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In 1935, Erdos proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

Consider an elliptic quadric $$Q^-(3,q)$$ in $$\mathrm{PG}(3,q)$$ . Let $$\mathcal {E}$$ and $$\mathcal {T}$$ denote the set of all lines of $$\mathrm{PG}(3,q)$$ which meet $$Q^-(3,q)$$ in 0 and 1 point, respectively. In this paper, we characterize the minimum size $$(\mathcal {T}\cup \mathcal {E})$$ -blocking sets and give a different proof for the characterization of minimum size $$\mathcal {E}$$ -blocking sets in $$\mathrm{PG}(3,q)$$ which works for all q. We also discuss whether the main results of this paper (Theorems 1.6 and 1.7) can be extended to ovoids in $$\mathrm{PG}(3,q)$$ .

In this paper, we study a class of regular graphs, which is related to the Grassmann graph. This class of graphs is called the doubled Grassman graph. The Grassmann graph is the class of graphs, which is defined similar to the Johnson graph. This time, however, we are concerned with the subspaces of a vector space, rather than with the subsets of a set. The doubled Grassmann graph is constructed from the Grassmann graph. This graph, was discovered by Biggs and Gardiner. In this paper, we determine the full automorphism group of the doubled Grassmann graph. For determining the automorphism group of the doubled Grassmann graph, we will use the methods used by Mirafzal (Proc. Indian Acad. Sci. (Math Sci.) 129(3) (2019), Art. 34, 8).

The link between Vandiver’s conjecture and Gauss sums is well known since the papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459, 1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Angles and Nuccio (Acta Arith 142(3):199–218, 2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver’s conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the p-th cyclotomic field (Sur la p-ramification abelienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order p of $${\mathbb {F}}_\ell ^\times $$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in §5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as $$\ell $$ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.

In this paper, almost unit 1-stable range condition for a commutative ring with unity is defined and similar results which are already proved for the weakly unit 1-stable range are proved. Several properties of rings satisfying almost unit 1-stable range are discussed. The characterizations for rings such as presimplifiable semilocal rings and idealization R(M) of a ring R and an R-module M, satisfying almost unit 1-stable range is also given in this paper. Few examples are given to show that almost unit 1-stable range condition is different from other stable range conditions.

Congruences for partitions have received a great deal of attention in literature. Recently, Bringmann et al. (Electron J. Combin. 22(3) (2015) Paper 3.17, 16 pp.) studied overpartitions with restricted odd differences. In this paper, we present a number of Ramanujan-type congruences for these restricted overpartition functions.

The aim of this paper is to state some results on an $$\alpha $$ -nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271–283), for any fixed automorphism $$\alpha $$ of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p 2) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p 2) of order 2|PSL(2,p 2)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p 3. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.

In this paper, we establish the approximate controllability results for a non-autonomous functional differential equation using the theory of linear evolution system, Schauder fixed point theorem, and by making use of resolvent operators. The results obtained in this paper, improve the existing ones in this direction, to a considerable extent. An example is also given to illustrate the abstract results.

This is the second of a series of papers devoted to the study of h-p spectral element methods for three dimensional elliptic problems on non-smooth domains. The present paper addresses the proof of the main stability theorem. We assume that the differential operator is a strongly elliptic operator which satisfies Lax–Milgram conditions. The spectral element functions are non-conforming. The stability estimate theorem of this paper will be used to design a numerical scheme which give exponentially accurate solutions to three dimensional elliptic problems on non-smooth domains and can be easily implemented on parallel computers.

This is the first of a series of papers devoted to the study of h-p spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming h-p spectral element method whose proof is given in a separate paper.

For proving the existence of a regularization parameter under a Morozov- type discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems, it is required to impose additional nonlinearity assumptions on the forward operator. Lipschitz continuity of the Freechet derivative and requirement of the Lipschitz constant to depend on a source condition is one such restriction (Ramlau P, Numer. Funct. Anal. Optim. 23(1&22 )( 2003) 147-172). Another nonlinearity condition considered by Scherzer (Computing, 51 (1993) 45-60) was by requiring the forward operator to be close to a linear operator in a restricted sense. A seemingly natural nonlinear assump- tion which appears in many applications which attracted attention in various contexts of the study of nonlinear problems is the so-called η-condition. However, a Morozov-type discrepancy principle together with η-condition does not seem to have been studied, except in a recent paper by the author (Bull. Aust. Math. Soc. 79 (2009) 337-342), where error estimates under a general source condition is derived, by assuming the existence of the parameter. In this paper, the existence of the parameter satisfying a Morozov-type discrepancy principle is proved under the η-condition on the forward operator, by assuming the source condition as in the papers of Scherzer (Computing, 51 (1993) 45-60) and Ramlau (Numer. Funct. Anal. Optim. 23(1&22 )( 2003) 147-172). This source condition is, in fact, a special case of the source condition in the author's paper (Bull. Aust. Math. Soc. 79 (2009) 337-342).

This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic spaceH n by discrete subgroups Г, i.e. ℳ =H n/Г. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green’s function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green’s function, and hence the eigenfunctions, near infinity. This information is used to construct anS-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between theS-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of theS-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group Г. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles.

Let p be a prime, $$\chi $$ denote a Dirichlet character modulo p. For any integer x ( $$1\le x\le p-1$$ ), $${\bar{x}}$$ denotes the integer inverse of x such that $$x{\bar{x}}\equiv 1(\bmod \, p)$$ , we study the following mean value of a kind of character sums with generalized Dirichlet L-functions $$\begin{aligned} {\mathop {\mathop {\sum }\limits _{\chi (-1)=1}}\limits _{\chi \ne \chi _0}} \left| \sum _{x=1}^{p-1}\chi (x+{\bar{x}})\right| ^2|L(1,\chi ,a)|^2, \end{aligned}$$ where $$\chi _0$$ is the principal character modulo p, and $$L(1,\chi ,a)$$ is the generalized Dirichlet L-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula.

In this paper, we study the Mexican hat wavelet transform (MHWT) of generalized function space $$\mathcal {G}'$$ . The space $$\mathcal {G}'$$ consists of purely entire functions with certain advance conditions developed by Howell (J. Math. Anal. Appl. 180 (1993) 79–92; 187 (1994). An inversion formula is also established.

Let N be a sufficiently large integer. In this paper, it is proved that, with at most $$O(N^{4/27+\varepsilon })$$ exceptions, all even positive integers up to N can be represented in the form $$p_1^2+p_2^2+p_3^3+p_4^3+p_5^6+p_6^6$$ , where $$p_1,p_2,p_3,p_4,p_5,p_6$$ are prime numbers. This gives a large improvement of a recent result $$O(N^{127/288+\varepsilon })$$ due to Liu (Proc. Indian Acad. Sci. (Math. Sci.) 130(1) (2020) Article ID. 8).

Alspach et al. (J. Austral. Math. Soc. 56(3) (1994) 391–402) constructed an infinite family of tetravalent graphs M(a; m, n) and proved that if $$n\ge 9$$ be odd and $$a^3\equiv 1 (\mathrm{mod}~n)$$ , then M(a; 3, n) is half-arc-transitive. In this paper, we show that if $$a^3\equiv 1 (\mathrm{mod}~n)$$ , then M(a; 3, n) is an infinite family of tetravalent half-arc-transitive Cayley graphs for all integers n except 7 and 14.

Let $$R=K[x_1,\ldots ,x_n]$$ be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d. Bandari and Herzog (Eur. J. Combin. 34 (2013) 752–763) conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) $${\text {height}}(I)=n-1$$ ; (ii) I contains at least $$n-3$$ pure powers of the variables $$x_1^d,\ldots ,x_{n-3}^d$$ ; (iii) I is a monomial ideal in at most four variables.

In this paper, we consider the symmetric diagonal Diophantine equation $$x^5+ky^q+k'z^r=u^5+kv^q+k'w^r$$ , where k and $$k'$$ are nonzero rational numbers. We prove that if $$2 \le q,r \le 5$$ and $$(q,r) \ne (5,5)$$ this equation admits infinitely many nontrivial rational solutions.

