependent RNA polymerase (RdRp). Finally, the molecular dynamics simulation demonstrated the stability of the protein-ligand interaction and offered insights into the specific amino acid residues participating in the binding process.

The spectrum of asymptotic Cayley trees

We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.