Exponential Parameterization Research Articles

Thin twigs decompose faster than thick ones under stagnant and flowing water: a double exponential decay model parameterization

Thin twigs decompose faster than thick ones under stagnant and flowing water: a double exponential decay model parameterization

Dynamical dark energy confronted with multiple CMB missions

The measurements of the cosmic microwave background (CMB) have played a significant role in understanding the nature of dark energy. In this article, we investigate the dynamics of the dark energy equation of state, utilizing high-precision CMB data from multiple experiments. We begin by examining the Chevallier–Polarski–Linder (CPL) parametrization, a commonly used and recognized framework for describing the dark energy equation of state. We then explore the general Exponential parametrization, which incorporates CPL as its first-order approximation, and extensions of this parametrization that incorporate nonlinear terms. We constrain these scenarios using CMB data from various missions, including the Planck 2018 legacy release, the Wilkinson Microwave Anisotropy Probe (WMAP), the Atacama Cosmology Telescope (ACT), and the South Pole Telescope (SPT), as well as combinations with low-redshift cosmological probes such as Baryon Acoustic Oscillations (BAO) and the Pantheon sample. While the ΛCDM cosmology remains consistent within the 68% confidence level, we observe that the extensions of the CPL parametrization are indistinguishable for Planck data. However, for ACT and SPT data, the inclusion of additional terms begins to reveal a peak in wa,DE that was previously unconstrained.

Additive manufacturing of non-uniform scaffolds for mass center control

In addition to solid components, additive manufacturing can shape parts with internal structures or lattices to control specific mechanical properties such as the relation between strength and mass, inertia, and the center of mass. This study shows a strategy to produce scaffolds based on linear mapping, exponential parameterization, and Voronoi's diagram. The approach provides layer by layer, a net of lines denser on a specific layer point that can be defined to control the layer center of mass. This method is compatible with current CAD software and three-dimensional printing procedure stages; thus, it is easily reproducible. Furthermore, different linear mapping rulings and parameterizations were tested to evaluate the method's capabilities for modifying the center of mass of a symmetrical NACA 0015 profile. The numerical and experimental results show that it is possible to accommodate the center of mass between −8 and 18 mm to the NACA's theoretical center of mass.

Cool skin effect and warm skin phenomenon observed by shipboard radiometer in the Northwest Pacific

Sea surface temperature (SST) is an important variable in the study of ocean boundary layers and heat exchange. The accurate simulation and measurement of skin effects are vital to air–sea model processing and satellite SST retrieval. Shipboard measurements from eleven cruises in the Northwest Pacific between August 2015 and October 2018 were used to estimate the cool skin effect and compare model results. The temperature difference ΔT between the sea surface skin temperature (SSTskin), as measured by an infrared radiometer, and the sea surface depth temperature (SSTdepth) at around 4 meters showed a mean difference and a standard deviation of the same 0.2 K, with a total of 5-min 39909 measurements. Both daytime and nighttime ΔT values were compared to physical model simulations and were found to have relatively larger mean values. A set of new coefficients for an exponential parameterization of the cool skin effect was derived in the research area, which performed well in comparison to previous empirical models. In nighttime observations from two summer cruises, the reverse process of heat flux transfer from the air to the sea in the form of a warm skin was distinguished. There were 667 positive ΔT values out of the 1917 nighttime observations, with magnitudes ranging from around 0 to 0.3 K. A high proportion of the cases of the warm skin phenomenon occurred when the air was very humid and much warmer than the sea surface.

Feature selection based on double-hierarchical and multiplication-optimal fusion measurement in fuzzy neighborhood rough sets

