Non-integer Values Research Articles

Tubings, chord diagrams, and Dyson–Schwinger equations

AbstractWe give series solutions to single insertion place propagator‐type systems of Dyson–Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson–Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and noninteger values of the insertion parameter . We remark on interesting combinatorial connections and properties.

Tilted-Plane Structure of the Energy of Finite Quantum Systems.

The piecewise linearity condition on the total energy with respect to the total magnetization of finite quantum systems is derived using the infinite-separation-limit technique. This generalizes the well-known constancy condition, related to static correlation error, in approximate density functional theory. The magnetic analog of Koopmans' theorem in density functional theory is also derived. Moving to fractional electron count, the tilted-plane condition is derived, lifting certain assumptions in previous works. This generalization of the flat-plane condition characterizes the total energy surface of a finite system for all values of electron count N and magnetization M. This result is used in combination with tabulated spectroscopic data to show the flat-plane structure of the oxygen atom, among others. We find that derivative discontinuities with respect to electron count sometimes occur at noninteger values. A diverse set of tilted-plane structures is shown to occur in d-orbital subspaces, depending on chemical coordination. General occupancy-based total-energy expressions are demonstrated thereby to be necessarily dependent on the symmetry-imposed degeneracies.

Universal scaling in real dimension

The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its scaling theory is controlled by a single parameter, the spectral dimension ds, which plays the role of the relevant parameter on complex geometries. The graph under consideration allows us to tune the value of the spectral dimension continuously also to noninteger values and to find the universal exponents as continuous functions of the dimension. By means of extensive numerical simulations, we probe the scaling exponents of a simple instance of O(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O({{{{{{{\\mathcal{N}}}}}}}})$$\\end{document} symmetric models on the LRDG showing quantitative agreement with the theoretical prediction of universal scaling in real dimensions.

Confinement of N-Body Systems and Non-integer Dimensions

The squeezing process of a three-dimensional quantum system by use of an external deformed one-body oscillator potential can also be described by the d-method, without external field and where the dimension can take non-integer values. In this work we first generalize both methods to N particles and any transition between dimensions below 3. Once this is done, the use of harmonic oscillator interactions between the particles allows complete analytic solutions of both methods, and a direct comparison between them is possible. Assuming that both methods describe the same process, leading to the same ground state energy and wave function, an analytic equivalence between the methods arises. The equivalence between both methods and the validity of the derived analytic relation between them is first tested for two identical bosons and for squeezing transitions from 3 to 2 and 1 dimensions, as well as from 2 to 1 dimension. We also investigate the symmetric squeezing from 3 to 1 dimensions of a system made of three identical bosons. We have in all the cases found that the derived analytic relations between the two methods work very well. This fact permits to relate both methods also for large squeezing scenarios, where the brute force numerical calculation with the external field is too much demanding from the numerical point of view, especially for systems with more than two particles.

Symmetry breaking in an extended O(2) model

Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we study an extended-O(2) model in two dimensions that differs from the ordinary O(2) model by the addition of an explicit symmetry breaking term −hqcos(qφ). Its coupling hq allows to smoothly interpolate between the O(2) model (hq=0) and a q-state clock model (hq→∞). In the latter case, a q-state clock model can also be defined for noninteger values of q. Thus, such a limit can also be considered as an analytic continuation of an ordinary q-state clock model to noninteger q. In previous work, we established the phase diagram for noninteger q in the infinite coupling limit (hq→∞). We showed that there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we seek to establish the phase diagram at finite values of the coupling using Monte Carlo and tensor methods. We show that for noninteger q, the second-order phase transition at low temperature and crossover at high temperature persist to finite coupling. For integer q=2, 3, 4, we know there is a second-order phase transition at infinite coupling (i.e. the well-known clock models). At finite coupling, we find that the critical exponents for q=3, 4 vary with the coupling, and for q=4 the transition may turn into a Berezinskii-Kosterlitz-Thouless transition at small coupling. We comment on the similarities and differences of the phase diagrams with those of quantum simulators of the Abelian-Higgs model based on ladder-shaped arrays of Rydberg atoms. Published by the American Physical Society 2024

Review: Fractal Geometry in Precipitation

Rainfall, or more generally the precipitation process (flux), is a clear example of chaotic variables resulting from a highly nonlinear dynamical system, the atmosphere, which is represented by a set of physical equations such as the Navier–Stokes equations, energy balances, and the hydrological cycle, among others. As a generalization of the Euclidean (ordinary) measurements, chaotic solutions of these equations are characterized by fractal indices, that is, non-integer values that represent the complexity of variables like the rainfall. However, observed precipitation is measured as an aggregate variable over time; thus, a physical analysis of observed fluxes is very limited. Consequently, this review aims to go through the different approaches used to identify and analyze the complexity of observed precipitation, taking advantage of its geometry footprint. To address the review, it ranges from classical perspectives of fractal-based techniques to new perspectives at temporal and spatial scales as well as for the classification of climatic features, including the monofractal dimension, multifractal approaches, Hurst exponent, Shannon entropy, and time-scaling in intensity–duration–frequency curves.

A Language and Its Dimensions: Intrinsic Dimensions of Language Fractal Structures

The present paper introduces a novel object of study, a language fractal structure; we hypothesize that a set of embeddings of all n‐grams of a natural language constitutes a representative sample of this fractal set. (We use the term Hailonakea to refer to the sum total of all language fractal structures, over all n). The paper estimates intrinsic (genuine) dimensions of language fractal structures for the Russian and English languages. To this end, we employ methods based on (1) topological data analysis and (2) a minimum spanning tree of a data graph for a cloud of points considered (Steele theorem). For both languages, for all n, the intrinsic dimensions appear to be noninteger values (typical for fractal sets), close to 9 for both of the Russian and English language.

Analysis of a discrete time fractional-order Vallis system

Vallis system is a model describing nonlinear interactions of the atmosphere and temperature fluctuations with a strong influence in the equatorial part of the Pacific Ocean. As the model approaches the fractional order from the integer order, numerical simulations for different situations arise. To see the behavior of the simulations, several cases involving integer analysis with different non-integer values of the Vallis systems were applied. In this work, a fractional mathematical model is constructed using the Caputo derivative. The local asymptotic stability of the equilibrium points of the fractional-order model is obtained from the fundamental production number. The chaotic behavior of this system is studied using the Caputo derivative and Lyapunov stability theory. Hopf bifurcation is used to vary the oscillation of the system in steady and unsteady states. In order to perform these numerical simulations, we apply Grünwald–Letnikov tactics with Binomial coefficients to obtain the effects on the non-integer fractional degree and discrete time vallis system and plot the phase diagrams and phase portraits with the help of MATLAB and MAPLE packages.

Heat transfer in a wet porous moving inclined longitudinal fin exposed to convection and radiation in the presence of shape-dependent hybrid nanofluid: Adomian decomposition Sumudu transformation approach

This research paper investigates the thermal examination of a non-integer-ordered mobile fin situation with a hybrid nanofluid flowing past it at an unceasing speed U. The combination approach is utilized to evaluate the physical and thermal characteristics of the hybrid nanofluid, excluding the thermal conductivity and dynamic viscosity. This governing equation is transformed into a dimensionless form and the Adomian Decomposition Sumudu Transform method is employed to solve the conundrum of a moving fin immersed in a hybrid nanofluid. The reliability of this research has been confirmed by verifying the results through a comparison with numerical simulations. The influence of diverse factors on the thermal overview for varying noninteger values alpha is analyzed and presented in graphical representations. Moreover, the analysis reveals that the highest heat flux occurs in the scenario where nanoparticles with spherical-platelet shapes are combined. This is succeeded by the spherical-cylindrical combination, and finally, the spherical-spherical pairing, in terms of heat flux magnitude. Here, we discovered that the temperature at the fin tip decreases by 11.19141 % for s-s shaped nanoparticles, 10.45649 % for s-c shaped nanoparticles and 10.21162 % for s-p shaped nanoparticles as the wet parameter (m2) increases by a factor of 400, and also when the radiation parameter is increased by 400 %, the temperature of the fin tip drops by 13.11313 % for s-s shape nanoparticles, 11.97076 for s-c shape nanoparticles and 11.653899 % for s-p shape nanoparticles.

