Total embedding distributions of Ringel ladders

The total embedding distributions of a graph consists of the orientable embeddings and non-orientable embeddings and are known for only a few classes of graphs. The orientable genus distribution of Ringel ladders is determined in [E.H. Tesar, Genus distribution of Ringel ladders, Discrete Mathematics 216 (2000) 235–252] by E.H. Tesar. In this paper, using the overlap matrix, we obtain nonhomogeneous recurrence relation for rank distribution polynomial, which can be solved by the Chebyshev polynomials of the second kind. The explicit formula for the number of non-orientable embeddings of Ringel ladders is obtained. Also, the orientable genus distribution of Ringel ladders is re-derived.     

A relative maximum genus graph embedding and its local maximum genus

A relative embedding of a connected graph is an embedding of the graph in some surface with respect to some closed walks, each of which bounds a face of the embedding. The relative maximum genus of a connected graph is the maximum of integerk with the property that the graph has a relative embedding in the orientable surface withk handles. A polynomial algorithm is provided for constructing relative maximum genus embedding of a graph if the relative tree of the graph is planar. Under this condition, just like maximum genus embedding, a graph does not have any locally strict maximum genus.     

Log‐concavity of genus distributions for circular ladders

A well‐known conjecture in topological graph theory says that the genus distribution of every graph is log‐concave. In this paper, the genus distribution of the circular ladder is re‐derived, using overlap matrices and Chebyshev polynomials, which facilitates proof that this genus distribution is log‐concave.     

On the embedding of variational inequalities

This work is devoted to the approximation of variational inequalities with pseudo-monotone operators. A variational inequality, considered in an arbitrary real Banach space, is first embedded into a reflexive Banach space by means of linear continuous mappings. Then a strongly convergent approximation procedure is designed by regularizing the embedded variational inequality. Some special cases have also been discussed.

     

On the problem of field embedding

A cohomological interpretation of solvability conditions for the problem of field embedding is given. Embedding conditions are formulated in these terms in cases when there are additional restrictions on the constructed field (e.g., fixation of the branching points of the field in the number case).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 137–140, August, 1968.     

On embedding well-separable graphs

Call a simple graph H of order nwell-separable, if by deleting a separator set of size o(n) the leftover will have components of size at most o(n). We prove, that bounded degree well-separable spanning subgraphs are easy to embed: for every γ>0 and positive integer Δ there exists an n₀ such that if n>n₀, Δ(H)?Δ for a well-separable graph H of order n and δ(G)?(1-1/2(χ(H)-1)+γ)n for a simple graph G of order n, then H⊂G. We extend our result to graphs with small band-width, too.     

On Birget''s regular embedding

This paper gives another construction of (S)_reg, and a different proof of the uniqueness of coded normal forms.     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.     

On an embedding of certainp-groups

It is shown that ap-group with cyclic centre can be embedded in a finite group as a normal subgroup contained in its Frattini subgroup if and only if it is either an extraspecial 2-group of order at least 2⁷ or the central product of a cyclic groupQ of order ≧4 and an extraspecial groupE of order ≧2⁵, amalgamating Ω₁ (Q) and the centre ofE.     

Blowing up points and embedding flat stable planes in the nonorientable compact surface of genus one

We show that the point set of every flat stable plane embeds in the point set of the real projective plane. Connectedness of lines or of the point space is not assumed. We give two largely independent proofs; the first one is more conceptual, while the second one is more direct, and shorter. The first proof uses a new construction called blowing up a point, i.e., replacing it with its line pencil; this amounts to adding a cross cap. This construction seems to be of interest in its own right.     

On the Toeplitz embedding of an arbitrary matrix

It has been shown by Delosme and Morf that an arbitrary block matrix can be embedded into a block Toeplitz matrix; the dimension of this embedding depends on the complexity of the matrix structure compared to the block Toeplitz structure. Due to the special form of the embedding matrix, the algebra of matrix polynomials relative to block Toeplitz matrices can be interpreted directly in terms of the original matrix and therefore can be extended to arbitrary matrices. In fact, these polynomials turn out to provide an appropriate framework for the recently proposed generalized Levinson algorithm solving the general matrix inversion problem.     

On the uniqueness of embedding a residual design

We prove that if a residual $\text{2-(k(k+λ?1)}\text{λ,k,λ)}$ design R has more than one embedding into a symmetric design then k ? λ(λ?1)². If equality holds then R has exactly two embeddings and the corresponding derived design is in both cases λ ? 1 identical copies of the design of points and lines of PG(3, λ ? 1). Using the main proposition from which these results follow we also prove that if a symmetric2-(v,k, λ) design has an axial non-central or central non-axial automorphism then k?λ(λ² ? 2λ + 2).     

On geometric embedding problems and semiabelian groups

The focus of interest in this paper are geometric Galois realizations of finite groups. A tool to construct such realizations are geometric embedding problems which are introduced. Then the class of semiabelian groups having geometric Galois realizations is studied and characterized in several ways. In particular, a table of small non-semiabelian groups is computed. At the end, geometric Galois realizations of some small non-semiabelian groups are constructed. The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft