A Domain Embedding Method Using the Optimal Distributed Control and a Fast Algorithm

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. The method is based on formulating the problem as an optimal distributed control problem inside a disc in which the arbitrary domain is embedded. The optimal distributed control problem inside the disc is solved rapidly using a fast algorithm developed by Daripa et al. [3,7,10–12]. The arbitrary domains can be simply or multiply connected and the proposed method can be applied, in principle, to a large number of elliptic problems. Numerical results obtained for Dirichlet problems associated with the Poisson equation in simply and multiply connected domains are presented. The computed solutions are found to be in good agreement with the exact solutions with moderate number of grid points in the domain.     

Subspaces of Normed Riesz Spaces

It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)p(y) for all x and y with -yxy. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.     

B-Splines with Arbitrary Connection Matrices

We consider a space of Chebyshev splines whose left and right derivatives satisfy linear constraints that are given by arbitrary nonsingular connection matrices. We show that for almost all knot sequences such spline spaces have basis functions whose support is equal to the support of the ordinary B-splines with the same knots. Consequently, there are knot insertion and evaluation algorithms analogous to de Boors algorithm for ordinary splines.     

A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p q and it is also independent of the classification of the arc-transitive graphs of order pq for p q.     

A Lower Bound on the Greedy Weights of Product Codes

A greedy 1-subcode is a one-dimensional subcode of minimum (support) weight. A greedy r-subcode is an r-dimensional subcode with minimum support weight under the constraint that it contain a greedy (r - 1)-subcode. The r-th greedy weight e _r is the support weight of a greedy r-subcode. The greedy weights are related to the weight hierarchy. We use recent results on the weight hierarchy of product codes to develop a lower bound on the greedy weights of product codes.     

A geometric formula for Haefliger knots

Haefliger has shown that a smooth embedding of the (4k−1)-sphere in the 6k-sphere can be knotted in the smooth sense. In this paper, we give a formula with which we can detect the isotopy class of such a Haefliger knot. The formula is expressed in terms of the geometric characteristics of an extension, analogous to a Seifert surface, of the given embedding. In particular, the Hopf invariant associated to the extension plays a crucial role. This leads us to a new characterisation of Haefliger knots.     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

Grünbaum colorings of even triangulations on surfaces

A Grünbaum coloring of a triangulation G is a map c : such that for each face f of G, the three edges of the boundary walk of f are colored by three distinct colors. By Four Color Theorem, it is known that every triangulation on the sphere has a Grünbaum coloring. So, in this article, we investigate the question whether each even (i.e., Eulerian) triangulation on a surface with representativity at least r has a Grünbaum coloring. We prove that, regardless of the representativity, every even triangulation on a surface has a Grünbaum coloring as long as is the projective plane, the torus, or the Klein bottle, and we observe that the same holds for any surface with sufficiently large representativity. On the other hand, we construct even triangulations with no Grünbaum coloring and representativity , and 3 for all but finitely many surfaces. In dual terms, our results imply that no snark admits an even map on the projective plane, the torus, or the Klein bottle, and that all but finitely many surfaces admit an even map of a snark with representativity at least 3.     

Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

This paper studies the geometric decay property of the joint queue-length distribution {p(n ₁,n ₂)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c ₁,c ₂,d ₁ and d ₂, an upper bound of the decay rate is derived in the sense