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In China, Xuan paper has been the paper of choice as artwork support and for conservation, for several centuries. However, little is known about its material properties, especially given the many grades of sized and unsized Xuan paper. In addition, there is a lack of information on its degradation. In this research, a selection of contemporary unsized Xuan papers was investigated, representing diverse raw materials. Seven out of twelve contemporary unsized Xuan papers were determined to be approximately neutral and contain > 2% alkaline reserve, indicating good durability. Viscometry was used to determine the degree of polymerisation (DP) as none of the samples gave significant reactions to the phloroglucinol spot test. The average DP of ten contemporary unsized Xuan papers is ~ 1700, excluding two papers that have presumably been sun-bleached, and that exhibit significantly lower DP. Using X-ray fluorescence, it can be demonstrated that Ca and Si are the dominant elements and interestingly, Ca content is directly correlated with ash content and with alkaline reserve. Accelerated degradation was performed at two sets of environmental conditions, i.e. 90 °C, 30% RH and 60 °C, 70% RH, and the established degradation rates agreed with the Collections Demography model of paper degradation meaning that degradation of Xuan papers proceeds in the same way as other types of paper. This research gives fundamental insights into contemporary unsized Xuan papers, which exhibit good stability during accelerated degradation despite the low starting DP in the context of the samples used in this study. Our findings may inform methods of Xuan paper production, selection of Xuan paper for conservation purposes, as well as preventive conservation of Xuan paper-based artefacts.

Necessary optimality conditions for a bilevel optimization problem are given in the paper by Kohli (J Optim Theory Appl 152: 632–651, 2012). Recently, the same author corrected his results in the note (J Optim Theory Appl 181:706–707, 2019). In this work, we have pointed out that some of the new modifications are wrong. We correct the flaws and present an alternative proof for the main result.

There are similarities between Drinfeld modules and abelian varieties. The purpose of this paper is to investigate these similarities in terms of supersingularity. More specifically, we provide explicit formulas of Hasse invariants (or, equivalently, of supersingular polynomials) for Drinfeld modules, which is a continuation of the 2013 paper by Ahmad El-Guindy and Matthew A. Papanikolas. We present several supersingular Drinfeld modules as an application.

This paper is concerned with some stochastic properties of a novel rock–paper–scissors model. Firstly, the global existence of an unique positive solution of the stochastic model is obtained. Then we demonstrate the positive solution of the model is stochastically bounded. Besides, some sufficient conditions for population to be stochastically permanent and extinct are derived with the use of some appropriate Lyapunov functions. At last, some numerical simulations are carried out to illustrate our theoretical analysis results.

The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.

International audience; We report a pyrolysis-gas chromatography/mass spectrometry (Py-GC/MS) method capable of analyzing traditional East Asian papers. The method proposed is based on rapid and easy single step Py-GC/MS analysis that can be carried out with a minimum amount of matter, in the few g range. Three reference papers manufactured from kozo (Broussonetia kazinoki Siebold & Zucc.), mitsumata (Edgeworthia chrysantha Lindl.) and gampi (Wikstroemia sikokiana Franch. & Sav.) with the traditional hand paper making processes were examined. The method allows discrimination between terpenic and steroid compounds, which were revealed as chemical markers of origin of the plant fibers. Each paper investigated was found to have characteristic pyrolysis fingerprints that were unique to the traditional handmade paper, demonstrating the potential for differentiation of these biochemical components of fiber plants on East Asian papers towards identification and conservation of cultural heritage. The investigation on Py-GC/MS was extended to liquid extraction followed by GC/MS analysis to characterize the biochemical components of fiber plants. The main contribution of this study is to provide molecular criteria for discriminating plant species used for traditional East Asian hand papermaking. Py-GC/MS complements efficiently microscope identification especially for adverse cases. A case study of archaeological Chinese paper painting artefacts was thereafter successfully investigated to address informative potential and efficiency of the criteria of identification on ancient and degraded East Asian paperworks.

We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ ” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).