In fuzzy neighborhood rough sets (FNRSs), uncertainty measurement performs mainly classification-hierarchical and multiplication-simple fusion, so the corresponding feature selection has advancement space. This paper aims to improve uncertainty measurement and feature selection via FNRSs. Two measurement strategies regarding class-hierarchical fusion and multiplication-optimal fusion are proposed, and three measure-based heuristic feature selection algorithms are developed. Concretely, fuzzy neighborhood self-information (FNSI) and joint entropy (FNJE) constitute two bases of heterogeneous fusion, and their multiplication fusion induces both the existing measure FNSIJE (which is based on classification-level fusion) and a new measure CFNSIJE (which is based on class-level fusion); furthermore, FNSIJE and CFNSIJE are extended to the optimal measures FNSIJEE and CFNSIJEE, respectively, by exponential parameterization. The four types of fusion measures acquire their calculation algorithms and granulation nonmonotonicity and systematically motivate four heuristic feature selection algorithms, i.e., the current FNSIJE-FS and the new CFNSIJE-FS, FNSIJEE-FS, and CFNSIJEE-FS. By using examples and experiments, relevant uncertainty measurement and granulation nonmonotonicity are validated, while the novel selection algorithms demonstrate better classification performances. This study establishes the hierarchical fusion and exponential expansion to acquire robust uncertainty measurement and optimal feature selection, and the measurement, nonmonotonicity, and selection have strong generalization for information fusion and rough-set learning.

Scalar thermal radiation using the adding-doubling method.

The scalar radiative transfer equation in the presence of thermal radiation source is solved in detail, using the adding-doubling method; Planck functions within any given layer are assumed to possess constant, linear, or exponential parameterizations with optical thickness. The radiance profile in any zenith direction is calculated directly in terms of matrix inversions. The inputs to the model are the inherent optical properties (layer total single-scattering albedos, scattering phase functions, and optical thickness) along with temperature and altitude profiles, and the top of the atmosphere and ground surface boundary conditions. The algorithm is implemented in a state-of-the-art MATLAB program, with the cosmic microwave background as the source at the upper boundary and a Lambertian surface reflection at the lower boundary. The simulations are validated against the VLIDORT discrete ordinate radiative transfer model. Results are compared in detail for cases with linear and exponential Planck function parameterizations.

On the ratio of the components of the atmospheric vertical electric current density in fair weather

This work presents the results of ground-based and tethered balloon observations of altitude profiles of the components of the atmospheric vertical electric current density. The magnitude of the conduction current density was observed in the range 0.2 - 2 pAm−2. The ratio of the negative to positive component of the conduction current density averaged 1.6 outside the layer, where their dependence on the height was noticeable. In the framework of developed numerical model with reasonable values of the parameters it is found that the ratio of the convection current density to the density of the total vertical atmospheric electric current in the atmospheric boundary layer mainly falls in the range 0.2 - 0.6 and tends to increase under strong convection and low electrical conductivity. An exponential parameterization of the dependence of the electromotive force on the ground-level electrical conductivity is proposed.

A trust-region augmented Hessian implementation for state-specific and state-averaged CASSCF wave functions.

In this work, we present a one-step second-order converger for state-specific (SS) and state-averaged (SA) complete active space self-consistent field (CASSCF) wave functions. Robust convergence is achieved through step restrictions using a trust-region augmented Hessian (TRAH) algorithm. To avoid numerical instabilities, an exponential parameterization of variational configuration parameters is employed, which works with a nonredundant orthogonal complement basis. This is a common approach for SS-CASSCF and is extended to SA-CASSCF wave functions in this work. Our implementation is integral direct and based on intermediates that are formulated in either the sparse atomic-orbital or small active molecular-orbital basis. Thus, it benefits from a combination with efficient integral decomposition techniques, such as the resolution-of-the-identity or the chain-of-spheres for exchange approximations. This facilitates calculations on large molecules, such as a Ni(II) complex with 231 atoms and 5154 basis functions. The runtime performance of TRAH-CASSCF is competitive with the other state-of-the-art implementations of approximate and full second-order algorithms. In comparison with a sophisticated first-order converger, TRAH-CASSCF calculations usually take more iterations to reach convergence and, thus, have longer runtimes. However, TRAH-CASSCF calculations still converge reliably to a true minimum even if the first-order algorithm fails.

Coupled cluster downfolding methods: The effect of double commutator terms on the accuracy of ground-state energies.

