Groups \(S_n \times S_m\) in construction of flag-transitive block designs

In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

Invariance and normality of projections in the dual of Banach algebras

We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.

Obstructions to the existence of compact Clifford–Klein forms for tangential symmetric spaces

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

$$\alpha $$-Dirac-harmonic maps from closed surfaces

Abstract$$\alpha $$ α -Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $$\alpha $$ α -harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For $$\alpha >1$$ α > 1 , the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing $$\alpha $$ α -harmonic maps for $$\alpha >1$$ α > 1 and then letting $$\alpha \rightarrow 1$$ α → 1 . The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $$\alpha $$ α -Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By $$\varepsilon $$ ε -regularity and suitable perturbations, we can then show that such a sequence of perturbed $$\alpha $$ α -Dirac-harmonic maps converges to a smooth coupled $$\alpha $$ α -Dirac-harmonic map.

On the existence problem for a fixed point of a generalized contracting multivalued mapping

We discuss the still unresolved question, posed in [S. Reich, Some Fixed Point Problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 57:8 (1974), 194–198], of existence in a complete metric space X of a fixed point for a generalized contracting multivalued map Φ: X⇉X having closed values Φ(x)⊂X for all x∈X. Generalized contraction is understood as a natural extension of the Browder–Krasnoselsky definition of this property to multivalued maps: ∀x,u∈X h(φ(x),φ(u))≤ η(ρ(x,u)), where the function η:R_+→R_+ is increasing, right continuous, and for all d>0, η(d)<d (h(•,•) denotes the Hausdorff distance between sets in the space X). We give an outline of the statements obtained in the literature that solve the S. Reich problem with additional requirements on the generalized contraction Φ. In the simplest case, when the multivalued generalized contraction map Φ acts in R, without any additional conditions, we prove the existence of a fixed point for this map.

Balanced Equi-$n$-Squares

We define a $d$-balanced equi-$n$-square $L=(l_{ij})$, for some divisor $d$ of $n$, as an $n \times n$ matrix containing symbols from $\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. We show how to construct a $d$-balanced equi-$n$-square from a partition of a Latin square of order $n$ into $d \times (n/d)$ subrectangles. In graph theory, $L$ is equivalent to a decomposition of $K_{n,n}$ into $d$-regular spanning subgraphs of $K_{n/d,n/d}$. We also study when $L$ is diagonally cyclic, defined as when $l_{(i+1)(j+1)}=l_{ij}+1$ for all $i,j \in \mathbb{Z}_n$, which correspond to cyclic such decompositions of $K_{n,n}$ (and thus $\alpha$-labellings). We identify necessary conditions for the existence of (a) $d$-balanced equi-$n$-squares, (b) diagonally cyclic $d$-balanced equi-$n$-squares, and (c) Latin squares of order $n$ which partition into $d \times (n/d)$ subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed $d \geq 1$ when $n$ is sufficiently large, and we resolve the existence problem completely when $d \in \{1,2,3\}$. Along the way, we identify a bijection between $\alpha$-labellings of $d$-regular bipartite graphs and what we call $d$-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either $d$ or $0$ filled cells in each row and column. We use $d$-starters to construct diagonally cyclic $d$-balanced equi-$n$-squares, but this also gives new constructions of $\alpha$-labellings.

Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.