This paper discusses Helmedag’s article concerning Faustmann’s formula (Helmedag in Eur J For Res, 2018. https://doi.org/10.1007/s10342-018-1101-8). He computed the present value of a fully regulated forest, including standing timber and theoretical land values. He showed that the optimal rotation in a fully regulated forest would always be the one that maximized the sustainable forest net revenues. Helmedag concluded that the discount rate would have no significance for the optimal rotation in a fully regulated forest, while his solution would still fulfil the optimality condition implied by Faustmann’s formula. Here, we refute this assertion. In fact, the assumption of a fully regulated forest has no impact on the optimal rotation period, or may even reduce it. We illustrate this by appropriately considering the actual costs of achieving a fully regulated forest with altered rotation. One must not ignore the alterations of the financial flows when changing the underlying rotation age in an established fully regulated forest. When the opportunity costs of the transition period are included, the optimal rotation becomes the same as that of a single even-aged forest, or shorter, depending on the transition regime. Under the optimal transition regime, the diminishing marginal rate of return for extending the rotation period in a fully regulated forest matches the discount rate, when we achieve the Faustmann rotation. We conclude that the optimal rotation period is independent of the status as fully regulated forest, provided efficient harvest operations during the transition period.

In this short article, we mainly give some examples to prove that the main results of Section 4 of the paper [W. S. Liao and J. Wu, Ann. Func. Anal. 6 (2015), no. 3, 191–202] are not true. Then we give the corrected matrix inequalities.

The dating of documents is one of the pending issues to be resolved in many fields. Although there are a variety of analytical methodologies focused on the inks of documents to estimate the dating of documents, the analysis of documents itself attracted little attention. A non-destructive method based on Fourier transformed infrared (FTIR) was proposed in this research to estimate the dating of documents. First, FTIR spectra of journals dated from 1940 to 1980, naturally aged and conserved in the China Agriculture University library were collected. Second, the least squares support vector machines (LS-SVM) was applied to distinguish documents of different dating, and the accuracy was 99.26%. In order to find interested wavenumber that influence the dating process of documents, sparse partial least squares (sPLS) was applied to select informative variables. The average of selected variables was 483 after 100 runs, and the selected variables were focused on the absorption peaks of inorganic components and cellulose. Splicing sPLS with LS-SVM (sPLS–LS-SVM) built model to see the effective of selected variables. Average accuracy of sPLS–LS-SVM model was 99.34%, even the best result could reach 100.00% after 100 circle times. The present work indicates that the possibility of FTIR combined with chemometrics can estimate the dating of documents accurately. Additionally, the wavenumber which influence the dating of documents are mostly focused on cellulose and inorganic components.

Very recently Bordelles, Dai, Heyman, Pan and Shparlinski studied asymptotic behaviour of the quantity $$\begin{aligned} S_f(x) := \sum _{n\leqslant x} f\left( \left[ \frac{x}{n}\right] \right) , \end{aligned}$$and established some asymptotic formulas for $$S_f(x)$$ under three different types of assumptions on f. In this short note we improve some of their results.

In this work, some counterexamples are given to refute some results reported in the paper by Guo and Li [8] (J Optim Theory Appl 162,(2014), 821-844). We correct the faulty in some of their theorems and we present alternative proofs. Moreover, we extend the definition of approximately pseudo-dissipative in the setting of metrizable topological vector spaces.

Studying of paper carriers of information has become increasingly important in forensic practice. This is caused by growing number of bogus or in other manner fraudulently alternated documents. An identification of individual paper sheets and studying their differences are the parts of forensic analysis of multiple sheets documents. This paper focuses on the possibilities of processing of mathematical and statistical data obtained by measurements of FT-IR spectral characteristics of individual paper sheets in a studied document. The spectral data were used to create data matrix which contains values of absorbance and interval of wavenumber. The method of differentiation is based on the comparison of maximum absorbance values within the range of wavenumbers from 1562 to 1215 cm−1. This spectrum range is specific in occurrence of absorption bands that belong to calcium carbonate CaCO3—an element of paper added into it in relatively high amounts in the form of filler which content or percentage occurrence may be different considering to the paper producer or production technology. Based on the above-said, the paper sheets originating from different producers or from various production technology units can be differentiated one from another using a robust statistical analysis.

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

What can be learned from data about human survival at extreme age? In this rejoinder we give our views on some of the issues raised in the discussion of our paper Rootzen and Zholud (Extremes 20(4), 713–728, 2017).

In this short note, we show that Theorem 4.3 of Liu and Yu (Monatshefte Math 195:173–176, 2021) is a consequence of Lemma 2 of Ballester-Bolinches and Esteban-Romero (J Aust Math Soc 75:181–191, 2003).

We thank Holger Rootzen and Dmitrii Zholud’s for their stimulating work, that led us to further investigate the problem of best fitting the human life span distribution. Their assertion “human life is unlimited but short” is based on their conclusion that the exponential model is the best to fit. We provide results based on standard Extreme Value approaches, the Block Maxima and Peaks-over-Threshold, and on the whole data available in the IDL database (IDL A, B validations and GRG). We verify that negative values for the extreme value index are more likely, supporting the conclusion that models with finite endpoint seem better to fit to the human life span distribution.

O. Einstein (2008) proved Bollob\'as-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants., Comment: 6 pages. Improved bound for Theorem 4

In this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32---49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on $$\mathbb {R}^n$$Rn. Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases.

We give comments for the paper from Liu et al. (2019) about the Item Response Theory (IRT) model under consideration, the justification to compute Marginal likelihood, about what we learn with the data analysis performed and finally, about the computational issues in this paper.

This is an invited discussion of review paper “Nonparametric Bayesian Inference in Applications” by Peter Muller, Fernando A. Quintana and Garritt L. Page.

In this paper we consider, discuss and slightly complement recent fixed point results for mappings in b-metric spaces established by Sintunavarat (J Fixed Point Theory Appl 18:397–416, 2016). Thus, all our results are with shorter proofs. In addition, an application to integral equations is given to illustrate the usability of our approach.

This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper (LeCrone et al. J Evolut Equ 14:509–533 2014). The improvement is explained in “The improvement” section and proved in “Proof of the main result” section. The importance of the extension is demonstrated in “Application to the Navier–Stokes equations” section with an application to the Navier–Stokes system in critical \(L_q\)-spaces.

In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13---27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem.

Amongst all forest utilization phases, harvesting plays a key role as the beginning phase of the utilization cycle and has a significant effect on the following phases. In this research, in order to examine the reduction of costs with respect to present and planned operations, the starting time of the harvesting or cutting phase and work flow were recorded. The starting time of each phase of the entire operation was recorded as well. Before harvesting the marked tree, the appropriate felling direction was determined and the azimuth of the correct direction was taken. The results indicate that there was not a meaningful relationship between the differences in direction of a leaning tree compared with the felling direction. However, there was a significant relationship between the difference of the defined and felling direction and the difference in tree leaning and defined direction. The same trend was observed between tree leaning and felling directions and tree leaning and defined directions. On the other hand, there was no significant relationship between tree leaning and felling directions and the defined and tree felling directions. In addition, this research shows that with an increase in tree volume, the time for tree cutting rose. Furthermore, when the difference of tree leaning and felling direction decreased, the time to cut the tree declined. The regression equation is Y = 168.9–0.14 (the difference between tree leaning and felling direction) + 0.7 (volume).

Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.

A criteria due to Toivo Nieminen concerning selfadjoint linear operators in complex Hilbert spaces is extended to the case of linear relations.

We summarize the papers published by Einstein in the Annalen der Physik in the years 1902-04 on the derivation of the properties of thermal equilibrium on the basis of the mechanical equations of motion and of the calculus of probabilities. We point out the line of thought that led Einstein to an especially economical foundation of the discipline, and to focus on fluctuations of the energy as a possible tool for establishing the validity of this foundation. We also sketch a comparison of Einstein's approach with that of Gibbs, suggesting that although they obtained similar results, they had different motivations and interpreted them in very different ways., Comment: 22 pages, submitted to JSP

In this note, we give affirmative answers to two open problems posed by Agarwal et al. in the paper (J Optim Theory Appl 163(2):482---490, 2014).

This note concerns about the conclusion of a lemma of a published paper of this journal.

For any discrete valuation ring R, any R-linear mapping u from an R-module E into an R-module F and any \(y_0\in F\), a necessary and sufficient condition for the solvability of the equation \(u(x)=y_0\) is established, and an application of this result is presented.

We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s method.

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

This special issue of METRIKA contains mainly a collection of papers presented at the 3rd Workshop on Goodness-of-Fit and Change-Point problems, which was supported by the German Research Foundation (DFG), and was held at the Church House of the Protestant Academy Baden in Bad Herrenalb in the Black Forest near Karlsruhe, Germany, 8–10 September, 2017. The first workshop in this series took place in 2012 at the University of Sevilla, and the second was held at the National and Kapodistrian University of Athens, 2015. There is already a follow-up event that will take place in 2019 at the University of Trento, Italy. We firmly believe that this series of workshops will become a regular international meeting

Stochasticity plays an important role in the evolutionary dynamic of cyclic dominance within a finite population. To investigate the stochastic evolution process of the behaviour of bounded rational individuals, we model the Rock-Scissors-Paper (RSP) game as a finite, state dependent Quasi Birth and Death (QBD) process. We assume that bounded rational players can adjust their strategies by imitating the successful strategy according to the payoffs of the last round of the game and then analyse the limiting distribution of the QBD process for the game stochastic evolutionary dynamic. The numerical experiments results are exhibited as pseudo colour ternary heat maps. Comparisons of these diagrams shows that the convergence property of long run equilibrium of the RSP game in populations depends on population size and the parameter of the payoff matrix and noise factor. The long run equilibrium is asymptotically stable, neutrally stable and unstable respectively according to the normalised parameters in the payoff matrix. Moreover, the results show that the distribution probability becomes more concentrated with a larger population size. This indicates that increasing the population size also increases the convergence speed of the stochastic evolution process while simultaneously reducing the influence of the noise factor.