Downfolding coupled cluster techniques have recently been introduced into quantum chemistry as a tool for the dimensionality reduction of the many-body quantum problem. As opposed to earlier formulations in physics and chemistry based on the concept of effective Hamiltonians, the appearance of the downfolded Hamiltonians is a natural consequence of the single-reference exponential parameterization of the wave function. In this paper, we discuss the impact of higher-order terms originating in double commutators. In analogy to previous studies, we consider the case when only one- and two-body interactions are included in the downfolded Hamiltonians. We demonstrate the efficiency of the many-body expansions involving single and double commutators for the unitary extension of the downfolded Hamiltonians on the example of the beryllium atom, and bond-breaking processes in the Li2 and H2O molecules. For the H2O system, we also analyze energies obtained with downfolding procedures as functions of the active space size.

Exponential parameterization to fit reduced data

This paper discusses the topic of interpolating data points Qm in arbitrary Euclidean space with Lagrange cubics γ^L and exponential parameterization which is governed by a single parameter λ ∈ [0, 1] and replaces a discrete set of unknown knots {ti}i=0m (ti ∈ I) with new values {t^i}i=0m (t^i∈I^). To compare γ with γ^L specific mapping ϕ:I→I^ must also be selected. For certain applications and theoretical considerations ϕ should be an injective mapping(e.g. in length estimation of γ which fits Qm). We justify two sufficient conditionsyielding ϕ as injective and analyze their asymptotic representatives for Qm getting dense. The algebraic constraints established here are also geometrically visualized with the aid of Mathematica 2D and 3D plots. Examples illustrating the latter and numerical investigation for the convergence rate in length estimation of γ are also supplemented. The reparameterization issue has its applications among all in computer graphics and robot navigation for trajectory planning once e.g. a new curve γ˜=γ∘ϕ controlled by the appropriate choice of interpolation knots and ϕ (and/or possibly Qm) needs to be constructed.

Extended vibrational coupled cluster: Stationary states and dynamics.

For the first time, equations are derived for computing stationary vibrational states with extended vibrational coupled cluster (EVCC) and for propagating nuclear wave packets using time-dependent EVCC (TDEVCC). Expressions for energies, properties, and auto-correlation functions are given. For TDEVCC, convergence toward the ground state for imaginary-time propagation is shown, as well as separability in the case of non-interacting subsystems. The analysis focuses substantially on the difference between bra and ket parameterizations for EVCC and TDEVCC compared to normal vibrational coupled cluster (VCC) and time-dependent VCC (TDVCC). A pilot implementation is presented within a new full-space framework that offers easy access to completely general, albeit not efficient, implementations of alternative VCC variants, such as EVCC. The new methods were tested on 35 three- and six-mode molecular systems. Both EVCC[k] and TDEVCC[k] showed good, hierarchical convergence toward the exact limit. This convergence was generally better than for normal VCC[k] and TDVCC[k] and better still than for (time-dependent) vibrational configuration interaction, though this should be balanced with the higher computational complexity of EVCC. The results highlight the importance of exponential parameterizations and separability in general, as seen, in particular, for the TDEVCC bra parameterization, which is in contrast to the partially linear one of TDVCC. With the results being rooted in the general structures of coupled cluster (CC) theory, they are expected to be relevant to other applications of both normal and extended CC theory as well.

Exact parameterization of fermionic wave functions via unitary coupled cluster theory.

A formal analysis is conducted on the exactness of various forms of unitary coupled cluster (UCC) theory based on particle-hole excitation and de-excitation operators. Both the conventional single exponential UCC parameterization and a factorized (referred to here as "disentangled") version are considered. We formulate a differential cluster analysis to determine the UCC amplitudes corresponding to a general quantum state. The exactness of conventional UCC (ability to represent any state) is explored numerically, and it is formally shown to be determined by the structure of the critical points of the UCC exponential mapping. A family of disentangled UCC wave functions is proven to exactly parameterize any state, thus showing how to construct Trotter-error-free parameterizations of UCC for applications in quantum computing. From these results, we construct an exact disentangled UCC parameterization that employs an infinite sequence of particle-hole or general one- and two-body substitution operators.