In this paper, we look at the notion of cohomological triviality of fibrations of homogeneous spaces of affine algebraic groups defined over $${\mathbb {C}}$$ and use topological methods, primarily the theory of covering spaces. This is made possible because of the structure theory of affine algebraic groups. Further, we generalize our results for arbitrary connected algebraic groups and their homogeneous spaces. As an application of our methods, we give a structure result for quasi- reductive algebraic groups (i.e., algebraic groups whose unipotent radical is trivial), up to isogeny.

In this paper, we establish a non-asymptotic lower bound for the running maximum of an asymptotically almost negatively associated (AANA) family of random variables. The obtained result is applied to characterize the Kolmogorov–Feller weak law of large numbers for these sequences. We also provide a simplified proof of the necessity part in the Baum–Katz law of large numbers in the AANA setting.

An n-color composition is one in which a part of size m can come in m colors (denoted by subscripts). Compositions that read the same when read forwards or backwards are said to be palindromic. In this paper, we study the number of n-color palindromic compositions whose parts have subscripts belonging to a particular arithmetic progression. That is, the subscripts are of the form $$\ell a + b$$ , where $$\ell $$ and b are fixed positive integers and $$a\ge 0$$ is arbitrary. Among our results, we derive an explicit formula for the generating function and provide a connection with Riordan arrays. Finally, we describe bijections between certain restricted classes of palindromic n-color compositions and subsets of ordinary compositions and ternary words.

In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formula for representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power theory, $$\alpha $$ -norm, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example.

In this paper, we rewrite two forms of an Euler–Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter. We also use the Weierstrass–Enneper representation of minimal surfaces to obtain some identities involving these Dirichlet series and one complex parameter.

In this paper, we give proof of certain recent conjectural supercongruences on sums involving fourth power of some rising factorials.

Let $$G=(V,E)$$ be a graph with the vertex-set V and the edge-set E. Let N(v) denote the set of neighbors of the vertex v of G. The graph G is called irreducible whenever for every $$v,w \in V$$ if $$v \ne w$$ , then $$N(v)\ne N(w).$$ In this paper, we present a method for finding automorphism groups of connected bipartite irreducible graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite irreducible graphs, including a class of graphs which are derived from Grassmann graphs. Let $$a_0$$ be a fixed positive integer. We show that if G is a connected non-bipartite irreducible graph such that $$c(v,w)=|N(v)\cap N(w)|=a_0$$ when v, w are adjacent, whereas $$c(v,w) \ne a_0$$ , when v, w are not adjacent, then G is a stable graph, that is, the automorphism group of the bipartite double cover of G is isomorphic with the group $${\mathrm{Aut}}(G) \times {\mathbb {Z}}_2$$ . Finally, we show that the Johnson graph J(n, k) is a stable graph.

Let $$G=(V,E)$$ be a graph. A subset $$S\subseteq V$$ is a 2-dominating set of G if each vertex in $$V-S$$ is adjacent to at least two vertices in S. The 2-domination number of G is the cardinality of the smallest 2-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs $$P(5k+1,3)$$ , $$P(5k+2,3)$$ and $$P(5k+3,3)$$ is $$4k+2$$ , $$4k+3$$ and $$4k+4$$ , respectively. This proves one conjecture due to Bakhshesh et al. (Proc. Indian Acad. Sci. (Math. Sci.) 128 (2018) 17).

To every finite simple graph G, we associate the so-called edge ideal I(G), which is a square-free quadratic monomial ideal generated by the monomials corresponding to the edges of G. In this paper, we determine all the initial graded Betti numbers of edge ideals of triangular graphs and alternate triangular graphs in terms of the underlying graph. We also compute the regularity of edge ideals of triangular and alternate triangular graphs.

C R Rao has made seminal contributions in many areas in statistics. A review of some of his fundamental contributions towards the development of the theory and application of Mahalanobis distance in classification problems is presented here. The review is based on information provided in some of Rao’s famous research and autobiographical papers.

