Explicit Expressions Research Articles

New Circular Distribution with an Application to Biology

In this paper, we introduce a new circular distribution known as the wrapped new weighted exponential distribution (WNWE) is introduced. We derive an explicit expression for its probability density function and establish closed-form expressions for the distribution function, characteristic function, and trigonometric moments. Furthermore, we discuss the properties of the proposed model. We employ the method of maximum likelihood estimation to estimate the parameters. To demonstrate the applicability of the proposed distribution, we analyze a real dataset consisting of 50 starhead top minnows.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Approximate averaged derivative moments and stochastic response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises

Stochastic response prediction is one of the principal topics in diverse branches of engineering, such as aerospace engineering, mechanical engineering, and vehicle engineering. However, the challenge of addressing high-dimensional partial differential equations or evaluating high-dimensional domain integrals in determining averaged derivative moments has limited the research. This study proposes a method to predict the stochastic response of high-degree-of-freedom (DOF) quasi-non-integrable Hamiltonian systems under Poisson white noises. The original finite DOF system is reduced to a one-dimensional one governing the Hamiltonian by applying the stochastic averaging method, and the associated reduced generalized Fokker–Planck–Kolmogorov (GFPK) equation is derived. Then, the two-step generalized elliptical coordinate transformation is extended to evaluate averaged derivative moments that are the key to obtaining the stationary probability density function (SPDF) for probabilistic information prediction. Moreover, the perturbation method is introduced to address the high-order GFPK equation. Finally, an 8-DOF strongly nonlinear system under purely additive Poisson white noises is given as an example to highlight the accuracy of the proposed method. Results from the proposed method show good agreement with those from Monte Carlo simulations, which indicates that the proposed method can be applied to assess the response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises. In addition, if the potential energy of an arbitrary finite n-DOF system is the same form as that in Appendix B, pertinent high-dimensional domain integrals are converted into explicit expressions when n is even.

An innovative treatment of anharmonic and Morse potentials to determine the spectroscopic constants of diatomic molecules

In this work, we develop an operational method to determine the explicit expressions of the spectroscopic constants ω e , ω e x e , ω e y e , and ω e z e of diatomic systems using the solutions of the Schrödinger equation up to the second order of polynomial anharmonic and Morse potentials with the help of the Floquet theorem combined with the resonating average method. As an example, we performed numerical calculations of the above-mentioned constants for H2, LiH, CO, and NO molecules. We present and discuss our results compared to those of other authors available in the literature.

Reversible negative compressibility metamaterials inspired by Braess’s Paradox

Negative compressibility metamaterials have attracted significant attention due to their distinctive properties and promising applications. Negative compressibility has been interpreted in two ways. Regarding the negative compressibility induced by a uniaxial load, it can only occur abruptly when the load reaches a certain threshold. Hence, it can be termed as transient negative compressibility. However, fabrication and experiments of such metamaterials have rarely been reported. Herein, we demonstrate them. Inspired by Braess’s paradox, a novel mechanical model is proposed with reversible negative compressibility. It shows multiple types of force responses during a loading-unloading cycle, including transient negative compressibility and hysteresis. Phase diagrams are employed to visualize the relationship between force responses and system parameters. Besides, explicit expressions for the conditions and intensity of negative compressibility are obtained for design and optimization. The model replacement method inspired by compliant mechanism design is then introduced to derive specific unit cell structures, thus avoiding intuition-based approaches. Additive manufacturing technology is utilized to fabricate the prototypes, and negative compressibility is validated via simulations and experiments. Furthermore, it is demonstrated that metamaterials with transient negative compressibility can be activated through electrical heating and can function as actuators, thereby possessing machine-like properties. The proposed mechanical metamaterial and the introduced design methodology have potentials to impact micro-electromechanical systems, force sensors, protective devices, and other applications.

