In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from ‘strong stability’ forms of the corresponding (pure) extremal results. These results hold for the α-spectral radius defined using the α-norm for any α > 1; the usual spectral radius is the case α = 2. Our results imply that any hypergraph Turán problem ...

The line shape of resonances in the overlapping regime is studied by using the eigenvalues and eigenfunctions of the effective Hamiltonian of an open quantum system. A generalized expressioñ q k (E) for the Fano parameter of the resonance state k is derived that contains the interaction of the state k with neighboured states l = k via the continuum. It is energy dependent since the coupling coe...

Finally, Theorem 1.2 is quite straightforward to deduce from Corollary 5.4. Alternatively , it is not diicult to prove this theorem directly using Theorem 2.1. We conclude by presenting the rst of these proofs. Proof of Theorem 1.2. From the list given in Corollary 5.4 of binary 3{connected matroids whose set of non-essential elements has rank two, we eliminate the ma-troids under (i) and (iii)...

The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions R♦ (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of R♦. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left R♦-module R ♦ is shown to split into three disjoint sets of cardinalities 9, 9 and 3 accor...

Cubic bipartite graphs with girth at least 6 correspond to symmetric combinatorial (v3) configurations. In 1887 V. Martinetti described a simple reduction method which enables one to reduce each combinatorial (v3) configuration to one from the infinite set of so-called irreducible configurations. The aim of this paper is to show that a slightly extended set of reductions enables one to reduce e...

We consider the following three problems: 1. Let U be a q-subset of GF(q 2) with the properties 0; 1 2 U and u ? v is a square for all u; v 2 U. Does it follow that U consists of the elements of the subbeld GF(q)? Here q is odd. 2. Let f : GF(q) ! GF(q) be any function, and let be the set of diierence quotients (directions, slopes). What are the possibilities for jD f j? 3. Let B be a subset of...

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 in...

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane as vertices and an adjacency resembling that of star Cayley graphs. The graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copi...

Wedescribe amethod that we believemay be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed.We demonstrate the power of the me...

The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it computes sufficient conditions to have an algebraic counterpart related by tropicalization. We also provide sufficient conditions in a geometric construction to...