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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.

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).

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

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

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

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.

We present some rigorous results about the bending behaviour of paper. By adapting these results to the Mobius strip, we obtain some qualitative properties of developable Mobius strips which minimize the bending energy. We also provide some numerical simulations which illustrate and strengthen the analytic results.

We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients., Comment: 8 pages

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

The archimedean solids Cubus simus (snub cube) and Dodecaedron simum (snub dodecahedron) cannot be constructed by ruler and compass. We explain that for general reasons their vertices can be constructed via paper folding on the faces of a cube, respectively dodecahedron, and we present explicit folding constructions. The construction of the Cubus simus is particularly elegant. We also review and prove the construction rules of the other Archimedean solids., Comment: 13 pages, 12 figures, v2: as published in Geometriae Dedicata

*Correspondence: wadade@se.kanazawa-u.ac.jp 3Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan Full list of author information is available at the end of the article The purpose of this note is to clarify the novelty of the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’ which was published in the J. Inequal. Appl. :,  []. After this paper was published, the authors were informed of the references [–], and [], the results of which partly overlap with those of []. In the paper [], the authors established the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces H n p p,q(R); see Section  in [] for the precise definition of H n p p,q(R). The main theorem in [] is stated as follows. Theorem A [, Theorem .] Let n ∈ N,  < p

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.

An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.

We will revisit the main theorem in [1] in which given a Turing machine program P and an input X, a formula Ω(X,P ) is constructed, about which it is asserted, among other things, that it is true iff the machine P accepts an input X in less than 2|X| steps. We claim that this is not true. Counterexample. As X we take a word 11 having length 2. The input inscribed on the tape before the computation starts consists of symbols 01100. . . (followed by zeros). We construct a Turing machine P accepting all nonempty words and rejecting the empty word: instruction M(0) : q10 → q3R, instruction M(1) : q30 → q2L, instruction M(2) : q31 → q0L, instruction M(3) : q00 → q00, instruction M(4) : q20 → q20. We show how P operates on input X. Before the first step, the head scans a cell with symbol 0 and sees 0. Step 1. Perform the instruction M(0) : q10 → q3R, after which the head scans cell 1 and sees 1. Step 2. Perform the instruction M(2) : q31 → q0L, after which the head scans cell 0 and sees 0. Step 3. Perform the instruction M(3) : q00 → q00, after which the head scans cell 0 and sees 0. Thus the Turing machine constructed accepts input X in less than 2|X| = 4 steps. Now consider a formula Ω(X,P ) constructed given a program P and an input X. Our goal is to check the value of a formula χ(0)&R(P ) → χ(ω) for the following values of variables

We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) + + ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions.

We considered in Example 3.1 of the paper [1] an S-structure on R2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.

for their paper “Preconditioning Indefinite Systems in Interior Point Methods for Optimization”, published in Volume 28, pages 149–171. This paper describes a class of indefinite preconditioners for reduced KKT systems arising in quadratic and nonlinear optimization with interior point methods. Spectral analysis of preconditioners is given and the improvements resulting from the use of primal-dual regularization are demonstrated. Computational results are reported for the application of the preconditioner to a variety of medium-size convex quadratic programming problems. Work on the paper began during the Summer of 2001, about a year after the authors had first met when Zilli invited Gondzio to give a series of lectures on interior point methods in Padova in September, 2000. The authors had complementary research backgrounds: Bergamaschi and Zilli worked on the theory and implementation of iterative methods for linear and nonlinear systems of equations [2], while Gondzio worked on the design and implementation of interior point methods for large-scale optimization [3]. This was essential since together, the three authors could tackle in depth a problem which required expertice in several areas. At that time Bergamaschi and Zilli were impressed with the recent development of Lukysan and Vlycek [6], who applied conjugate gradients to indefinite systems, and with the later analysis of Keller, Gould and Wathen [4] of indefinite constraint preconditioners. They convinced Gondzio to look into the issue. In the meantime, Rozlozn´ok and Simoncini [7] came up with an improved understanding of issues connected with indefinite preconditioning (in the context of saddle point problems arising in partial differential equation); these developments led the authors to take a closer look at the problem. Starting in July, 2001, the authors began to incorporate into Gondzio’s interior point code HOPDM (higher order primal dual methods) four iterative techniques: conjugate gradients, BiCGstab, GMRES and QMR. A description of these schemes can be found in Kelley’s book [5] and the references therein. The authors realized that the key difficulty in the solution of augmented systems of Newton equations in interior point methods for quadratic and nonlinear programming (otherwise known as reduced KKT systems) consisted in the loss of sparsity caused by the presence of the Hessian matrix Q in the system � (Q + � 1

