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We correct the proof of Theorem 2 in the mentioned paper concerning finite-difference equilibrium solutions. [ABSTRACT FROM AUTHOR]

The concept of Γ-hypersemigroup has been introduced in Turk J Math 2020; 44 (5): 1835-1851 in which it has in which it has been shown that various results on Γ-hypersemigroups can be obtained directly as corollaries of more general results from the theory of le-semigroups (i.e. lattice ordered semigroups having a greatest element) or poe-semigroups. As a continuation of the paper mentioned above, in the present paper, the concept of ordered Γ-hypersemigroups has been introduced, and their relation to lattice ordered semigroups is given. It has been shown that although the results on ordered Γ-hypersemigroups cannot be obtained as corollaries to the corresponding results of le or poe-semigroups, still the main idea comes from the le-semigroups or poe-semigroups, and the proofs go along the lines of the le or poe-semigroups. [ABSTRACT FROM AUTHOR]

In this note, we correct an inaccuracy of Eq. (13) in the proof of Theorem 1 in the paper (Higham and Kloeden, 2007). We also give a complete proof of the theorem by applying the Burkholder–Davis–Gundy Inequality. [ABSTRACT FROM AUTHOR]

In this note, we correct the statement of Theorem 12 from the paper in the title above (McLean et al., 2020) and fill some gaps in the proof. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the existence of radially symmetric positive solutions in of the elliptic system: , where α and γ are positive constants (β will be allowed to be negative). This system has trivial solutions of the form and where ϕ and ψ are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters has been studied recently by several authors [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453–458; T.C. Lin, J. Wei, Ground states of N coupled nonlinear Schrödinger equations in , , Comm. Math. Phys. 255 (2005) 629–653; T.C. Lin, J. Wei, Ground states of N coupled nonlinear Schrödinger equations in , , Comm. Math. Phys., Erratum, in press; L. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, preprint; B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in , preprint; J. Yang, Classification of the solitary waves in coupled nonlinear Schrödinger equations, Physica D 108 (1997) 92–112]. For , perhaps the most general existence result has been proved in [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453–458] under conditions which are equivalent to ours. Motivated by some numerical computations, we return to this problem and, using our approach, we give a more detailed description of the regions of parameters for which existence can be proved. In particular, based also on numerical evidence, we show that the shape of the region of the parameters for which existence of solution can be proved, changes drastically when we pass from dimensions to dimension . Our approach differs from the ones used before. It relies heavily on the spectral theory for linear elliptic operators. Furthermore, we also consider the case which has to be treated more extensively due to some lack of compactness for even functions. This case has not been treated before. [Copyright &y& Elsevier]

Abstract: Ludics [J.-Y. Girard, Locus solum, Math. Structures in Comput. Sci. 11 (2001) 301–506] is a recent proposal of analysis of interaction, developed by abstracting away from proof-theory. It provides an elegant, abstract setting in which interaction between agents (proofs/programs/processes) can be studied at a foundational level, together with a notion of equivalence from the point of view of the observer. An agent should be seen as some kind of black box. An interactive observation on an agent is obtained by testing it against other agents. In this paper we explore what can be observed interactively in this setting. In particular, we characterize the objects that can be observed in a single test: the primitive observables of the theory. Our approach builds on an analysis of the geometrical properties of the agents, and highlights a deep interleaving between two partial orders underlying the combinatorial structures: the spatial one and the temporal one. [Copyright &y& Elsevier]

This paper presents a new proof method about the paper (Yuan et al. in Appl. Math. Model. 47:811–825, 2017). In the proof, the global convergence of the Polak–Ribière–Polak algorithm is established without these two assumptions d k T g k < 0 and g k + 1 d k ≤ − σ 1 g k T d k which are needed in the above paper. This means that this paper has the same results under weaker conditions. More dimension functions for practical problems are tested to show the performance of the modified algorithm and the normal algorithm. An application of the fact engineering model is done to show the effectiveness of the given conjugate gradient algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a class of cellular neural networks with time-varying delays. Applying differential inequality strategies without assuming the boundedness conditions on the activation functions, we obtain a new sufficient condition that ensures that all solutions of the considered neural networks converge exponentially to the zero equilibrium point. We give an example to illustrate the effectiveness of the theoretical results. The results obtained in this paper are completely new and complement the previously known studies of Tang (Appl. Math. Lett. 21:872–876, 2008). [ABSTRACT FROM AUTHOR]

On pages 255–256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan's Notebooks: Part III (1991) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an elementary proof of this identity. In the process, we also give an elementary proof of three Ramanujan's P - Q identities of level 15. Further, using the P - Q identities we prove four Ramanujan type Eisenstein series of level 15 due to Cooper and Ye (Trans. Am. Math. Soc.368 (2016) 7883–7910), where they have proved using the theory of modular forms. [ABSTRACT FROM AUTHOR]