Given a convex body $$K$$K, consider the smallest number $$N$$N so that there is a point $$P\in \partial K$$PźźK such that every circle centred at $$P$$P intersects $$\partial K$$źK in at most $$N$$N points. In 1946 Erdźs conjectured that $$N=2$$N=2 for all $$K$$K, but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $$N=\infty $$N=ź and that there are convex bodies for which $$N = 6$$N=6.

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

In this commentary, we try to make clearer the state of the art concerning the relation between mechanical contact interactions and the different notions of stresses. We emphasize the importance of the concept of virtual displacements. Its role has been recognized in Mechanics and in Continuum Mechanics long ago (see e.g., Vailati in Il principio dei lavori virtuali da Aristotele a Erone d’Alessandria, 113–128, 1987; Russo in The forgotten revolution, Springer, Berlin, 2003, or Cosserat and Cosserat in Sur la Theorie des Corps Deformables, Herman, Paris, 1909; Cosserat and Cosserat in Note sur la theorie de l.action euclidienne, Gauthier-Villars, Paris, 1908), and it is central as well when starting with an expression of the power expended by internal stresses and deducing the form of contact interactions as when starting with some form of the contact interactions and developing a representation theorem for these contact interactions based on the Cauchy tetrahedron construction.

In this note we correct an error in the paper (Ariza-Ruiz et al. in J Optim Theory Appl 167:409–429, 2015).

Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

The traditional measurement of quality indices cannot meet the requirements of new products, precise and intelligent control. Prediction models of pulp components are established based on analyzing the correlation between cellulose, hemicellulose, lignin content and whiteness, Kappa number, intrinsic viscosity, respectively, in the chlorine dioxide bleaching process. The results show that the models can simulate the varieties of components in bleaching process and judge the quality of bleached pulp more directly than traditional quality indices. The models have a practical significance for paper products requiring the concerned component and match to the tendency of precise and intelligent control of the bleaching process.

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S$$\end{document}R0S corresponding to the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0$$\end{document}R0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.

This paper studies an SEIR-type epidemic model with time delay and vaccination control. The vaccination control is applied when the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0>1$$\end{document}R0>1. The vaccination strategy is expressed as a state delayed feedback which is related to the current and previous state of the epidemic model, and makes the model become a linear system in new coordinates. For the presence and absence of vaccination control, we investigate the nonnegativity and boundedness of the model, respectively. We obtain some sufficient conditions for the eigenvalues of the linear system such that the nonnegativity of the epidemic model can be guaranteed when the vaccination strategy is applied. In addition, we study the stability of disease-free equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_01$$\end{document}R0>1. Finally, we use the obtained theoretical results to simulate the vaccination strategy to control the spread of COVID-19.

Månsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a remedy, Türkan and Özel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE ($$ \hat{k}_{q4} $$ k ^ q 4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation.

New analytical solutions for the static deflection of anisotropic composite beams resting on variable stiffness elastic foundations are obtained by the means of the Homotopy Analysis Method (HAM). The method provides a closed-form series solution for the problem described by a non-homogeneous system of coupled ordinary differential equations with constant coefficients and one variable coefficient reflecting variable stiffness elastic foundation. Analytical solutions are obtained based on two different algorithms, namely conventional HAM and iterative HAM (iHAM). To investigate the computational efficiency and convergence of HAM solutions, the preliminary studies are performed for a composite beam without elastic foundation under the action of transverse uniformly distributed loads considering three different types of stacking sequence which provide different levels and types of anisotropy. It is shown that applying the iterative approach results in better convergence of the solution compared with conventional HAM for the same level of accuracy. Then, analytical solutions are developed for composite beams on elastic foundations. New analytical results based on HAM are presented for the static deflection of composite beams resting on variable stiffness elastic foundations. Results are compared to those reported in the literature and those obtained by the Chebyshev Collocation Method in order to verify the validity and accuracy of the method. Numerical experiments reveal the accuracy and efficiency of the Homotopy Analysis Method in static beam problems.