A new strategy was proposed to determine the Kappa number of bleached Eucalyptus globulus kraft pulps based on FT-Raman spectroscopy. Before modeling, smoothing (Savitzky-Golay), baseline correction (air-PLS), and Gaussian fitting were executed for obtaining data, including peak intensities and areas, from the Raman spectra. Two different Raman peaks, namely 1096 cm−1 and 2895 cm−1, were preset to normalize the lignin-related peak of 1600 cm−1. Therefore, two data types and two internal standard peaks were utilized to calculate ratios of selected peak intensities and peak areas. Then, they were interpreted as independent variables in the establishment of predicting model via Regression analysis. Among four different models, comparison of evaluations suggested that the value of peak intensity was superior to the value of area as an inputting value and the peak located at 2895 cm−1 performed marginally better than 1096 cm−1 as an internal standard. Statistical evaluation of the optimal predictive model, which presented as R is 0.91, RMSE is 0.46 and MAE is 0.36, indicated that the model is reliable. Therefore, the model has the high potential of accurately monitoring the Kappa number of bleached pulp and then could be helpful to reducing the extra chemical and energy consumptions in the pulp and paper industry.

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.

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.

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.

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.

This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).

Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.

In the present paper the method of Dwivedi and Trivedi in solving the dual integral equations, has been modified by presenting a very comprehensive and lucid solution, based on the analysis suggested by Fox.

We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function forBFtheory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds., Annales Henri Poincaré, 21 (12), ISSN:1424-0661, ISSN:1424-0637

We prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3-dimensional Euler equations. More precisely, the loss of regularity in a local smooth solution of the Euler equations implies the growth without bound of the deformation tensor as the critical time approaches; equivalently, if the deformation tensor remains bounded the existence of a smooth solution is guaranteed.

The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.

The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.

By determining the germinating power of potatoes of the Bintje variety it could be clearly demonstrated that seed potatoes (also when stored without sprouts at 2 C)-undergo changes in their germinating power and exhibit senility symptoms about twelve months after being harvested.

The theory of constructive formation of plane algebraic curves in Newton's writings is discussed in § 1: the apparatus by which Newton forms the curves, Newton's theorems on forming unicursal curves, his theory of conics, and his theory of (m, n) correspondence. Special Cremona plane and space transformations obtained by Newton's organic method are dealt with in § 2. The article ends with § 3, which shows two different directions in the theory of the constructive formation of plane algebraic curves in the XVIII-XIXth centuries. A synopsis is appended.

Let H be a real or complex Hilbert space and let L p(1≦p

Not all symmetries are on a par. For instance, within Newtonian mechanics, we seem to have a good grasp on the empirical significance of boosts, by applying it to subsystems. This is exemplified by the thought experiment known as Galileo’s ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin, but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore. But the significance of gauge symmetries seems less clear. For example, can gauge transformations in Yang-Mills theory—taken as mere descriptive redundancy—exhibit a similar relational empirical significance as the boosts of Galileo’s ship? This question has been debated in the last fifteen years in philosophy of physics. I will argue that the answer is ‘yes’, but only for a finite subset of gauge transformations, and under special conditions. Under those conditions, we can mathematically identify empirical significance with a failure of supervenience: the state of the Universe is not uniquely determined by the intrinsic state of its isolated subsystems. Empirical significance is therefore encoded in those relations between subsystems that stand apart from their intrinsic states.

