Convergence Of Expansion Research Articles

A q-Analog of the Class of Completely Convex Functions and Lidstone Series

This paper introduces a q-analog of the class of completely convex functions. We prove specific properties, including that q-completely convex functions have convergent q-Lidstone series expansions. We also provide a sufficient and necessary condition for a real function to have an absolutely convergent q-Lidstone series expansion.

Renormalization of nuclear chiral effective field theory with nonperturbative leading-order interactions

We extend the renormalizability study of the formulation of chiral effective field theory with a finite cutoff, applied to nucleon-nucleon scattering, by taking into account non-perturbative effects. We consider the nucleon-nucleon interaction up to next-to-leading order in the chiral expansion. The leading-order interaction is treated non-perturbatively. In contrast to the previously considered case when the leading-order interaction was assumed to be perturbative, new features related to the renormalization of the effective field theory are revealed. In particular, more severe constraints on the leading-order potential are formulated, which can enforce the renormalizability and the correct power counting for the next-to-leading order amplitude. To illustrate our theoretical findings, several partial waves in the nucleon-nucleon scattering, $^3P_0$, $^3S_1-{^3D_1}$ and $^1S_0$ are analyzed numerically. The cutoff dependence and the convergence of the chiral expansion for those channels are discussed.

Multi-parameter identification of concrete dam using polynomial chaos expansion and slime mould algorithm

This paper presents a novel methodology that combines polynomial chaos expansion and slime mould algorithm for multi-parameter identification of concrete dams. This methodology not only incorporates the merits of low computational cost in the polynomial chaos expansion and fast convergence of slime mould algorithm, but also considers the priori uncertainty in the input parameters by introducing statistical probability theory. By considering two examples with different complexity, this paper verifies the effectiveness of the proposed method with a univariate simply supported beam model, followed by a complex multivariate dam model to demonstrate its practicability in real engineering problems. In addition, parameter sensitivity analysis of the dam model is conducted at an extremely low cost by polynomial chaos expansion based on Sobol’ indices. Furthermore, the conventional parameter identification methods based on optimization methods directly combined with the finite element model are employed for comparison, highlighting two distinct advantages of the proposed method: (i) the proposed method improves the computational efficiency by nearly 52 times while ensuring a high accuracy, and (ii) the classical non-population optimization algorithm, Bayesian optimization, is used for comparison, revealing the outstanding performance of slime mould algorithm in terms of convergence speed and robustness. The application of the proposed algorithm is not only limited to dams, but also it can be extended to any structure.

Hausdorff dimension of sets defined by almost convergent binary expansion sequences

AbstractIn this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set \begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*} is determined for any $ \alpha\in[0,1] $ . This completes a question considered by Usachev [Glasg. Math. J.64 (2022), 691–697] where only the dimension for rational $ \alpha $ is given.

Extremal Inhomogeneous Gibbs States for SOS-Models and Finite-Spin Models on Trees

We consider $${\mathbb {Z}}$$ -valued p-SOS-models with nearest neighbor interactions of the form $$|\omega _v-\omega _w|^p$$ , and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any symmetries of the tree. This generalises the results of Gandolfo et al. (J. Stat. Phys. 148:999–1005, 2012) about the Ising model, and shows that the latter behaviour is robust. We treat three different types of extensions: non-compact state space gradient models, models without spin-symmetry, and models in small random fields. We give a detailed construction and full proofs of the extremality of the low-temperature states in the set of all Gibbs measures, analysing excess energies relative to the ground states, convergence of low-temperature expansions, and properties of cutsets.

Hologenome analysis reveals independent evolution to chemosymbiosis by deep-sea bivalves

BackgroundBivalves have independently evolved a variety of symbiotic relationships with chemosynthetic bacteria. These relationships range from endo- to extracellular interactions, making them ideal for studies on symbiosis-related evolution. It is still unclear whether there are universal patterns to symbiosis across bivalves. Here, we investigate the hologenome of an extracellular symbiotic thyasirid clam that represents the early stages of symbiosis evolution.ResultsWe present a hologenome of Conchocele bisecta (Bivalvia: Thyasiridae) collected from deep-sea hydrothermal vents with extracellular symbionts, along with related ultrastructural evidence and expression data. Based on ultrastructural and sequencing evidence, only one dominant Thioglobaceae bacteria was densely aggregated in the large bacterial chambers of C. bisecta, and the bacterial genome shows nutritional complementarity and immune interactions with the host. Overall, gene family expansions may contribute to the symbiosis-related phenotypic variations in different bivalves. For instance, convergent expansions of gaseous substrate transport families in the endosymbiotic bivalves are absent in C. bisecta. Compared to endosymbiotic relatives, the thyasirid genome exhibits large-scale expansion in phagocytosis, which may facilitate symbiont digestion and account for extracellular symbiotic phenotypes. We also reveal that distinct immune system evolution, including expansion in lipopolysaccharide scavenging and contraction of IAP (inhibitor of apoptosis protein), may contribute to the different manners of bacterial virulence resistance in C. bisecta.ConclusionsThus, bivalves employ different pathways to adapt to the long-term co-existence with their bacterial symbionts, further highlighting the contribution of stochastic evolution to the independent gain of a symbiotic lifestyle in the lineage.

Decay constant of [formula omitted] accurate up to [formula omitted

In this paper, we evaluate, up to QCD next-to-next-to-next-to-leading order, the Bc⁎ decay constant, which, within the nonrelativistic QCD (NRQCD) framework, is factorized as the product of the short-distance coefficient (SDC) and the long-distance matrix element. For the first time, the renormalization constant and the anomalous dimension for the NRQCD vector current composed of cb¯, which are functions of the charm quark mass mc, the bottom quark mass mb and the factorization scale μΛ, are obtained analytically at O(αs2) and O(αs3). The SDC is calculated up to O(αs3) in perturbative expansion. Explicitly, the O(αs2) correction to the SDC is analytically calculated in terms of logarithmic and polylogarithmic functions of r≡mb/mc, and the O(αs3) correction to the SDC is numerically calculated at a series of values of r, ranging from 2.1 to 4.0. Surprisingly, we find that the nontrivial part of the SDC at O(αs3) can be well estimated by a linear function of r. In addition, we find that the O(αs2) and O(αs3) corrections to the decay constant and decay width are considerable and very significant, which indicates a very poor convergence for the perturbative expansion.

Contrasting patterns of disparity suggest differing constraints on the evolution of trilobite cephalic structures during the Cambrian ‘explosion’

AbstractTrilobites are an abundant group of Palaeozoic marine euarthropods that appear abruptly in the fossil recordc. 521 million years ago. Quantifying the development of morphological variation (or ‘disparity’) through time in fossil groups like trilobites is critical in understanding evolutionary radiations such as the Cambrian ‘explosion’. Here, I use geometric morphometrics to quantify ‘cumulative disparity’ in functionally‐important structures within the trilobite cephalon across their initial radiation during Cambrian Series 2. Overall cephalic disparity increased rapidly and attained a maximum within several million years. This pattern is dominated by the cephalic outline (in particular the genal spines), reflecting rapid, convergent expansion to the extremes of morphospace in a few early families. In contrast, removing the outline and focusing on structures such as the glabella and eye ridges (associated with feeding and vision, respectively) showed a more gradual increase in disparity, closer in line with taxonomic diversity and supporting the hypothesis of a relatively accurate trilobite fossil record. These contrasting patterns suggest that disparity in different structures was constrained in different ways, with extrinsic (ecological) factors probably having the major impact on overall disparity. It also implies that patterns of disparity in isolated substructures cannot necessarily be taken individually as representative of overall morphologies.

Uniform Convergent Expansions of the Error Function in Terms of Elementary Functions

We derive a new analytic representation of the error function {text {erf}},, z in the form of a convergent series whose terms are exponential and rational functions. The expansion holds uniformly in z in the double sector vert arg (pm z)vert <pi /4. The expansion is accompanied by realistic error bounds.

Hot and dense QCD shear viscosity at leading log

The leading-order weak-coupling shear viscosity of QCD was computed almost 20 years ago, and the extension to next-to-leading order is 4 years old. But these results have never been applied at finite baryon chemical potential μ, despite the fact that intermediate-energy heavy ion collisions and merging neutron stars may explore the Quark-Gluon Plasma in a regime where baryon chemical potentials are large. Here we extend the leading-log shear viscosity calculation to finite μ, and we argue that the convergence of the weak-coupling expansion, while questionable for achievable plasmas, should be better at μ > T than at μ = 0.

On the Analytical Solution of Fractional SIR Epidemic Model

This article presents the solution of the fractional SIR epidemic model using the Laplace residual power series method. We introduce the fractional SIR model in the sense of Caputo’s derivative; it is presented by three fractional differential equations, in which the third one depends on the first coupled equations. The Laplace residual power series method (LRPSM) is implemented in this research to solve the proposed model, in which we present the solution in a form of convergent series expansion that converges rapidly to the exact one. We analyze the results and compare the obtained approximate solutions to those obtained from other methods. Figures and tables are illustrated to show the efficiency of the LRPSM in handling the proposed SIR model.