The influence functional in open holography: entanglement and Rényi entropies

Open quantum systems are defined as ordinary unitary quantum theories coupled to a set of external degrees of freedom, which are introduced to take on the rôle of an unobserved environment. Here we study examples of open quantum field theories, with the aid of the so-called Feynman-Vernon Influence Functional (“IF”), including field theories that arise in holographic duality. We interpret the system in the presence of an IF as an open effective field theory, able to capture the effect of the unobserved environment. Our main focus is on computing Rényi and entanglement entropies in such systems, whose description from the IF, or “open EFT”, point of view we develop in this paper. The issue of computing the entanglement-Rényi entropies in open quantum systems is surprisingly rich, and we point out how different prescriptions for the IF may be appropriate depending on the application of choice. A striking application of our methods concerns the fine-grained entropy of subsystems when including gravity in the setup, for example when considering the Hawking radiation emitted by black holes. In this case we show that one prescription for the IF leads to answers consistent with unitary evolution, while the other merely reproduces standard EFT results, well known to be inconsistent with unitary global evolution. We establish these results for asymptotically AdS gravity in arbitrary dimensions, and illustrate them with explicit analytical expressions for the IF in the case of matter-coupled JT gravity in two dimensions.

Nonstationary random vibration analysis of hysteretic systems with fractional derivatives by FFT-based frequency domain method

Nonstationary random vibration problems of nonlinear fractional systems are challenging and drawing increasing attention. This study presents an efficient numerical method for the random vibration analysis of large-scale nonlinear fractional systems subjected to fully nonstationary excitations. Firstly, the fast Fourier transform (FFT)-based frequency domain method originally proposed for the nonstationary random vibration analysis of linear integer-order systems is extended to linear fractional systems, in which the explicit expression of dynamic responses of fractional systems is derived based on the load superposition principle and Newmark method. Secondly, an equivalent linearization analysis framework is constructed to solve the nonstationary responses of hysteretic systems with fractional derivatives, in which the nonstationary response analyses of equivalent linear systems are accomplished by the FFT-based frequency domain method. The presented method can account for nonstationary excitations with arbitrary forms of evolutionary power spectrum, even of the non-separable one. Two numerical examples are used to confirm its superior performance in terms of accuracy and computational efficiency.

A variable-node element model and its application to damage analysis of RAC using the 2D base force element method

The existing methods find it difficult to resolve the inconsistency in the nodal displacements of nodes at the interface of dense and sparse element models. Therefore, this paper proposes a new variable-node element model using the base force element method (BFEM) and successfully applies it to study the mechanical properties of recycled aggregate concrete (RAC). First, the variable-node element model with an arbitrary change of node number in the mid-edges was established, and the explicit expressions of the 2 × 2 nodal compliance matrix and nodal displacements were given. Subsequently, the triangular and quadrilateral elements of single-sides and double-sides of hanging element models were developed based on the variable-node element model. In addition, a double-notched model of RAC with different aggregate types but the same aggregate distribution position was established, and numerical simulations of uniaxial tension were performed. The results show that the compliance matrix and nodal displacement are independent of the element characteristic without Gaussian integration. The variable-node hanging element model has high numerical accuracy and applicability, and the nodes between the sparse and dense meshes do not require any construction measures. The variable-node elements ensure the coordination of the nodal displacement at the interfaces. The aggregate shape has a significant effect on the damage mode, maximum principal strain, principal stress distribution, and stress concentration, while has a negligible effect on RAC tensile strength when the aggregate distribution is the same.

TMD factorisation for diffractive jets in photon-nucleus interactions

Using the colour dipole picture and the colour glass condensate effective theory, we study the diffractive production of two or three jets via coherent photon-nucleus interactions at high energy. We consider the hard regime where the photon virtuality and/or the transverse momenta of the produced jets are much larger than the saturation momentum Qs of the nuclear target. We show that, despite this hardness, the leading-twist contributions are controlled by relatively large parton configurations, with transverse sizes R ~ 1/Qs, which undergo strong scattering and probe gluon saturation. We demonstrate that these leading-twist contributions admit transverse-momentum dependent (TMD) factorisation, in terms of quark and gluon diffractive TMD distribution functions, for which we obtain explicit expressions from first principles. We go beyond our previous work by evaluating the contributions involving the quark diffractive distributions and by establishing that their DGLAP evolution emerges via controlled calculations within the colour dipole picture. We find the same expression for the quark diffractive TMD in two different processes (semi-inclusive diffraction and the diffractive production of quark-gluon dijets), thus demonstrating its universality.