Identification of the extended standard linear solid material model by means of experimental dynamical measurements

A novel algebraic procedure for the non-parametric identification of the material model by means of dynamical test measurements is proposed. An extended Standard Linear Solid (SLS) material model is taken into account to model the material linear visco-elastic behavior. It consists of the series arrangement of fractional Kelvin model elements adopting real parameters and integer and non-integer order time differential operators, and of hysteretic Kelvin model elements adopting complex parameters and integer order time differential operators. Hysteretic Kelvin model elements are introduced to take account of the material hysteretic behavior. The material E(j⋅ω) complex modulus, is analytically modeled as the ratio of pseudo polynomials (non-integer power terms) in the j⋅ω Fourier variable. A multi-step, iterative, material model identification technique is here proposed to identify the unknown polynomial coefficients and the non-integer exponent values starting from E(j⋅ω) material discrete estimates from input-output dynamical measurements made on a beam specimen at different ω frequency values. Computational, nonphysical SLS elements resulting from the application of the identification procedure can be found and eliminated, so that a low order optimal model result. Some results obtained by applying the proposed identification technique with real experimental measurements are shown and discussed.

Applicability of noninteger bessel type orbital basis sets: numerical and analytical approaches

Recently, usefulness of the noninteger principal quantum numbers for Bessel type orbitals was discussed by Weniger in (Adv Quant Chem 83:209–237, 2021). In this study, we analyzed the applicability and numerical accuracy of basis sets of noninteger Bessel type orbitals to electronic structure calculations. Both numerical and analytical approaches are applied to two-electron atomic systems. The results of the numerical test demonstrated the potential of the noninteger values of principal quantum number for the improving of Bessel type functions approach in use of LCAO methods. Nevertheless, the analytical approach is still not suitable and in development and needs to be investigated further.The performance of the presented basis functions is also compared to the numerical Hartree–Fock results.

Revisiting and redefining return rate for determination of the precise growth status of a species.

Growth curve models play an instrumental role in quantifying the growth of biological processes and have immense practical applications across all disciplines. The most popular growth metric to capture the species fitness is the "Relative Growth Rate" in this domain. The different growth laws, such as exponential, logistic, Gompertz, power, and generalized Gompertz or generalized logistic, can be characterized based on the monotonic behavior of the relative growth rate (RGR) to size or time. Thus, in this case, species fitness can be determined truly through RGR. However, in nature, RGR is often non-monotonic and specifically bell-shaped, especially in the situation when a species is adapting to a new environment [1]. In this case, species may experience with the same fitness (RGR) for two different time points. The species precise growth and maturity status cannot be determined from this RGR function. The instantaneous maturity rate (IMR), as proposed by [2], helps to determine the correct maturity status of the species. Nevertheless, the metric IMR suffers from severe drawbacks; (i) IMR is intractable for all non-integer values of a specific parameter. (ii) The measure depends on a model parameter. The mathematical expression of IMR possesses the term "carrying capacity" which is unknown to the experimenter. (iii) Note that for identifying the precise growth status of a species, it is also necessary to understand its response when the populations are deflected from their equilibrium position at carrying capacity. This is an established concept in population biology, popularly known as the return rate. However, IMR does not provide information on the species deflection rate at the steady state. Hence, we propose a new growth measure connected with the species return rate, termed the "reverse of relative of relative growth rate" (henceforth, RRRGR), which is treated as a proxy for the IMR, having similar mathematical properties. Finally, we introduce a stochastic RRRGR model for specifying precise species growth and status of maturity. We illustrate the model through numerical simulations and real fish data. We believe that this study would be helpful for fishery biologists in regulating the favorable conditions of growth so that the species can reach a steady state with optimum effort.

