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Mathematics is often treated as different from other disciplines, since arguments in the field rely on deductive proof rather than empirical evidence as in the natural sciences. A mathematical paper can therefore, at least in principle, be replicated simply by reading it. While this distinction is sometimes taken as the basis to claim that the results in mathematics are therefore certain, mathematicians themselves know that the published literature contains many mistakes. Reading a proof is not easy, and checking whether an argument constitutes a proof is surprisingly difficult. This article uses peer review of submissions to mathematics journals as a site where referees are explicitly concerned with checking whether a paper is correct and therefore could be published. Drawing on 95 qualitative interviews with mathematics journal editors, as well as a collection of more than 100 referee reports and other correspondence from peer review processes, this article establishes that while mathematicians acknowledge that peer review does not guarantee correctness, they still value it. For mathematicians, peer review 'adds a bit of certainty', especially in contrast to papers only submitted to preprint servers such as arXiv. Furthermore, during peer review there can be disagreements not just regarding the importance of a result, but also whether a particular argument constitutes a proof or not (in particular, whether there are substantial gaps in the proof). Finally, the mathematical community is seen as important when it comes to accepting arguments as proofs and assigning certainty to results. Publishing an argument in a peer-reviewed journal is often only the first step in having a result accepted. Results get accepted if they stand the test of time and are used by other mathematicians. [ABSTRACT FROM AUTHOR]

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

We give a corrected proof of the last theorem (Theorem 4.11) of Regularized Riesz energies of submanifolds, published in Math. Nachr. 291 (2018), no. 8–9, 1356–1373. Namely, we prove that the Riesz energy EΩ(z) of a compact body Ω is Möbius invariant if and only if the dimension of Ω is even and z=−2dimΩ. [ABSTRACT FROM AUTHOR]

There exists a mistake in the proof of Theorem 4.2. We present a new proof of this theorem, which shows that the main results of the paper are still true. [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]

We correct a mistake that appeared in the first section of the original article, which appeared in Tran. Amer. Math. Soc. 370 (2018), 99-130. Namely, Corollary 1.16 was false as stated and was subsequently used in later proofs in the paper. In this note it is argued that all the relevant statements after Corollary 1.16 can be saved with at most minor modifications. In particular, all the main results of the original paper remain valid as stated, but some intermediate claims are slightly modified or proved more directly without Corollary 1.16. [ABSTRACT FROM AUTHOR]

In his seminal paper introducing the fine structure of L, Jensen (Ann Math Log 4:229–308, 1972) proved that under V = L any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen's result that is straight-forward and accessible to those without a knowledge of Jensen's fine structure theory. The proof here instead uses hyperfine structure, a very natural and simpler alternative to fine structure theory introduced by Friedman and Koepke (Bull Symb Log 3:453–468, 1997). [ABSTRACT FROM AUTHOR]

In a short fragment published for the first time with his lost notebook in 1988 (Ramanujan, The lost notebook and other unpublished papers (Narosa, New Delhi, 1988)), Ramanujan offered two beautiful identities, associated, respectively, with the classical circle and divisor problems. In fact, they are analogues, with an additional variable, but not generalizations, of classical identities associated with these two famous problems. After Ramanujan's death in 1920, the lost notebook and fragments of papers of Ramanujan were sent to Hardy. We do not have any official record of what was included in this mailing, but it is likely that the aforementioned fragment was included in this parcel. If so, then it is possible that Ramanujan wrote it at the end of his life in either 1919 or 1920. On the other hand, from a paper that Hardy published in 1915 (Hardy, Quart. J. Math. (Oxford) 46 (1915) 263–283) on the circle problem, we are aware that by early in his stay in England, Ramanujan had a strong interest in these problems, and so the fragment may emanate from this period. Two of the present authors and Kim published a proof of the identity connected with the circle problem in 2013 (Berndt, Kim and Zaharescu, Adv. Math. 236 (2013) 24–59). In this paper, a proof of the second identity is given for the first time. [ABSTRACT FROM AUTHOR]

