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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, we study the spectra of Schrodinger operators on zigzag carbon nanotubes, which are broken by abrasion or during refining process. Throughout this paper, we assume that the carbon nanotubes are broken periodically and we deal with one of those models. Making use of the Floquet–Bloch theory, we examine the spectra of the Schrodinger operators and compare the spectra of the broken case and the pure unbroken case.

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.

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 $$pd(G_1)=pd(G_2)$$ .

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 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.

In this paper, we study the noncommutative Orlicz space \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\), which generalizes the concept of noncommutative Lp space, where \(\mathcal {M}\) is a von Neumann algebra, and φ is an Orlicz function. As a modular space, the space \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\) possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace \(E_{\varphi }(\tilde {\mathcal {M}},\tau )=\overline {\mathcal {M}\bigcap L_{\varphi }(\tilde {\mathcal {M}},\tau )}\) in \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\), which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function φ satisfies the Δ2-condition, then \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\) is uniformly monotone, and convergence in the norm topology and measure topology coincide on the unit sphere. Hence, \(E_{\varphi }(\tilde {\mathcal {M}},\tau )=L_{\varphi }(\tilde {\mathcal {M}},\tau )\) if φ satisfies the Δ2-condition.

A graph is called integral, if all of its eigenvalues are integers. In this paper, we give some results about integral pentavalent Cayley graphs on abelian or dihedral groups.

The purpose of the present paper is to investigate the influence of strictly convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss the properties of the group of isometries and the exponential maps on a complete Finsler manifold admitting strictly convex functions.

If X is a geodesic metric space and $$x_1,x_2,x_3\in X$$ , a geodesic triangle $$T=\{x_1,x_2,x_3\}$$ is the union of the three geodesics $$[x_1x_2]$$ , $$[x_2x_3]$$ and $$[x_3x_1]$$ in X. The space X is $$\delta $$ -hyperbolic (in the Gromov sense) if any side of T is contained in a $$\delta $$ -neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by $$\delta (X)$$ the sharp hyperbolicity constant of X, i.e. $$\delta (X)=\inf \{\delta \ge 0: \, X \, \text { is }\delta \text {-hyperbolic}\}$$ . In this paper, we characterize the lexicographic product of two graphs $$G_1\circ G_2$$ which are hyperbolic, in terms of $$G_1$$ and $$G_2$$ : the lexicographic product graph $$G_1\circ G_2$$ is hyperbolic if and only if $$G_1$$ is hyperbolic, unless if $$G_1$$ is a trivial graph (the graph with a single vertex); if $$G_1$$ is trivial, then $$G_1\circ G_2$$ is hyperbolic if and only if $$G_2$$ is hyperbolic. In particular, we obtain the sharp inequalities $$\delta (G_1)\le \delta (G_1\circ G_2) \le \delta (G_1) + 3/2$$ if $$G_1$$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

Let A be a nonempty finite subset of an additive abelian group G and let r and h be positive integers. The generalized h-fold sumset of A, denoted by $$h^{(r)}A$$ , is the set of all sums of h elements of A, where each element appears in a sum at most r times. The direct problem for $$h^{(r)}A$$ is to find a lower bound for $$|h^{(r)}A|$$ in terms of |A|. The inverse problem for $$h^{(r)}A$$ is to determine the structure of the finite set A for which $$|h^{(r)}A|$$ is minimal with respect to some fixed value of |A|. If $$G = \mathbb {Z}$$ , the direct and inverse problems are well studied. In case of $$G = \mathbb {Z}/p\mathbb {Z}$$ , p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271–279). In this paper, we express the generalized sumset $$h^{(r)}A$$ in terms of the regular and restricted sumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case $$G = \mathbb {Z}$$ .

The purpose of this paper is to derive some inequalities for dimension of the c-nilpotent multiplier of finite dimensional Lie algebras and their factor Lie algebras. We further obtain an inequality between dimensions of c-nilpotent multiplier of Lie algebra L and tensor product of a central ideal by its abelianized factor Lie algebra.

In this paper, by using L p ( 1 < p≤2) weak convergence method on backward stochastic differential equations (BSDEs) with non-uniformly Lipschitz coefficients, we obtain the limit theorem of g-supersolutions. As applications of this theorem, we study the decomposition theorem of ${\mathcal {E}}_{g}$ -supermartingale, the nonlinear decomposition theorem of Doob-Meyer’s type and so on. Furthermore, by using the decomposition theorem of ${\mathcal {E}}_{g}$ -supermartingale, we provide some useful characterizations of an ${\mathcal {E}}^{g}$ -evaluation by the generating function g(t;ω;y;z) without the assumption that g is continuous with respect to t. Our results generalize the known results in Ph. Briand et al., Electronic Commun. Probab. 5 (2000) 101–117; L Jiang, Ann. Appl. Probab. 18 (2008) 245–258; S Peng, Probab. Theory Relat. Fields 113 (1999) 473–499; S Peng, Modelling derivatives pricing with their generating functions (2006) http://arxiv.org/abs/math/0605599 and E Rosazza Gianin, Insur. Math. Econ. 39 (2006) 19–34.