Borel–Laplace sum rules with τ decay data, using OPE with improved anomalous dimensions

We perform numerical analysis of double-pinched Borel–Laplace QCD sum rules for the strangeless semihadronic τ-decay data. The D = 0 contribution to the theoretical contour integral in the sum rules is evaluated by the (truncated) Fixed Order perturbation theory method (FO) and by the Principal Value (PV) of the Borel integration. We use for the full Adler function the operator product expansion (OPE) with the terms ∼〈O D 〉 of dimension D = 2n where 2 ≤ n ≤ 5 for the (V+A)-channel, and 2 ≤ n ≤ 7 for the V-channel data. In our previous works Ayala, Cvetič and Teca (2021, Eur. Phys. J. C 81 930), Ayala, Cvetič and Teca (2022, Eur. Phys. J. C 82 362), only the (V+A)-channel data was analysed. In this work, the analysis of a new set of V-channel data is performed as well. Further, a renormalon-motivated construction of the D = 0 part of the Adler function is improved in the u = 3 infrared renormalon sector, by involving the recently known information on the two principal noninteger values of the effective leading-order anomalous dimensions. Additionally, the OPE of the Adler function has now the D = 6 contribution with the principal anomalous dimension (), and terms of higher dimension (with zero anomalous dimension). Cross-checks of the obtained extracted values of α s and of the condensates were performed by reproduction of the (central) experimental values of several double-pinched momenta a (2,n). The averaged final extracted values of the () coupling are: , corresponding to .

Evolving Tangent Hyperbolic memristor based 6D chaotic model with fractional order derivative: Analysis and applications

The nonlinear dynamics of fractional order systems and memristor based circuits have attracted an escalating attention in recent years. This paper reports on a novel six dimensional nonlinear dynamical system with a tangent hyperbolic memristor circuit and amalgamated image encryption. The dynamical system is analyzed using standard tools, including phase portraits, equilibrium points, Eigenvalues, Routh–Hurwitz stability criterion, Lyapunov exponents and bifurcation diagrams. The analysis suggests that the developed system is chaotic in nature and has exciting new 2D and 3D trajectories that have not been reported before. The chaotic system is numerically solved for different values of the fractional order q, and the results of equilibrium points and Lyapunov exponents are mentioned in tables. Fractional order circuits are designed for q=1 and q=0.99 utilizing the approximation fractional order technique of the transfer function. The developed circuit is generalized in a way that the circuit can work for both the integer and non-integer fractional order values by just introducing the fractional block to make it work for q=0.99, and removing it with a conventional integrator capacitor to work for q=1. The circuit simulation and numerical results were compared, and signifying that they were both in good agreement with each other. Lastly, the random number generated from the chaotic system is utilized to scramble the image via the scheme of Amalgamated Image Encryption. The scrambled image was tested using different image security test algorithms to support the idea that the chaotic system and image can together form an advantageous key.

Conversion of a beam carrying fractionnal angular momentum in High-Harmonics Generation

Exotic light fields combining non-trivial spin and angular momentum may not be eigenstates of either the spin or orbital angular momenta operators. For these fields, it is relevant to define a Generalized Angular Momentum operator of which they are eigenvectors. Their associated eigenvalues can take, depending on the case, non-integer values. We report that this new quantity is conserved via non-linear phenomena, such as High Harmonic Generation.

Pulse respiration quotient as a measure sensitive to changes in dynamic behavior of cardiorespiratory coupling such as body posture and breathing regime.

Objective: In this research we explored the (homeo)dynamic character of cardiorespiratory coupling (CRC) under the influence of different body posture and breathing regimes. Our tool for it was the pulse respiration quotient (PRQ), representing the number of heartbeat intervals per breathing cycle. We obtained non-integer PRQ values using our advanced Matlab® algorithm and applied it on the signals of 20 healthy subjects in four conditions: supine position with spontaneous breathing (Supin), standing with spontaneous breathing (Stand), supine position with slow (0.1Hz) breathing (Supin01) and standing with slow (0.1Hz) breathing (Stand01). Main results: Linear features of CRC (in PRQ signals) were dynamically very sensitive to posture and breathing rhythm perturbations. There are obvious increases in PRQ mean level and variability under the separated and joined influence of orthostasis and slow (0.1Hz) breathing. This increase was most pronounced in Stand01 as the state of joint influences. Importantly, PRQ dynamic modification showed greater sensitivity to body posture and breathing regime changes than mean value and standard deviation of heart rhythm and breathing rhythm. In addition, as a consequence of prolonged supine position, we noticed the tendency to integer quantization of PRQ (especially after 14min), in which the most common quantization number was 4:1 (demonstrated in other research reports as well). In orthostasis and slow breathing, quantization can also be observed, but shifted to other values. We postulate that these results manifest resonance effects induced by coupling patterns from sympathetic and parasympathetic adjustments (with the second as dominant factor). Significance: Our research confirms that cardiorespiratory coupling adaptability could be profoundly explored by precisely calculated PRQ parameter since cardiorespiratory regulation in healthy subjects is characterized by a high level of autonomic adaptability (responsiveness) to posture and breathing regime, although comparisons with pathological states has yet to be performed. We found Stand01 to be the most provoking state for the dynamic modification of PRQ (cardiorespiratory inducement). As such, Stand01 has the potential of using for PRQ tuning by conditioning the cardiorespiratory autonomic neural networks, e.g., in the cases where PRQ is disturbed by environmental (i.e., microgravity) or pathologic conditions.

Investigation of time-fractional mathematical model of COVID-19 with nonsingular kernel

In this article, a mathematical model of COVID-19 is investigated using the Atangana–Baleanu in sense of Caputo fractional operator. Mathematical analysis and modeling has led the results (allow policymakers to understand and predicts the dynamics of infectious disease under several different scenarios) about various nonpharmaceutical involvements to restrict the spread of pandemic disease worldwide. The present investigation meant to study worldwide research activity on mathematical modeling of spread and control of several infectious diseases with a known history of serious outbreaks. The existence of a unique solution is studied using a fixed point theorems. The stability of the solution is carried out through the concept of Ulam–Hyers stability. The considered model is computationally analyzed through the Adams–Bashforth technique. A fresh investigation with the proposed epidemic model is brought and the obtain results are define using plots which shows the performance of the classes of the consider model. The results show that the proposed scheme is very insistent and obvious to operate for the system of nonlinear equations. One can see a quick stability of all the compartments as the order decrease to noninteger values as compared to integer-order θ = 1. All theoretical results are simulated and validated through numerical simulations.

On rationality of C-graded vertex algebras and applications to Weyl vertex algebras under conformal flow

Using the Zhu algebra for a certain category of C-graded vertex algebras V, we prove that if V is finitely Ω-generated and satisfies suitable grading conditions, then V is rational, i.e., it has semi-simple representation theory, with a one-dimensional level zero Zhu algebra. Here, Ω denotes the vectors in V that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element ωμ parameterized by μ∈C and prove that for certain non-integer values of μ, these vertex algebras, which are non-integer graded, are rational, with a one-dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate C-graded Weyl vertex algebras of arbitrary ranks.

Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger β.

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β_{eff}(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

Relativistic high-order harmonic generation of spatiotemporal optical vortices

It is well known that light can possess spin angular momentum (SAM) and orbital angular momentum (OAM) in the longitudinal direction. Interesting novel light with transverse SAM or OAM has also been discovered recently, with the latter being called spatiotemporal optical vortices. Here we present a route to generate spatiotemporal optical vortices by means of high harmonic generation from relativistic laser plasma interactions. Three-dimensional particle-in-cell simulations show the evidence of edge dislocations in wave front structures and circulating energy flow in space-time. Further calculations of the SAM, OAM and total angular momentum confirm the harmonics carrying pure transverse OAM with noninteger values. The results open up a possibility to generate intense spatiotemporal optical vortices in the extreme ultraviolet region with attosecond duration.