Makeenko and Migdal (Phys Lett B 88(1):135–137, 1979) gave heuristic identities involving the expectation of the product of two Wilson loop functionals associated to splitting a single loop at a self-intersection point. Kazakov and Kostov (Nucl Phys B 176(1):199–215, 1980) reformulated the Makeenko–Migdal equations in the plane case into a form which made rigorous sense. Nevertheless, the first rigorous proof of these equations (and their generalizations) was not given until the fundamental paper of Lévy (2017). Subsequently Driver, Kemp, and Hall Commun. Math. Phys. 351(2), 741–774 (2017) gave a simplified proof of Lévy's result and then with Driver, Gabriel, Kemp, and Hall Commun. Math. Phys. 352(3), 967–978 (2017) we showed that these simplified proofs extend to the Yang–Mills measure over arbitrary compact surfaces. All of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts. The goal of this article is to give a rigorous functional integral proof of the Makeenko–Migdal equations guided by the original heuristic machinery invented by Makeenko and Migdal. Although this stochastic proof is technically more difficult, it is conceptually clearer and explains "why" the Makeenko–Migdal equations are true. It is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing Yang–Mill's fields. [ABSTRACT FROM AUTHOR]

In this paper, we extend the work by the first author and Q. Chen [Duke Math. J. 162 (2013), pp. 1149-1169] to classify n-dimensional (n ≥ 5) complete D-flat gradient steady Ricci solitons. More precisely, we prove that any n-dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well. [ABSTRACT FROM AUTHOR]

The first statement of Lemma 11 in our recent paper [KG18] (Math. Comp. 87 (2018), no. 314, 2611-2640) is incorrect: For the sequence {Gk}k of nested admissible partitions produced by the adaptive discontinuous Galerkin method (ADGM) we have G+ := ∪k≥0∩j≥kGj, and Ω+ := (∪ E: E ∈ G+}). In the first line of the proof of [KG18, Lemma 11 on p. 2620], we used that |Ω| = |interior(Ω\Ω+)| + |Ω+|, where \vert\cdot \vert denotes the Lebesgue measure. This, however, is not true in general, since there are counter examples where Ω+ is dense in Ω and 0 = |interior(Ω\Ω+)| < |Ω\Ω+| Below, we present the required minor modifications to complete the proof of the main result stating convergence of the ADGM of [KG18] and address some typos regarding the broken dG-norm. A corrected full version of the article is available at arXiv:1909.12665v2. [ABSTRACT FROM AUTHOR]

The homogeneous perfect sets introduced by Wen and Wu [Hausdorff dimension of homogeneous perfect sets, Acta. Math. Hungar.107 (2005) 35–44] is an important class of Moran sets. In this paper, we obtain the Assouad dimension and Assouad spectrum formulas for homogeneous perfect set under suitable condition. In the proof an Assouad spectrum formula for a large class of fractal sets is established. [ABSTRACT FROM AUTHOR]

This paper provides certain computations with transfer associated with projective bundles of \mathrm {Spin} vector bundles. One aspect is to revise the proof of the main result of [Trans. Amer. Math. Soc.349 (1997), pp. 4385-4399] which says that all fourfold products of the Ray classes are zero in symplectic cobordism. [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

For a graph G = (V,E), a hypergraph H is called a Berge-G if there is a bijection f : E(G) → E(H) such that e ⊆ f(e) for all e ∈ E(G). The family of Berge-G hypergraphs is denoted by B(G). The maximum number of edges in an n-vertex r-graph with no subhypergraph isomorphic to any Berge-G is denoted by exr(n, B (G)). Gyárfás [SIAM J. Discrete Math., 33 (2019), pp. 383-- 392] showed that for n ≥ 6, ex3(n, B (K4)) = [n/3] [n+1/3] [n+2/3]. However, we found an error in the proof of the result when n ≥ 7. A recent result due to Gerbner, Methuku, and Palmer [European J. Combin., 86 (2020), 103082] implies that for n ≥ 9, ex3(n, B (K4)) = [n/3] [n+1/3] [n+2/3]. In this paper we prove the remaining cases n = 7 and n = 8 for the completeness of the conclusion. [ABSTRACT FROM AUTHOR]

In this paper we give an elementary proof of the uniqueness of solutions to a gas-disk interaction system with diffusive boundary condition. The existence of near-equilibrium solutions for this type of system with various boundary conditions has been extensively studied in Aoki et al. (ESAIM:M2AN 42:26–275, 2008), Caprino et al. (Commun Math Phys 264: 167–189, 2006; Math Models Methods Appl Sci 17(9):1369–1403, 2007), Cavallaro (Rend Mat Ser VII 27:123–145, 2007), Chen and Strauss (Arch Ration Mech Anal 211:879–910, 2014; Commun Math Phys 338:139–168, 2015), Koike (J Stat Phys 172:795–823, 2018a; Kinet. Relat. Model 11(3): 441–467, 2018b) and Sisti and Ricciuti (SIAM J Math Anal 46(6):3579–3611, 2014). However, the uniqueness has been an open problem, even for solutions near equilibrium. Our work gives the first rigorous proof of the uniqueness among solutions that are only required to be locally Lipschitz; in particular, it holds for solutions far from equilibrium states. [ABSTRACT FROM AUTHOR]