To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which is homeomorphic to the complexified complement in the realizable case. Further we generalize the classical theorem of Salvetti and show that this space has the homotopy type of a cell complex defined in terms of the oriented matroid., Comment: 19 pages, 3 figures. Third version: some more typos fixed, minor changes, final version. To appear in Proceedings Mathematical Sciences. In the second version: typos fixed, exposition improved, contact information updated. arXiv admin note: text overlap with arXiv:1110.1520

In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs GP (n,2), where n ≥ 5. We also present some basic results for GP (n, k), where n ≥ 5 and k ≥ 3.

A subgroup H of a group G is said to be \(\mathcal {M}^{*}\)-supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is \(\mathcal {M}\)-supplemented in G. In this paper, we prove as follows: Let E be a normal subgroup of a group G. Suppose that every maximal subgroup of every non-cyclic Sylow subgroup P of F∗(E) is \(\mathcal {M}^{*}\)-supplemented in G, then \(E\leq Z_{\mathcal {U}{\Phi }}(G)\).

Recently several authors investigated structures of generalized bi-circular projections in spaces where the descriptions of the group of surjective isometries are known. Following the same idea in this paper we give complete descriptions of generalized 3-circular projections for symmetric norms on \(\mathbb {C}^{n}\) and \(\mathbb {M}_{m \times n}(\mathbb {C})\).

A graph G is called (k,d)∗-choosable if for every list assignment L satisfying ∣L(v)∣ ≥k for all v∈V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without intersecting i-cycles for all i=3,4,5 is (3,1)∗-choosable.

In this paper, we study the null controllability of the viscous Camassa–Holm equation on the one-dimensional torus. By using a moving distributed control, we obtain that the system is null controllable for a given data with certain regularity.

The Wiener-type invariants of a simple connected graph G = (V, E) can be expressed in terms of the quantities $W_{f}=\sum_{\{u,v\}\subseteq V}f(d_{G}(u,v))$ for various choices of the function f(x), where d G (u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a connected graph to be Hamiltonian, a connected graph to be traceable, and a connected bipartite graph to be Hamiltonian in terms of the Wiener-type invariants.

In this paper, the optimal control problem for the extended Fisher–Kolmogorov equation is studied. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.

In this paper, we present some identities involving convolutions of Dirichlet characters and the Mobius function, which are related to a well known identity of Ramanujan, Hardy and Littlewood.

Denote by $$C_m$$ the cyclic group of order m. Let $${\mathcal {R}}(C_m)$$ be its real representation ring, and $$\Delta (C_m)$$ its augmentation ideal. In this paper, we give an explicit $${\mathbb {Z}}$$ -basis for the n-th power $$\Delta ^{n}(C_m)$$ and determine the isomorphism class of the n-th augmentation quotient $$\Delta ^n(C_m)/\Delta ^{n+1}(C_m)$$ for each positive integer n.

In this article, we consider the following fractional Hamiltonian systems: $$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$ where $$\alpha \in (1/2, 1)$$ , $$\lambda >0$$ is a parameter, $$L\in C(\mathbb {R}, \mathbb {R}^{n\times n})$$ and $$W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})$$ . Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all $$t\in \mathbb {R}$$ , that is, $$L(t) \equiv 0$$ is allowed to occur in some finite interval $$\mathbb {I}$$ of $$\mathbb {R}$$ . Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on $$\mathbb {R} \setminus \mathbb {I}$$ as $$\lambda \rightarrow \infty ,$$ and converge to $$\tilde{u}$$ in $$H^{\alpha }(\mathbb {R})$$ ; here $$\tilde{u} \in E_{0}^{\alpha }$$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $$\mathbb {I}$$ . Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

In this paper, we establish some weak and strong convergence theorems for a new iterative algorithm under some suitable conditions to approximate the common fixed point of three infinite families of multi-valued generalized nonexpansive mappings in a uniformly convex Banach spaces. Our results generalize and improve several previously known results of the existing literature.

In this paper, we prove that some power series with rational coefficients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the p-adic analogues of these results and show that these results have analogues in the field of p-adic numbers.

We study a diffusive predator–prey model with modified Leslie–Gower and Holling-II schemes with $$D=0$$ . We establish the existence of traveling wave solutions connecting a positive equilibrium and a boundary equilibrium via the ‘shooting method’, and the non-existence by the ‘eigenvalue method’. It should be emphasized that a threshold value $$c^*=\sqrt{4\alpha }$$ is found in our paper.

In this paper, we give conditions under which the powers of the multiplication operator \(M_{z}\) are reflexive on a Banach space of functions analytic on a plane domain.

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.

In this paper, we provide combinatorial meanings to two generalized basic series with the aid of associated lattice paths. These results produce two new classes of infinite 3-way combinatorial identities. Five particular cases are also discussed. These particular cases provide new combinatorial versions of Gollnitz–Gordon identities and Gollnitz identity. Seven q-identities of Slater and five q-identities of Rogers are further explored using the same combinatorial object. These results are an extension of the work of Goyal and Agarwal (Utilitas Math. 95 (2014) 141–148), Agarwal and Rana (Utilitas Math. 79 (2009) 145–155), and Agarwal (J. Number Theory 28 (1988) 299–305).