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We consider discrete dynamical systems coming from the models of evolution of populations playing rock - paper - scissors game . Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved.

In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

This paper presents an example of how paper folding can be used in a geometry class to support conceptual understanding. Specifically, it explains an activity that constructs a rhombus and explores its attributes by using paper folding. The steps of constructing a rhombus are described and some discussion questions are given to consolidate understanding. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

We discuss the recent paper by A. Dar\'os and L.K. Arruda (On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation, J. Diff. Equat., 266 (2019), 1946-1968). Our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space., Comment: 10 pages and 2 figures

The purpose of this note is to discuss how paper folding can be used to find the exact trigonometric ratios of the following four angles: 22.5°, 67.5°, 27°, and 63°.

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers “Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production” by Tao et al. (2019) [2] and “Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation” by Frassu and Viglialoro (2021) [1] . These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures

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.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

This work contains largely afterthoughts, relating to the paper immediately preceding it. We correlate and interpret our contributions in that paper, relative to those of an earlier publication by Aspley, He and McCuan. We propose specific laboratory experiments, suggested by formal predictions of those two papers.

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

Leonetti proved that whenever $${\mathcal {I}}$$ is an ideal on $${\mathbb {N}}$$ such that there exists an uncountable family of sets that are not in $${\mathcal {I}}$$ with the property that the intersection of any two distinct members of that family is in $${\mathcal {I}}$$ , then the space $$c_{0,{\mathcal {I}}}$$ of sequences in $$\ell _\infty $$ that converge to 0 along $${\mathcal {I}}$$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $$\ell _\infty / c_{0,{\mathcal {I}}}$$ does not even embed into $$\ell _\infty $$ .

In Fritz John's famous paper (1979), he discovered that for the wave equation $\Box u=|u|^p$, where $1, Comment: 11 pages, 3 figures

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).

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

In T. Kruse and A. Popier, we established existence and uniqueness of solutions of backward stochastic differential equations in under a monotonicity condition on the generator and in a general filtration. There was a mistake in the case . Here we give a corrected proof. Moreover the quasi-left continuity condition on the filtration is removed.

In this paper we correct an error made in our paper [Blouhi, T.; Caraballo, T.; Ouahab, A. Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion. Stoch. Anal. Appl. 34 (2016), no. 5, 792-834]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown.

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4

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.

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

In Vukicevic and Stevanovic (2013), Theorem 2 is erroneous when m = 3 k + 1 for some integer k . The correct one is given now. That is, P I ( G ) ≥ m 2 − 3 m + 2 when m = 3 k + 1 for some integer k , where G is a connected bicyclic graph with m edges.

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

This review is devoted to recent developments in blood flow modelling. It begins with the discussion of blood rheology and its non-Newtonian properties. After that we will present some modelling methods where blood is considered as a heterogeneous fluid composed of plasma and blood cells. Namely, we will describe the method of Dissipative Particle Dynamics and will present some results of blood flow modelling. The last part of this paper deals with one-dimensional global models of blood circulation. We will explain the main ideas of this approach and will present some examples of its application.

In this paper we generalize two results of Khan and Majumdar on the weak sequential convergence of functions in the Lebesgue-Bochner space L X 1 (Ω) and on a Fatou-like convergence theorem.

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 preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.

The present study, which is a continuation of the previous paper, augments a recent work on the use of phylogenetic networks. We develop techniques to characterize the topology of various X-trees and binary trees of biological and phylogenetic interests. We have obtained the results for various k-level X-trees and phylogenetic networks with variants of Zagreb, Szeged, Padmakar-Ivan, Schultz and Atom Bond Connectivity topological indices.