Kaplansky (Ann. Math. 46(3) (1945) 474–479) introduced the notion of completely mixed games. Fifty years later in 1995, he wrote another beautiful paper where he gave a set of necessary and sufficient conditions for a skew symmetric matrix game to be completely mixed. In this work, we attempt to answer when bimatrix games will be completely mixed. In particular, we give a set of necessary and sufficient condition for a bimatrix game (A, B) to be completely mixed when A and B are skew symmetric matrices of order 3. We give several examples to show the sharpness of our result. We also formulate a conjecture when A and B are skew symmetric matrices of order 5.

Given a pair of smoothly bounded domains $$D_1, D_2 \subset \mathbb {C}$$ , the purpose of this paper is to obtain an inequality that relates the Caratheodory metrics on $$D_1, D_2, D_1 \cap D_2$$ and $$D_1 \cup D_2$$ .

In this paper, we discuss some properties of Euler totient function of associated Pell numbers which are repdigits in base 10.

In this paper, we answer questions raised by Bardakov and Gongopadhyay (Commun. Algebra 43(11) (2015) 4809–4824). We prove that the palindromic width of HNN extension of a group by proper associated subgroups is infinite. We also prove that the palindromic width of the amalgamated free product of two groups via a proper subgroup is infinite (except when the amalgamated subgroup has index two in each of the factors). Combining these results it follows that the palindromic width of the fundamental group of a graph of groups is mostly infinite.

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. We show that the cup product together with the degree $-1$ graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology forms a Gerstenhaber algebra. This generalizes a classical fact that the Hochschild cohomology of an associative algebra carries a Gerstenhaber algebra structure., Comment: 14 pages; final version; accepted for publication in Proc. Indian Acad. Sci. Math. Sci.; comments are still welcome

Let R be a ring. The prime intersection graph of ideals of R, denoted by $$G_{P}(R)$$ , is the graph whose vertex set is the collection of all non-trivial (left) ideals of R with two distinct vertices I and J are adjacent if and only if $$I\cap J\ne 0$$ and either one of I or J is a prime ideal of R. We discuss connectedness in $$G_{P}(R)$$ . The concepts of bipartition, planarity and colorability are interpreted. Finally, we introduce the idea of traversability in $$G_{P}({\mathbb {Z}}_n)$$ . The core part of this paper is observed in the ring $${\mathbb {Z}}_n$$ .

In an earlier work, we proved a generalized entropy ergodic theorem for finite nonhomogeneous Markov chains (NMC). In this paper, we establish a generalized strong law of large numbers for finite m-th order NMC. Then we deduce a generalized entropy ergodic theorem for finite m-th order NMC, under some assumptions on the continuity rate and of non-nullness. Explicit upper and lower bounds relating the generalized relative entropy density of the original finite non-null stationary sequence and its canonical m-order Markov approximation is obtained.

An edge-colored graph G is rainbow k-connected, if for every two vertices of G, there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k-connected coloring of G, $$rc_k(G)$$ , is the rainbow k-connection number of G. Let G and H be two connected graphs, where O is an orientation of G. Let $${\vec {e}}$$ be an oriented edge of H. The edge-comb product of G (under the orientation O) and H on $$\vec {e}$$ , $$G{}^o\rhd _{\vec {e}}H$$ , is a graph obtained by taking one copy of G and |E(G)| copies of H and identifying the i-th copy of H at the edge $${\vec {e}}$$ to the i-th edge of G, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.

Let R(n) denote the number of representations of an even integer n as the sum of two squares, two cubes and two sixth powers of primes, and by $$\mathcal {E}(N)$$ we denote the number of even integers $$n \leqslant N$$ such that the expected asymptotic formula for R(n) fails to hold. In this paper, it is proved that $$\mathcal {E}(N) \ll N^{\frac{127}{288} + \varepsilon }$$ for any $$\varepsilon >0$$ .

This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level N where $$N=2,3,5$$ and 7. The method is based on the observation that these Eisenstein series are integral over the graded algebra of modular forms on SL(2, Z) and their zeros are ‘controlled’ by those of $$E_4$$ and $$E_6$$ in the fundamental domain of $$\Gamma _0(N)$$ .