In this paper we analyze two stochastic versions of one of the simplest classes of contagion models, namely so-called SIS models. Several formulations of such models, based on stochastic differential equations, have been recently discussed in literature, mainly with a focus on the existence and uniqueness of stationary distributions. With applicability in view, the present paper uses the Fokker-Planck equations related to SIS stochastic differential equations, not only in order to derive basic facts, but also to derive explicit expressions for stationary densities and further characteristics related to the asymptotic behaviour. Two types of models are analyzed here: The first one is a version of the SIS model with external parameter noise and saturated incidence. The second one is based on the Kramers-Moyal approximation of the simple SIS Markov chain model, which leads to a model with scaled additive noise. In both cases we analyze the asymptotic behaviour, which leads to limiting stationary distributions in the first case and limiting quasistationary distributions in the second case. Finally, we use the derived properties for analyzing the decision problem of choosing the cost-optimal level of treatment intensity.

Funder: Cantab Capital Institute for the Mathematics of Information, We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT2 conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert–Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.

Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0

In this short paper, we study compact Lagrangian submanifolds of the homogeneous nearly Kahler 6-dimensional unit sphere $${\mathbb {S}}^6(1)$$. Following a strategy in Hu et al. (J Geom Phys 144:199–208, 2019), we shall establish an optimal pinching theorem in terms of the Ricci curvature so that a new characterization of the totally geodesic $${\mathbb {S}}^3(1)$$ and the Dillen–Verstraelen–Vrancken’s Berger sphere $$S^3$$ (described in J Math Soc Jpn 42:565–584, 1990) in $${\mathbb {S}}^6(1)$$ can be given.

This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to obtain the expression of certain q-orthogonal polynomials as solutions of q-difference equations. In addition, we show how to generalize Ramanujan’s integrals by the technique of q-difference equation. More over, we find two generalized q-Chu–Vandermonde formulas from the perspective of the method of q-difference equations.

Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^q$$\end{document}[0,1]q it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q-dimensional regular regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\subset \mathbb {R}^q$$\end{document}G⊂Rq and derive an explicit representation for the remainder term.

To give some comments on Suppes' paper 'Probability in Relativistic Par? ticle Theory' {Erkenntnis 16, 299-305) is no easy task for me. Suppes does not clearly say why he is looking for relativistic probability distributions. To know something about Suppes' motivations would have been impor? tant, however, since the mathematical problem he is trying to solve is not sufficiently determinate when it is not put into a physical frame. If he is, for example, looking for relativistic invariant equations for diffusion processes and the corresponding probability distributions, one should answer that the concept of relativistic invariant diffusion is not physically meaningful. Diffusion always takes place before a material background, a gas, fluid or solid body. This background defines (locally) a preferred inertial system, in which the gas, fluid or solid is at rest, and the existence of such a background (in a similar way as the existence of an ether for electrodynamics) destroys any relativistic invariance of the equa? tions governing the diffusion process. Therefore the theory of diffusion cannot give a reasonable motivation for Suppes' investigations in relativis? tic distributions. Suppes might also have been interested in quantum mechanical wave functions and in the probability distributions defined by them. When this was the case, why did he not look for the solutions of the Dirac equation and did not simply calculate the density function from these solutions? Suppes seemed to be surprised by the complete absence of relativistic probability distributions in the relevant literature. The reason is probably that there is no physical problem to be solved in the theory of special relativity for which probability distributions are needed. I shall give just one example. The probability distribution of velocities in a gas is given by

Since the outbreak of the pandemic Coronavirus Disease 2019, the world is in search of novel non-invasive methods for safer and early detection of lung diseases. The pulmonary pathological symptoms reflected through the lung sound opens a possibility of detection through auscultation and of employing spectral, fractal, nonlinear time series and principal component analyses. Thirty-five signals of vesicular and expiratory wheezing breath sound, subjected to spectral analyses shows a clear distinction in terms of time duration, intensity, and the number of frequency components. An investigation of the dynamics of air molecules during respiration using phase portrait, Lyapunov exponent, sample entropy, fractal dimension, and Hurst exponent helps in understanding the degree of complexity arising due to the presence of mucus secretions and constrictions in the respiratory airways. The feature extraction of the power spectral density data and the application of principal component analysis helps in distinguishing vesicular and expiratory wheezing and thereby, giving a ray of hope in accomplishing an early detection of pulmonary diseases through sound signal analysis.

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.