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail or-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail ar-field structure of the chain, can be used to solve such cases. FINITE MARKOV CHAIN; FINAL CLASS; CYCLICALLY MOVING SUBCLASS; TAIL ar-FIELD; ATOMIC SET; RECURRENCE; WHIRLPOOL

In the article ‘How long is my toilet roll-a simple exercise in mathematical modelling’ several models of increasing complexity are introduced and solved to calculate indirectly the length of paper on a toilet-roll. All these results are presented without errors. The authors of this comment believe the error analysis of measurements made in a laboratory is an important part of the experiment. The calculation of the absolute errors in indirect measurements provides us with information about the confidence that we must have in our experimental results. For that reason, the error calculation for all approximations has been made

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

In this short paper, using category theory, we argue that logical rules can be seen as fractions and logics as limit sketches.

In this short paper I report on a paper published in Nonlinear Analysis: Real World Applications in 2004. There is a major mistake early in that paper which makes most of its claims false. The class of reaction–diffusion systems considered in the paper has been the object of a renewed investigation in the past few years, by myself and others, and recent discoveries provide explicit counter-examples.

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

In the paper, by virtue of the binomial inversion formula, a general formula of higher order derivatives for a ratio of two differentiable function, and other techniques, the authors compute several sums in terms of the beta function and its partial derivatives, polygamma functions, the Gauss hypergeometric function, and a determinant. These results generalize known ones in combinatorics. This preprint has been formally published as "Feng Qi and Chuan-Jun Huang, Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales Serie A Matematicas, Vol. 114, Paper No. 191, 9 pages (2020); available online at https://doi.org/10.1007/s13398-020-00927-y."

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.

In this short paper, we extend the result of Shinbrot (1974) to an incompressible fluid with shear dependent viscosity. It is shown that a weak solution to the equations of non-Newtonian fluids lying in a space L q ( 0 , T ; L p ) satisfies an energy equality, where 2 r r − 1 ≤ p ≤ 2 r r − 2 and 1 p + 1 q ≤ r − 1 r , if r > 2 ; p ≥ 2 r r − 1 and r − 1 p + 1 q = r − 1 2 , if 2 ( n + 1 ) n + 2 r ≤ 2 . In particular, our result implies that the weak solution must satisfy the energy equality when r ≥ 3 n + 2 n + 2 , which is consistent with the known fact.

In this short paper we study the existence and uniqueness of solutions of free boundary problems coming from ecology in heterogeneous environments.

The purpose of this short paper is to describe a new proof of the Pythagorean Theorem that involves paper folding.

The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

In this short paper, we show how to deduce several types of generating functions from Srivastava {\it et al} [Appl. Set-Valued Anal. Optim. {\bf 1} (2019), pp. 187-201.] by the method of $q$-difference equations. Moreover, we build relations between transformation formulas and homogeneous $q$-difference equations., 10

In this paper we determine all Padovan numbers that are palindromic concatenations of two distinct repdigits.

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.

A (3,6)-fullerene is a plane cubic graph whose faces are only triangles and hexagons. A connected graph G with at least $$2n+2$$ vertices is said to be n-extendable if it has n independent edges and every set of n independent edges extends to a perfect matching of G. A graph G is said to be bicritical if for every pair of distinct vertices u and v, $$G-u-v$$ has a perfect matching. It is known that every (3,6)-fullerene is 1-extendable, but not 2-extendable. In this short paper, we show that a (3,6)-fullerene G is bicritical if and only if G has the connectivity 3 and is not isomorphic to one graph (2,4,2). As a surprising consequence we have that a (3,6)-fullerene is bicritical if and only if each hexagonal face is resonant.

We prove the existence of free objects in certain subcategories of Banach lattices, including p-convex Banach lattices, Banach lattices with upper p-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (Israel J. Math. 2019). In the p-convex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear p-summing maps., g Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. P. Tradacete gratefully acknowledges support by Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) through grants PID2020-116398GB-I00, MTM2016-76808-P, and MTM2016- 75196-P, as well as Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and from Consejo Superior de Investigaciones Científicas (CSIC), through “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). H. Jardón-Sánchez gratefully acknowledges support by Fundación María Cristina Masaveu Peterson through its Scholarships for Academic Excellence and Fundación “la Caixa” through its Postgraduate Studies in Europe Fellowships. V.G. Troitsky gratefully acknowledges support by Natural Sciences and Engineering Research Council of Canada