<Emphasis Type="Italic">E</Emphasis><Superscript>*</Superscript>-dense <Emphasis Type="Italic">E</Emphasis>-semigroups

We investigate E^*-dense semi\-groups as analogues of E-densesemigroupsfor semigroups with zero. We give a characterisation theorem forE^*-dense semigroups whose idempotents form a *-rectangularband. The construction methods of generalised Rees matrix semigroupsare employed to provide examples and illustrations. Our results areanalogous to those of Weipoltshammer for E-dense semigroups.     

The Riemannian <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L ² Riemannian metric—so-called because it induces an L ² topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L ¹-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L ² metric. We also give a user-friendly criterion for convergence (with respect to the L ² metric) in the manifold of metrics.     

Sobolev Inequalities on Forms and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript>-Cohomology on Complete Riemannian Manifolds

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some L ^p,q -estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the L ^p,q -cohomology and prove the L ^q -solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.     

Characterizations of <Emphasis Type="Italic">r</Emphasis>-Convex Functions

This paper discusses some properties of r-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of r-convexity is given without assuming differentiability. The concept of strict r-convexity is introduced. For a twice continuously differentiable function f, it is shown that the strict r-convexity of f is equivalent to a certain condition on ∇ ² f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.     

On Aleksandrov-Fenchel Inequalities for <Emphasis Type="Italic">k</Emphasis>-Convex Domains

In this lecture notes, we will discuss the classical Aleksandrov-Fenchel inequalities for quermassintegrals on convex domains and pose the problem of how to extend the inequalities to non-convex domains. We will survey some recent progress on the problem, then report some of our joint work [9] in which we generalize the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillion [8] for k =  1 case of the inequalities, uses the method of optimal transport.     

Integrals of <Emphasis Type="Italic">K</Emphasis> and <Emphasis Type="Italic">E</Emphasis> from lattice sums

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.     

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .      

<Emphasis Type="Italic">L</Emphasis>-fuzzy roughness of <Emphasis Type="Italic">n</Emphasis>-ary polygroups

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of ϑ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of ϑ-lower and T-upper L-fuzzy rough approximation operators is studied.     

Approximating the Distribution of Customers in <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Emphasis Type="Italic">/s</Emphasis> Queues

Approximation formulae are suggested for the mean and variance of customers in M/E _n /s queues. It is shown that the distributions can be approximated by using the mean and variance to fit Gamma functions. A brief comment on the more general E _m /E _n /s case is given.     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

On the <Emphasis Type="Italic">KO</Emphasis> Characteristic Cycle of a Spin<Superscript><Emphasis Type="Italic">c</Emphasis></Superscript> Manifold

We compute the KO-characteristic numbers of a characteristic submanifold of a Spin^c manifold in terms of its K-characteristic numbers. The proof is based on the geometry of the Thom class in K-theory and is simpler than the existing proofs of several previously known special cases.     

The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

Structure theorems of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-Azumaya algebras

Let k be a field and E(n) be the 2 ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R _M parameterized by symmetric matrices M in M _n (k). In this paper, we study the Azumaya algebras in the braided monoidal category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } and obtain the structure theorems for Azumaya algebras in the category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } , where M is any symmetric n×n matrix over k.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Geodesic loops and periodic geodesics on a Riemannian manifold diffeomorphic to <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Let M ⁿ be a closed 2-connected Riemannian manifold, such that π₃(M ⁿ ) ≠ { 0 }. In this paper we prove that either there exists a periodic geodesic on M ⁿ of length ≤ 6d, where d is the diameter of M ⁿ , or at each point p ∈ M ⁿ there exists a geodesic loop of length ≤ 2d.     

<Emphasis Type="Italic">p</Emphasis>-Fundamental tone estimates of submanifolds with bounded mean curvature

In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one \(C^2\)-foliations of open subsets \(\Omega \) of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of \(\Omega \).     

Gibbs <Emphasis Type="Italic">u</Emphasis>-states for the foliated geodesic flow and transverse invariant measures

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow. p]On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. p]On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ ^u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions.     

Nonlinear separation concerning <Emphasis Type="Italic">E</Emphasis>-optimal solution of constrained multi-objective optimization problems

This paper aims at investigating optimality conditions in terms of E-optimal solution for constrained multi-objective optimization problems in a general scheme, where E is an improvement set with respect to a nontrivial closed convex point cone with apex at the origin. In the case where E is not convex, nonlinear vector regular weak separation functions and scalar weak separation functions are introduced respectively to realize the separation between the two sets in the image space, and Lagrangian-type optimality conditions are established. These results extend and improve the convex ones in the literature.     

Homogenization of variational problems in manifold valued <Emphasis Type="Italic">BV</Emphasis>-spaces

This paper extends the result of Babadjian and Millot (preprint, 2008) on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a Γ-convergence analysis, we identify the homogenized energy in the space of functions of bounded variation. It turns out to be finite for BV-maps with values in the manifold. The bulk and Cantor parts of the energy involve the tangential homogenized density introduced in Babadjian and Millot (preprint, 2008), while the jump part involves an homogenized surface density given by a geodesic type problem on the manifold.