A Partial Information Decomposition for Multivariate Gaussian Systems Based on Information Geometry.

There is much interest in the topic of partial information decomposition, both in developing new algorithms and in developing applications. An algorithm, based on standard results from information geometry, was recently proposed by Niu and Quinn (2019). They considered the case of three scalar random variables from an exponential family, including both discrete distributions and a trivariate Gaussian distribution. The purpose of this article is to extend their work to the general case of multivariate Gaussian systems having vector inputs and a vector output. By making use of standard results from information geometry, explicit expressions are derived for the components of the partial information decomposition for this system. These expressions depend on a real-valued parameter which is determined by performing a simple constrained convex optimisation. Furthermore, it is proved that the theoretical properties of non-negativity, self-redundancy, symmetry and monotonicity, which were proposed by Williams and Beer (2010), are valid for the decomposition Iig derived herein. Application of these results to real and simulated data show that the Iig algorithm does produce the results expected when clear expectations are available, although in some scenarios, it can overestimate the level of the synergy and shared information components of the decomposition, and correspondingly underestimate the levels of unique information. Comparisons of the Iig and Idep (Kay and Ince, 2018) methods show that they can both produce very similar results, but interesting differences are provided. The same may be said about comparisons between the Iig and Immi (Barrett, 2015) methods.

Data-driven modeling of multi-stable origami structures: Extracting the global governing equation and exploring the complex dynamics

In recent years, multi-stable origami structures have garnered increasing attention for their applications in dynamic scenarios such as robotic arm motions, impact energy absorption, and spectrum gap regulation. Understanding the intricate working mechanisms and exploring the rich dynamics of these structures necessitate the development of dynamic models. However, existing dynamic modeling methods for origami structures are often cumbersome, and the resulting dynamic models often lack interpretability. To overcome these limitations, we propose a novel data-driven dynamic modeling approach based on B-spline Galerkin method and subset selection strategy. This approach directly captures the dynamics of multi-stable origami structures using measured data, eliminating the need for reliance on empirical or prior knowledge. To validate the effectiveness of the proposed approach, we first evaluate it on the Duffing system, which has explicit expressions, successfully reconstructing the governing equation. Subsequently, we apply this method to the dynamic modeling of the origami ball structure with tri-stability and the multi-cell stacked Miura-origami (SMO) structure with high dimensionality through simulation, showing favorable results. Finally, using experimental data collected from a bi-stable SMO structure prototype, we employ the proposed method to obtain a global model that can accurately predict different dynamic behaviors over a broad range of excitation frequencies, including intra-well periodic vibrations, inter-well periodic vibrations, and inter-well chaotic vibrations. Overall, our method showcases outstanding efficacy in formulating interpretable, manageable, and comprehensive dynamic models. It plays a pivotal role in delving into the intricate dynamics of multi-stable origami structures.

Augmented Lagrangian method for nonlinear circular conic programs: a local convergence analysis

In this paper, we analyse a local convergence of augmented Lagrangian method (ALM) for a class of nonlinear circular conic optimization problems. In light of the singular value decomposition, the Debreu theorem and the implicit function theorem, we prove that the sequence generated by ALM converges to a local minimizer in the linear convergence rate under the constraint nondegeneracy condition and the strong second-order sufficient condition, in which the ratio constant is proportional to 1 / τ , where τ is the associated penalty parameter with a given lower threshold. As a byproduct, we also derive explicit expressions of critical cone and its affine hull for the given nonlinear circular conic program.