Forward inclusive jet productions in pA collisions

Motivated by recent experimental LHC measurements on the forward inclusive jet productions and based on our previous calculations on forward hadron productions, we calculate a single inclusive jet cross section in $pA$ collisions at forward rapidity within the color glass condensate framework up to the next-to-leading order. Moreover, with the application of a jet algorithm and proper subtraction of the rapidity and collinear divergences, we further demonstrate that the resulting next-to-leading-order hard coefficients are finite. In addition, in order to deal with the large logarithms that can potentially spoil the convergence of the perturbative expansion and improve the reliability of the numerical predictions, we introduce the collinear jet function and the threshold jet function and resum these large logarithms hidden in the hard coefficients.

On expansions for nonlinear systems Error estimates and convergence issues

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied. First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation. Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input. Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion. Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

Approximation of functions by wavelet expansions with dilation matrix

In this paper, we obtain the degree of approximation of a function f in Lp(1 ? p ? ?) norm under general conditions of the pointwise and uniform convergence of wavelet expansions associated with the multiresolution analysis with dilation matrix. Our results show that the degree has the exponential decay (faster than any polynomial) for the function f in Lp(R) on a finite interval (a, b).

Convergence of Formal Solutions to the Second Member of the Fourth Painlevé Hierarchy in a Neighborhood of Zero

The second member of the fourth Painlevé hierarchy is considered. Convergence of certain power asymptotic expansions in a neighborhood of zero is proved. New families of power asymptotic expansions are found. Computations are carried out using a computer algebra system. Reference to a code that can be used for computing the Gevrey order of the formal expansion of the solution to the second-order differential equation in a symbolic computation packet is given.

Stability of ground state degeneracy to long-range interactions

We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g. power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (a) the stability of the gap, and (b) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density–density interactions with an exponent greater than 1.

Matrix group integrals, surfaces, and mapping class groups II: $$\textrm{O}\left( n\right) $$ and $$\textrm{Sp}\left( n\right) $$

Let w be a word in the free group on r generators. The expected value of the trace of the word in r independent Haar elements of \(\textrm{O}(n)\) gives a function \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) of n. We show that \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) has a convergent Laurent expansion at \(n=\infty \) involving maps on surfaces and \(L^{2}\)-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning \({\textrm{U}}\left( n\right) \), yet with some surprising twists. A priori to our result, \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) does not change if w is replaced with \(\alpha (w)\) where \(\alpha \) is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under \(\textrm{Aut}({\textbf{F}} _{r})\). As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of \({\mathcal {T}}r _{w}^{\textrm{O}}(n)\) as \(n\rightarrow \infty \), we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani. Our results are obtained more generally for a tuple of words \(w_{1},\ldots ,w_{\ell }\), leading to functions \({\mathcal {T}}r_{w_{1},\ldots ,w_{\ell }} ^{\textrm{O}}\). We also obtain all the analogous results for the compact symplectic groups \(\textrm{Sp}(n)\) through a rather mysterious duality formula.

Convergence of Wavelet Expansion at Generalized Continuous Points

Convergence of Wavelet Expansion at Generalized Continuous Points

Conditional Expanding of Functions by q-Lidstone Series

This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f(z) has such an expansion, in which case convergence is uniform on compact sets.

Convergence of series expansions in rovibrational configuration interaction (RVCI) calculations.

Rotational and rovibrational spectra are a key in astrophysical studies, atmospheric science, pollution monitoring, and other fields of active research. The ab initio calculation of such spectra is fairly sensitive with respect to a multitude of parameters and all of them must be carefully monitored in order to yield reliable results. Besides the most obvious ones, i.e., the quality of the multidimensional potential energy surface and the vibrational wavefunctions, it is the representation of the μ-tensor within the Watson Hamiltonian, which has a significant impact on the desired line lists or simulated spectra. Within this work, we studied the dependence of high-resolution rovibrational spectra with respect to the truncation order of the μ-tensor within the rotational contribution and the Coriolis coupling operator of the Watson operator. Moreover, the dependence of the infrared intensities of the rovibrational transitions on an n-mode expansion of the dipole moment surface has been investigated as well. Benchmark calculations are provided for thioformaldehyde, which has already served as a test molecule in other studies and whose rovibrational spectrum was found to be fairly sensitive. All calculations rely on rovibrational configuration interaction theory and the discussed high-order terms of the μ-tensor are a newly implemented feature, whose theoretical basics are briefly discussed.