In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham [Math. Comp. 70 (2001), pp. 873-891], generalizing arguments of Erdős and Pomerance [Math. Comp. 46 (1986), pp. 259-279] and Monier [Theoret. Comput. Sci. 12 (1980), 97-108]. These bounds are provided for both Jacobi symbol ±1 cases, providing evidence for the existence of several challenge pseudoprimes. [ABSTRACT FROM AUTHOR]

We will refer to the paper "Convergence rates for Kaczmarz-type algorithms", published in Numerical Algorithms, 79(1)(2018), 1-17. It was observed by Dr. Mokhtar Abbasi (Department of Mathematics, University of Qom, Iran) that in the proof of Theorem 7, related to the linear convergence rate of the MREK algorithm it exists an inequality which is not always true. [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 a recent paper Engle et al. (Commun Math Phys 354:231–246, 2017) showed that there is a unique state on the reduced holonomy–flux ∗ -algebra of homogeneous isotropic loop quantum cosmology, that is invariant under residual diffeomorphims. This result has been claimed to be true both for the Ashtekar–Bojowald–Lewandowski framework and for that introduced by the present author. Unfortunately, the uniqueness proof relies on an incorrect argument which spoils the second case. In our short note, we are going to patch this issue, this way keeping the nice uniqueness result in both cases. Moreover, we will even extend the underlying operator algebraic statements as this might help later for studying higher-dimensional models. [ABSTRACT FROM AUTHOR]

We study solutions to stationary Navier–Stokes system in a two dimensional exterior domain. We prove that any such solution with a finite Dirichlet integral converges to a constant vector at infinity uniformly. No additional conditions (on symmetry or smallness, etc.) are assumed. In the proofs we develop the ideas of the classical papers of Gilbarg and Weinberger (Ann Sc Norm Pisa (4) 5:381–404, 1978) and Amick (Acta Math 161:71–130, 1988). [ABSTRACT FROM AUTHOR]

This paper is a continuation of [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.]. We construct by-hand Legendrian surfaces for which specific properties of their gradient flow trees hold. These properties enable us to complete the proof in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.] that the Cellular DGA defined in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part I, preprint (2016), arXiv:1608.02984] is stable tame isomorphic to the Legendrian contact homology DGA defined in [T. Ekholm, J. Etnyre and M. Sullivan, The contact homology of Legendrian submanifolds in ℝ 2 n + 1 , J. Differential Geom.71(2) (2005) 177–305]. [ABSTRACT FROM AUTHOR]

This paper is a continuation of [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)]. For Legendrian surfaces in 1 -jet spaces, we prove that the Cellular DGA defined in [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)] is stable tame isomorphic to the Legendrian contact homology DGA, modulo the explicit construction of a specific Legendrian surface. In [Cellular computation of Legendrian contact homology for surfaces, to appear in Internat. J. Math.], we construct this surface, thereby completing Theorem 5.1 and the proof of the isomorphism. [ABSTRACT FROM AUTHOR]

In classical mathematics, the existence of a solution is of ten proven indirectly, nonconstructively, without an efficient method for constructing the corresponding object. In many cases, we can extract an algorithm from a classical proof, e.g., when an object is (nonconstructively) proven to be unique in a locally compact space (or when there are two such objects with a known lower bound on the distance between them). In many other practical situations, a (seemingly) natural formalization of the corresponding practical problem leads to a non-compact set. In this paper, we show that often, in such situations, we can extract efficient algorithms from classical proofs--if we explicitly take into account (implicit) knowledge about the situation. Specifically, we show that if we consistently apply Heisenberg's operationalism idea and define objects in terms of directly measurable quantities, then we get a Girard-domain type representation in which a natural topology is, in effect, compact--and thus, uniqueness implies computability. [ABSTRACT FROM AUTHOR]

This erratum concerns Lemma 3.1 in the original paper [Proc. Amer. Math. Soc. 147 (2019), pp. 1657-1669]. The statement of that lemma needs to be slightly changed since the statement of Claim 3.5 must be changed. The proof of the lemma is mostly still kept as the original one with some lines modified. This change does not effect the proof of the main results. [ABSTRACT FROM AUTHOR]