Total alkalinity change: The perspective of phytoplankton stoichiometry

AbstractMany biogeochemical processes change total alkalinity: this has been reported for carbonate precipitation and dissolution, uptake and release of various nitrogen‐containing compounds, uptake of phosphate, sulfate reduction combined with methane oxidation. However, the list is not exhaustive. Here we discuss additional processes, namely the uptake of Mg, K, Ca by phytoplankton, and calculate their contribution based on the explicit conservative expression for TA and an extended (compared to Redfield's C : N : P) stoichiometry of phytoplankton. These additional contributions are of the same order of magnitude as that of phosphate uptake and thus much smaller than the contribution from nitrate uptake and of opposite sign to the contribution by nitrate and phosphate uptake.

Hypergeometric expressions for type I Jacobi–Piñeiro orthogonal polynomials with arbitrary number of weights

For a general number p ≥ 2 p\geq 2 of measures, we provide explicit expressions for the Jacobi–Piñeiro and Laguerre of the first kind multiple orthogonal polynomials of type I, presented in terms of generalized hypergeometric functions.

Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds

We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.

N-order solutions to the Gardner equation in terms of Wronskians

$N$-order solutions to the Gardner equation (G) are given in terms of Wronskians of order $N$ depending on $2N$ real parameters. We get solutions expressed with trigonometric or hyperbolic functions. When one of the parameters goes to $0$, we succeed to get for each positive integer $N$, rational solutions as a quotient of polynomials in $x$ and $t$ depending on $2N$ real parameters. We construct explicit expressions of these rational solutions for the first orders.

Adaptive Sliding Mode Fixed-/Preassigned-Time Synchronization of Stochastic Memristive Neural Networks with Mixed-Delays

The paper addresses the fixed-/preassigned-time synchronization of stochastic memristive neural networks (MNNs) with uncertain parameters and mixed delays. Adaptive sliding mode control (ASMC) technology is mainly utilized. First, a proper sliding surface is constructed and the adaptive laws are given. Also, the synchronization control scheme is designed, which can ensure error system to realize fixed-time stability. Second, preassigned-time sliding mode control scheme is mainly provided to realize fast synchronization of MNNs. The presented theoretical methods can guarantee the error system convergence and stability for reaching and sliding mode within preassigned-time. And the synchronization criteria and explicit expression of settling time (ST) are acquired, where ST is not related with initial values and controller parameters but can be predefined perferentially. Finally, the calculation example is offered to interpret the practicability and availability of the innovations in this paper.

Induced moduli oscillation by radiation and space expansion in a higher-dimensional model

We investigate the cosmological expansion of the 3D space in a 6D model compactified on a sphere, beyond the 4D effective theory analysis. We focus on a case where the initial temperature is higher than the compactification scale. In such a case, the pressure for the compact space affects the moduli dynamics and induces the moduli oscillation even if they are stabilized at the initial time. Under some plausible assumptions, we derive the explicit expressions for the 3D scale factor and the moduli background in terms of analytic functions. Using them, we evaluate the transition times between different cosmological eras as functions of the model parameters and the initial temperature. Published by the American Physical Society 2024

Theory of coherent phonons coupled to excitons

The interaction of excitons with lattice vibrations underlies the scattering from bright to dark excitons as well as the coherent modulation of the exciton energy. Unlike the former mechanism, which involves phonons with finite momentum, the latter can be exclusively attributed to coherent phonons with zero momentum. We here lay down the microscopic theory of coherent phonons interacting with resonantly pumped bright excitons and provide the explicit expression of the corresponding coupling. The coupling notably resembles the exciton-phonon one, but with a crucial distinction: it contains the bare electron-phonon matrix elements rather than the screened ones. Our theory predicts that the exciton energy features a polaronic-like red-shift and monochromatic oscillations or beatings, depending on the number of coupled optical modes. Both the red-shift and the amplitude of the oscillations are proportional to the excitation density and to the square of the exciton-coherent-phonon coupling. We validate our analytical findings through comparisons with numerical simulations of time-resolved optical absorbance in resonantly pumped MoS2 monolayers.

Large N instantons from topological strings

The 1/N1/N expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large NN instantons. Explicit expressions for large NN instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large NN instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its 1/N1/N expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large NN instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov-Witten invariants. By focusing on the example of {\mathbb C}^3/{\mathbb Z}_3ℂ3/ℤ3, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.