A composite number is a positive integer that has at least one divisor integer other than 1 and itself. In this paper, we give a detailed structural description of a group if it has a unique composite conjugacy class size.

In this paper, we establish the weak laws of large numbers for the negative quadrant-dependent random sequences which extend the classic Kolmogorov–Feller weak law of large numbers. In addition, the moment convergence for the negative quadrant-dependent random sequences are also given.

In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs G consisting of two components. If $$G=G_1 \cup G_2$$ , then we give the bounds of the partition dimension of G for $$G_1 = P_n$$ or $$G_1=C_n$$ and also for $$pd(G_1)=pd(G_2)$$ .

The notation $$F\rightarrow (G,H)$$ means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number $$\hat{r}_c(G,H)$$ of graphs G and H is the minimum size of a connected graph F satisfying $$F\rightarrow (G,H)$$ . For $$m \ge 2,$$ the graph consisting of m independent edges is called a matching and is denoted by $$mK_2$$ . In 1981, Erdos and Faudree determined the size Ramsey numbers for the pair $$(mK_2, K_{1,t})$$ . They showed that the disconnected graph $$mK_{1,t} \rightarrow (mK_2,K_{1,t})$$ for $$ t,m \ge 1$$ . In this paper, we will determine the connected size Ramsey number $$\hat{r}_c(nK_2, K_{1,3})$$ for $$n\ge 2$$ and $$\hat{r}_c(3K_2, C_4)$$ . We also derive an upper bound of the connected size Ramsey number $$\hat{r}_c(nK_2, C_4),$$ for $$n\ge 4$$ .

It is known that semigroups are Ramsey algebras. This paper is an attempt to understand the role associativity plays in a binary system being a Ramsey algebra. Specifically, we show that the nonassociative Moufang loop of octonions is not a Ramsey algebra.

For any given two graphs G and H, the notation $$F\rightarrow $$ (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if $$F\rightarrow $$ (G, H) but $$F-e\nrightarrow (G,H)$$ , for every $$e \in E(F)$$ . The class of all Ramsey (G, H)-minimal graphs is denoted by $$\mathcal {R}(G,H)$$ . In this paper, we construct some infinite families of trees belonging to $$\mathcal {R}(P_3,P_n)$$ , for $$n=8$$ and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to $$\mathcal {R}(P_3,P_n)$$ , for $$n\ge 10$$ .

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faa di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.

In this paper, we study the homological theory in n-abelian categories. First, we prove some useful properties of n-abelian categories, such as $$(n+2)\times (n+2)$$ -lemma, 5-lemma and n-Horseshoes lemma. Secondly, we introduce the notions of right(left) n-derived functors of left(right) n-exact functors, n-(co)resolutions, and n-homological dimensions of n-abelian categories. For an n-exact sequence, we show that the long n-exact sequence theorem holds as a generalization of the classical long exact sequence theorem. As a generalization of $$\textsf {Ext}^*(-,-)$$ , we study the n-derived functor $$\textsf {nExt}^*(-,-)$$ of hom-functor $$\mathrm {Hom}(-,-)$$ . We give an isomorphism between the abelian group of equivalent classes of m-fold n-extensions $$\textsf {nE}^m(A,B)$$ of A, B and $$\textsf {nExt}_{\mathcal A}^m(A,B)$$ using n-Baer sum for $$m,n\ge 1$$ .

Let C be a bounded closed convex subset of a uniformly convex multi-Banach space X and let $${\mathfrak {I}}_{j} = \{T_j(t) : t\in G\}$$ be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we prove the strong mean ergodic convergence theorem for the almost-orbit of $$\mathfrak {I}$$ . Our results extend and unify many previously known results especially (Dong et al. On the strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in Banach space, preprint).

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold $${\mathbb {S}}^3\times {\mathbb {S}}^3$$ with a conformal vector field. There are three almost contact metric structures on a hypersurface of $${\mathbb {S}}^3\times {\mathbb {S}}^3$$ , and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in $${\mathbb {S}}^3\times {\mathbb {S}}^3$$ .