After a number of decades of research into the dynamics of rational belief, the belief revision theory community remains split on the appropriate handling of sequences of changes in view, the issue of so-called iterated revision. It has long been suggested that the matter is at least partly settled by facts pertaining to the results of various single revisions of one's initial state of belief. Recent work has pushed this thesis further, offering various strong principles that ultimately result in a wholesale reduction of iterated to one-shot revision. The present paper offers grounds to hold that these principles should be significantly weakened and that the reductionist thesis should ultimately be rejected. Furthermore, the considerations provided suggest a close connection between the logic of iterated belief change and the logic of evidential relevance. [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]

This note is an erratum for the paper On the Inequalities of Projected Volumes and the Constructible Region [SIAM J. Discrete Math., 33 (2019), pp. 694--711]. There is a mistake in the proof of Theorem 4 in Section 3. However, Theorem 4 is itself correct (a proof can be found in [Inequalities on Projected Volumes, arxiv.org/abs/1909.12858, 2019] by Leader, Randelovic, and Raty), and all the other results of the paper stand. [ABSTRACT FROM AUTHOR]

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsov's (2009) first-degree proof system ⊢B related to SIXTEEN3. The system GB is a standard Gentzen-type sequent calculus, FB is a four-place (horizontal) matrix sequent calculus, and QB is a quadruple (vertical) matrix sequent calculus. In contrast with GB, the calculus FB satisfies the subformula property, and the calculus QB reflects Odintsov's coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of ⊢B are proved. In addition, it is shown how the sequent systems for ⊢B can be extended to cut-free sequent calculi for Odintsov's LB, which is an extension of ⊢B by adding classical implication and negation connectives. [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]

Although Frege was eager to theoretically eliminate the judging subject from logic and mathematics, his system is permeated with notions that refer to subjective mental processes, such as grasping a thought, assuming, judging, and value. His semantic system depends on such notions, but since Frege in general shuns explaining them, his central conception of judgment and truth remains dark. In this paper it is proposed to fill out the gaps in Frege's explanations with the help of Husserl's phenomenological descriptions, especially those of the sixth Logical Investigation. This leads to a comparison between Frege's notion of judgment and Husserl's "Evidenz", and finally also to a phenomenological classification of Frege's remarks on truth. [ABSTRACT FROM AUTHOR]

In our paper "On solvable compact Clifford-Klein forms" [Proc. Amer. Math. Soc. 145 (2017), pp. 1819-1832] we have found a small gap in the proof of the main theorem. In this note we show that the result is correct and that the proof goes through after a slight modification. [ABSTRACT FROM AUTHOR]

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]

Students' difficulty with reasoning and justification in the context of proofs in geometry has been well documented. Studies reveal that many of the ideas that students are asked to prove are known to be true, and therefore, the process of such proof is likely to remain meaningless and purposeless. The implication of this claim is that students need to be asked to prove mathematical assertions that are not known to them to be true. This puts an enormous responsibility on teachers. The purpose of this paper is to provide examples of 'non-traditional conjectures' that can be explored in the dynamic geometry environment of computer construction programs. [ABSTRACT FROM AUTHOR]

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]

This note states the correction of arguments in the proof of Theorem 3.2 in the original paper. [ABSTRACT FROM AUTHOR]

In many cases Bayes' theorem is an appropriate algorithm for the aggregation of probabilistic evidence. As with other statistical procedures, there are restrictions that must be taken into account. In the present paper we shall comment on several approaches that have been devoted to one of these restrictions; the incorporation of uncertainty about the true state of a datum. A datum is a variable which can be partitioned into equivalence classes. These classes represent the possible data states which will be also called events. In Bayes' theorem an event is an item of information which will be used for revising the opinion about the relative likelihood of hypotheses. In any specific situation only one of the possible events will be the true event. An event may come from a source whose reporting or observational accuracy is not perfect. An example may illustrate the issue. A medical doctor wants to come to a diagnosis. To achieve this he considers several data. One datum might be the result of a medical test, which has three possible states: positive, negative, inconclusive. If the doctor reports the state of the datum to be positive, he may be wrong by whatever reasons. That is, the report of an event must not necessarily coincide with the actual or true event of the datum under consideration. This kind of uncertainty about the true event in any specific situation is a characteristic of the source. In most cases it will reduce the diagnostic impact of an event. Whenever the report of an event and the true event coincide imperfectly, measures of source inaccuracy must be incorporated into Bayes' theorem. The task can be considered as two stage probabilistic induction. The first step is induction from the reported to the actual event, the second is induction from the actual event to the creditation of hypotheses. This is the reason why Gettys and Willke (1969) speak about "cascaded inference." [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]