An improved sectional-Anosov closing lemma

We prove that all nonwandering points of a sectional-Anosov flow on a compact 3-manifold can be approximated by periodic points or by points for which the omega-limit set is a singularity. This improves the closing lemma in Morales (Mich. Math. J. 56(1):29?C53, 2008). We also describe a sectional-Anosov flow for which the recurrent points are not dense in the nonwandering set.     

On the sensitivity of sectional-Anosov flows

We study sectional-Anosov flows on compact 3-manifolds for which the maximal invariant and nonwandering sets coincide. We prove that every vector field close to one of these flows is sensitive with respect to initial conditions.     

Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:

the generalized Bethe tree B,

a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,

a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,

a graph obtained from the cycle C_r by attaching B, by its root, to each vertex of the cycle, and

a tree obtained from the path P_r by attaching B, by its root, to each vertex of the path,

is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

     

Topological parallelisms of the real projective 3-space

A parallelism of a projective 3-space Π is a family P of spreads such that each line of Π is contained in exactly one spread of P. A parallelism is said to be totally regular, if all its members are regular spreads. By a generalized line star with respect to an elliptic quadric Q of a classical projective 3-space we understand a set $\cal A$ of 2-secants of Q such that each non-interior point of Q is incident with exactly one line of $\cal A$ . From each generalized line star we can construct a totally regular parallelism which we do in essential by the Thas-Walker construction. A parallelisms of the real projective 3-space PG(3, ?) is called topological, if the operation of drawing a line parallel to a given line through a given point is continuous. Clifford parallelisms are topological. Using generalized line stars we exhibit examples of non-Clifford topological parallelisms and of non-topological parallelisms.     

On the complexity of generalized polynomials of <Emphasis Type="Italic">k</Emphasis>-valued functions

Specification of k-valued functions with generalized polynomials (for simple k) is considered. A generalized polynomial is a mod k polynomial in which each variable may also occur with one or several Post negations. The upper and lower estimates of the complexity of generalized polynomials are found for k-valued functions.     

Generalized linear functional equations on almost quasigroups I: Equations with at most two variables

Summary. We define almost quasigroups, a new class of groupoids which generalize quasigroups, and prove several representation theorems for them, essentially reducing them to loops (see Theorems 1, 2 and 9). Some well-known theorems on quasigroups are generalized, notably the theorems of A. A. Albert (Theorems 8, 9 and 10).¶We also define the normal form of equations and show that every generalized linear functional equation Eq on almost quasigroups is equivalent to a system consisting of several equations with at most one variable each, and one equation in the normal form, with the same number of variables as Eq. Eventually, the general solution of the generalized linear functional equations on almost quasigroups with at most two variables is given.¶We plan to solve other generalized linear functional equations in subsequent papers.     

在白噪声分析中的广义维里拉普拉斯

一类广义维里拉普拉斯，把在白噪声分析构架中通常定义的维里拉普拉斯作为特殊情形而包含。     

Comment on: “Fundamental flows with nonlinear slip conditions: exact solutions”, by R. Ellahi,T. Hayat,F. M. Mahomed and A. Zeeshan,Z. Angew. Math. Phys. 61 (2010) 877–888

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.     

Decomposition of generalized vector variational inequalities

A multicriteria optimization problem is called Pareto reducible if its weakly efficient solutions actually are efficient solutions for the problem itself or for at least one subproblem obtained from it by selecting certain criteria. The aim of this paper is to investigate a similar property within a special class of generalized vector variational inequalities, under appropriate generalized convexity assumptions.     

Monotonicity Formulas of Eigenvalues and Energy Functionals Along the Rescaled List’s Extended Ricci Flow

In this paper, we study monotonicity formulas of eigenvalues of Laplacian and entropies along the rescaled List’s extended Ricci flow, which is a weakly coupled system of second order and the motivation to study this flow stems from its connection to general relativity. The rescaled List’s extended Ricci flow, which is a generalized version including the extended Hamilton normalized flow, is equivalent to the List’s extended Ricci flow up to a homothetic rescale of the metric at each time. Moreover, we find natural entropies of \(\mathcal {F}_k\)-functional and \(\mathcal {W}_k\)-functional and show that they are monotonic.     

Abelianization and Nielsen realization problem of the mapping class group of a handlebody

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms to the mapping class group has no section when g is at least 5. In order to prove the above result, we show the vanishing of the first homology group with the real coefficient of the mapping class group of the handlebody with genus g at least 3 and a distinguished disk on its boundary.     

Hyper-Lagrangian submanifolds of Hyperkähler manifolds and mean curvature flow

In this article, we define a new class of middle dimensional submanifolds of a Hyperkähler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation. It is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.     

The generalized S-graphs of diameter 3

A graph is called a generalized S-graph if for every vertex v of G there exists exactly one vertex which is more remote from v than every vertex adjacent to v. A generalized S-graph of diameter 3 is called reducible if there is a pair of diametrical vertices v and such that v is also a generalized S-graph of diameter 3. Here we determine all irreducible generalized S-graphs of diameter 3.     

Laplacian spectral characterization of clover graphs

A clover graph is obtained from a 3-rose graph by attaching a path to the vertex of degree six, where a 3-rose graph consists of three cycles with precisely one common vertex. In this paper, it is proved that all clover graphs are determined by their Laplacian spectra.     

Approximation,solution operators and quantale-valued metrics

A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \)-approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \). In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \)-approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator.     

A d’Alembert Formula for Hopf Hypersurfaces

A Hopf hypersurface in complex hyperbolic space ${\mathbb{C}{\rm H}^n}$ is one for which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to ambient curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit parametrization.     

Spectral characterization of some weighted rooted graphs with cliques

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which

(1)

the edges connecting vertices at consecutive levels have the same weight,

(2)

each set of children, in one or more levels, defines a weighted clique, and

(3)

cliques at the same level are isomorphic.

These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.