On Pseudo-Holomorphic Curves from Two-Spheres into a Complex Grassmannian G（2, 5）

Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is non-±holomorphic. Furthermore, K = 1/3 if and only if s is totally real. We also prove that the Gaussian curvature K is either 1 or -4/3 if s is a non-degenerate holomorphic curve under some conditions.     

On minimal two-spheres immersed in complex Grassmann manifolds with parallel second fundamental form

In this paper, we study some properties of the linearly full conformal minimal immersions φ : S ² → G(k, n) with second fundamental form B. At first we compute the Laplacian of square length ||B||² of B and the relations of Gaussian curvature K and normal curvature K ^N . Then we obtain a necessary and sufficient condition of the parallel second fundamental form, and prove that K must be constant if B is parallel. Moreover, if it is not totally geodesic, K ≤?||B||²/2, especially, K =?||B||²/2 when it is holomorphic. We also consider the pseudo-holomorphic curve in G(k, n) with parallel second fundamental form and compute its Gaussian curvature and K?hler angle.     

On Darboux helices in Euclidean 4‐space

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yayl? et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ² + τ² = constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V₄‐slant helix are the same concepts.     

Integral complete 4-partite graphs

A graph G is called integral if all the roots of the characteristic polynomial P(G;x) are integers. In the paper the first known integral complete 4-partite graph K_p1,p2,p3,p4, where p₁<p₂<p₃<p₄, is constructed.     

The eccentric connectivity index of nanotubes and nanotori

Let G be a molecular graph. The eccentric connectivity index ξ^c(G) is defined as ξ^c(G)=∑_u∈V(G)deg_G(u)ε_G(u), where deg_G(u) denotes the degree of vertex u and ε_G(u) is the largest distance between u and any other vertex v of G. In this paper exact formulas for the eccentric connectivity index of TUC₄C₈(S) nanotube and TC₄C₈(S) nanotorus are given.     

Hamiltonicity of 4-connected graphs

Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of order n and σ5（G） 〉 n ＋ 3σ（G） ＋ 11, then G is Hamiltonian.     

Three-webs with covariantly constant curvature and torsion tensors

In this paper, we study a special class of multidimensional 3-webs with covariantly constant curvature and torsion tensors. In the first part, we prove that 3-webs of the class belong to G-webs, i.e., there is a subfamily of adapted frames whose components of curvature and torsion tensors are constant. The structure of the homogeneous space G/H carrying the 3-web is described. Structure equations of the G-group are found. In the second part, we find structure equations of the W ^??-web and finite equations of some special web classes.     

Almost equal group multiplications

In a recent paper entitled “A commutative analogue of the group ring” we introduced, for each finite group (G,⋅), a commutative graded Z-algebra R_(G,⋅) which has a close connection with the cohomology of (G,⋅). The algebra R_(G,⋅) is the quotient of a polynomial algebra by a certain ideal I_(G,⋅) and it remains a fundamental open problem whether or not the group multiplication ⋅ on G can always be recovered uniquely from the ideal I_(G,⋅).Suppose now that (G,×) is another group with the same underlying set G and identity element e∈G such that I_(G,⋅)=I_(G,×). Then we show here that the multiplications ⋅ and × are at least “almost equal” in a precise sense which renders them indistinguishable in terms of most of the standard group theory constructions. In particular in many cases (for example if (G,⋅) is Abelian or simple) this implies that the two multiplications are actually equal as was claimed in the previously cited paper.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

Co-calibrated G 2 structure from cuspidal cubics

We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G ₂ structure on the seven-dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G ₂ structure on SU(2, 1)/U(1). This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of certain seventh order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff?CWallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.     

On the geometry of locally conformally almost cosymplectic manifolds

We obtain the complete group of structure equations of a locally conformally almost cosymplectic structure (an lc $ ACy $ -structure in what follows) and compute the components of the Riemannian curvature tensor on the space of the associated G-structure. Normal lc $ ACy $ -structures are studied in more detail. In particular, we prove that the contact analogs of A. Gray’s second and third curvature identities hold on normal lc $ ACy $ -manifolds, while the contact analog of A. Gray’s first identity holds if and only if the manifold is cosymplectic.     

Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants

In a rotationally symmetric space ${{\overline M}}$ around an axis ${\mathcal{A}}$ (whose precise definition is satisfied by all real space forms), we consider a domain G limited by two equidistant hypersurfaces orthogonal to ${\mathcal{A}}$ . Let ${M \subset {\overline M}}$ be a revolution hypersurface generated by a graph over ${\mathcal{A}}$ , with boundary in ?G and orthogonal to it. We study the evolution M _t of M under the volume-preserving mean curvature flow requiring that the boundary of M _t rests on ?G and stays orthogonal to it. We prove that: (a) the generating curve of M _t remains a graph; (b) the flow exists as long as M _t does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of M, the flow is defined for every ${t\in [0,\infty[}$ and a sequence of hypersurfaces ${M_{t_n}}$ converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.     

A commutative analogue of the group ring

Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded Z-algebra R_G. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element.In the case when G is an elementary Abelian p-group it turns out that R_G is closely related to the symmetric algebra over F_p of the dual of G. We intend in subsequent papers to explore the close relationship between G and R_G in the case of a general (possibly non-Abelian) group G.Here we show that the Krull dimension of R_G is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1.     

4-regular claw-free IM-extendable graphs

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic to K_1,3. The kth power of a graph G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are , and T_r, r?2, where T_r is the graph with 4r vertices u_i,v_i,x_i,y_i, 1?i?r, such that for each i with 1?i?r, {u_i,v_i,x_i,y_i} is a clique of T_r and . We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are K₄×K₂, and .     

Structure of digraphs associated with quadratic congruences with composite moduli

We assign to each positive integer n a digraph G(n) whose set of vertices is H={0,1,…,n-1} and for which there exists a directed edge from a∈H to b∈H if . Associated with G(n) are two disjoint subdigraphs: G₁(n) and G₂(n) whose union is G(n). The vertices of G₁(n) correspond to those residues which are relatively prime to n. The structure of G₁(n) is well understood. In this paper, we investigate in detail the structure of G₂(n).     

A note on the signless Laplacian eigenvalues of graphs

In this paper, we consider the signless Laplacians of simple graphs and we give some eigenvalue inequalities. We first consider an interlacing theorem when a vertex is deleted. In particular, let G-v be a graph obtained from graph G by deleting its vertex v and κ_i(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that κ_i+1(G)-1?κ_i(G-v)?κ_i(G). Next, we consider the third largest eigenvalue κ₃(G) and we give a lower bound in terms of the third largest degree d₃ of the graph G. In particular, we prove that . Furthermore, we show that in several situations the latter bound can be increased to d₃-1.     

The maximum corank of graphs with a 2-separation

For a graph G=(V,E) with vertex-set V={1,2,…,n}, which is allowed to have parallel edges, and for a field F, let S(G;F) be the set of all F-valued symmetric n×n matrices A which represent G. The maximum corank of a graph G is the maximum possible corank over all A∈S(G;F). If (G₁,G₂) is a (?2)-separation, we give a formula which relates the maximum corank of G to the maximum corank of some small variations of G₁ and G₂.     

Removable edges of cycles in 5-connected graphs

Let G be a 5-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G?e; second, for each vertex x of degree 4 in G?e, delete x from G?e and then completely connect the 4 neighbors of x by K ₄. If multiple edges occur, we use single edge to replace them. The final resultant graph is denoted by G ? e. If G ? e is still 5-connected, then e is called a removable edge of G. In this paper, we investigate the distribution of removable edges in a cycle of a 5-connected graph. And we give examples to show some of our results are best possible in some sense.     

On the characteristic connection of gwistor space

We give a brief presentation of gwistor spaces, which is a new concept from G ₂ geometry. Then we compute the characteristic torsion T ^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T ^c is ?^c-parallel; this allows for the classification of the G ₂ structure with torsion and the characteristic holonomy according to known references. The case of an Einstein base manifold is envisaged.     

Truncation and augmentation of level-independent QBD processes

In the study of quasi-birth-and-death (QBD) processes, the first passage probabilities from states in level one to the boundary level zero are of fundamental importance. These probabilities are organized into a matrix, usually denoted by G.The matrix G is the minimal nonnegative solution of a matrix quadratic equation. If the QBD process is recurrent, then G is stochastic. Otherwise, G is sub-stochastic and the matrix equation has a second solution G_sto, which is stochastic. In this paper, we give a physical interpretation of G_sto in terms of sequences of truncated and augmented QBD processes.As part of the proof of our main result, we derive expressions for the first passage probabilities that a QBD process will hit level k before level zero and vice versa, which are of interest in their own right.The paper concludes with a discussion of the stability of a recursion naturally associated with the matrix equation which defines G and G_sto. In particular, we show that G is a stable equilibrium point of the recursion while G_sto is an unstable equilibrium point if it is different from G.