A Note on Modified Hermite Interpolation

We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {E}^n$$ based on reduced data$$Q_m = \{q_i\}_{i = 0}^m$$ in arbitrary Euclidean space $$\mathbb {E}^n$$. The respective interpolation knots $${\mathcal T} = \{t_i\}_{i = 0}^m$$ satisfying $$q_i = \gamma (t_i)$$ are assumed to be unknown. In our setting the substitutes $${\mathcal T}_{\lambda }=\{{\hat{t}}_i\}_{i = 0}^m$$ of $${{\mathcal {T}}}$$ are selected according to the so-called exponential parameterization governed by $$Q_m$$ and $$\lambda \in [0,1]$$. A modified Hermite interpolant $$\hat{\gamma }^H$$ introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit $$(\hat{{\mathcal {T}}}_{\lambda },Q_m)$$. The case of $$\lambda = 1$$ (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating $$\gamma {\in } C^4$$ by $${\hat{\gamma }}^H$$ [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkolazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4) the remaining $$\lambda \in [0,1)$$ render a linear convergence order in $${\hat{\gamma }}^H\approx \gamma $$ for any $$Q_m$$ sampled more-or-less uniformly. The related analysis relies on comparing the difference $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ in which $$\phi ^H$$ forms a special mapping between [0, T] and $$[0,{\hat{T}}]$$ with $${\hat{T}} = {\hat{t}}_m$$. In this paper: (a) several sufficient conditions enforcing $$\phi ^H$$ to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ [from Kozera and Wilkolazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica.

CP violation and quark-lepton complementarity of the neutrino mixing matrix

The comparative analysis of the neutrino mixing in the standard, cobimaximal and exponential parameterizations is performed. With the latest November 2018 data the logarithm of the mixing matrix is computed and the exact entries for the exponential matrix are obtained. The factorization of the real rotation and the CP violation in the exponential form of the mixing matrix is demonstrated. Quark-lepton complementarity hypothesis is reformulated, involving three mixing angles in the framework of the exponential parameterisation of the mixing matrix. It is shown that the cobimaximal parameterization, consistent with recent experimental data on neutrino mixing with the spread 3sigma , can provide exact quark-lepton complementarity based on the data for all three mixing angles. The dependence of the CP violation degree on the parameterization parameters in the standard and the exponential forms is studied with the help of the Jarlskog invariant.

CONVERGENCE ORDER IN TRAJECTORY ESTIMATION BY PIECEWISE-CUBICS AND EXPONENTIAL PARAMETERIZATION

This paper discusses the problem of estimating the trajectory of the unknown curve γ from the sequence of m+1 interpolation points in arbitrary Euclidean space En . The respective knots (in ascending order) are assumed to be unknown. Such Qm is coined reduced data. In our setting, a piecewise-cubic Lagrange interpolation is applied to fit Qm. Here, the missing knots Ƭm are replaced by their estimates in accordance with the exponential parameterization. The latter is controlled by a single parameter λ ∈ [0, 1]. This work analyzes the intrinsic asymptotics in approximating γ by ŷ3 based on the exponential parameterization and Qm. The multiple goals are achieved. Firstly, the existing result established for λ = 1 (i.e. for the cumulative chord parameterization) is extended to the remaining cases of λ ∈ [0, 1) and more-or-less uniformly sampled Qm. As demonstrated herein, a quartic convergence order α(1) = 4 in trajectory estimation drops discontinuously to the linear one α(λ) = 1, for all λ ∈ [0, 1). Secondly, the asymptotics derived in this paper is also analytically proved to be sharp with the aid of illustrative examples. Thirdly, the latter is verified in affirmative upon conducting numerical testing. Next, the necessity of more-or-less uniformity imposed on Qm is shown to be indispensable. In addition, several sufficient conditions for ŷ3 to be reparameterizable to the domain of γ are formulated. Lastly, the motivation for using the exponential parameterization with λ ∈ [0, 1) is also outlined.

Asymptotically safe f(R)-gravity coupled to matter II: Global solutions

Ultraviolet fixed point functions of the functional renormalisation group equation for f(R)-gravity coupled to matter fields are discussed. The metric is split via the exponential parameterisation into a background metric and a fluctuating part, the former is chosen to be the one of a four-sphere. Also when scalar, fermion and vector fields are included global quadratic solutions exist as in the pure gravity case for discrete sets of values for some endomorphism parameters defining the coarse-graining scheme. The asymptotic, large-curvature behaviour of the fixed point functions is analysed for generic values of these parameters. Examples for global numerical solutions are provided. A special focus is given to the question whether matter fields might destabilise the ultraviolet fixed point function. Similar to a previous analysis of a polynomial, small-curvature approximation to the fixed point functions different classes for such functions are found.

Exponential parameterization of wave functions for quantum dynamics: Time-dependent Hartree in second quantization

We derive equations for describing the time evolution of variational wave functions in linear and exponential parameterization with a second-quantization (SQ) formulation. The SQ formalism covers time-dependent Hartree (TDH), while exact states and approximate vibrational configuration interaction wave functions are described using state-transfer operators. We present detailed expressions for efficient evaluation of TDH in linear (L-TDH) and exponential (X-TDH) parametrization and an efficient implementation supporting linear scaling with respect to the number of degrees of freedom M when the Hamiltonian operator contains a constant number of terms per mode independently of the size of the system. The computational cost of the X-TDH method is reduced significantly compared to the L-TDH method for systems with many operator terms per mode such as is typical for accurate molecular potential-energy surfaces. Numerical results for L-TDH and X-TDH are presented which confirm the theoretical reduction of the M scaling compared to standard first-quantization formulations. Calculations on Henon-Heiles potentials with more than 105 dimensions and polycyclic aromatic hydrocarbons with up to 264 modes have been performed. Thus, the SQ formulation and the X-TDH method pave the way for studying the time-resolved quantum dynamics of large molecules.

A Modified Hermite Interpolation with Exponential Parameterization

This work discusses the problem of fitting a regular curve gamma based on reduced data pointsQ_m=(q_0,q_1,dots ,q_m) in arbitrary Euclidean space. The corresponding interpolation knots {mathcal T}=(t_0,t_1,dots ,t_m) are assumed to be unknown. In this paper the missing knots are estimated by {mathcal T}_{lambda }=(hat{t}_0,hat{t}_1,dots ,hat{t}_m) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter lambda in [0,1]. In order to fit (hat{mathcal T}_{lambda },Q_m), a modified Hermite interpolant hat{gamma }^H (a C^1 piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating gamma in C^4 by hat{gamma }^H is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for lambda =1 and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all lambda in [0,1). A slower linear convergence order in trajectory estimation is established for any lambda in [0,1) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Q_m and based on {mathcal T}_{lambda } together with some relevant applications are also briefly recalled in this paper.

Comparison of the exponential thermal transient parameterization methods with the SMTP method in the unipedicled DIEP flap computer modelling and simulation

The aim of this paper is to compare the spatial contrast of the image descriptors obtained via three different thermal transient parameterization methods in Active Dynamic Thermography. The thermal constants and amplitude values of the one- and two- exponential parametrization are compared to the Simplified Magnitude-Temporal Parametrization method (SMTP). The comparison is performed using the data obtained by simulating the cold stress ADT exam of the partial computer model of the Deep Inferior Epigastric Perforators (DIEP) skin flap that is used e.g. in one of the breast reconstruction procedures. The results prove that all of the described methods are capable of detecting flap perforators. The SMTP method proved itself as a method with comparable spatial contrast to 1-exponential parametrization. Moreover it has some additional advantages like the excitation of non-uniformity insensitivity and far less computational complexity.

The exponential parameterization of the neutrino mixing matrix as an SU(3) group element and an account for new experimental data

The exponential form of the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix for neutrinos is considered in the context of the fundamental representation of the SU(3) group. The logarithm of the mixing matrix is obtained. Based on the most recent experimental data on neutrino mixing, the exact values of the entries of the exponential matrix are calculated. The exact values for its real and imaginary parts are determined, respectively, in charge of the mixing without CP violation and of the pure CP violation effect. The hypothesis of complementarity for quarks and neutrinos is confirmed. The factorization of the exponential mixing matrix, which allows the separation of the mixing and of the CP violation itself in the form of the product of rotations around the real and imaginary axes, is demonstrated.