Infinite Families Of Examples Research Articles

Extremal fixed points and Diophantine equations

The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting’s theorem, Siegel’s theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples.

Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L, we are interested in finding the pairs (ϕ,ψ)∈Aut(V)×Aut(L), which are inducible by an automorphism in Aut(L˜). We connect the inducibility problem to the (2,3)-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut(V)×Aut(L) to be inducible lies in the (2,3)-cohomology group H(2,3)(L,V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2,3)-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2.

On tetravalent half-arc-transitive graphs of girth 5

A subgroup of the automorphism group of a graph Γ is said to be half-arc-transitive on Γ if its action on Γ is transitive on the vertex set of Γ and on the edge set of Γ but not on the arc set of Γ. Tetravalent graphs of girths 3 and 4 admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth 5. We show that, with two exceptions, all such graphs only have directed 5-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed 5-cycles, study some of their graph theoretic properties and prove that the 5-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth 5 admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth 5.

Interpolated family of non-group-like simple integral fusion rings of Lie type

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,[Formula: see text])), [Formula: see text] prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings [Formula: see text] interpolates to all integers [Formula: see text], providing (when [Formula: see text] is not prime-power) the first example of infinite family of non-group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring [Formula: see text] (with [Formula: see text] non-prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.

Block‐transitive designs based on grids

AbstractWe study point‐block incidence structures for which the point set is an grid. Cameron and the fourth author showed that each block may be viewed as a subgraph of a complete bipartite graph with bipartite parts (biparts) of sizes . In the case where consists of all the subgraphs isomorphic to , under automorphisms of fixing the two biparts, they obtained necessary and sufficient conditions for to be a 2‐design, and to be a 3‐design. We first reinterpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of . We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be ‐designs, for , and give infinite families of examples illustrating that block‐transitive, point‐primitive 2‐designs based on grids exist for all values of , and flag‐transitive, point‐primitive examples occur for all even . This approach also allows us to construct a small number of block‐transitive 3‐designs based on grids.

The graphs with a symmetrical Euler cycle

The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is a sequence (e1,e2,…,eℓ) of these ℓ edges, each occurring exactly once, such that ei, ei + 1 are incident with a common vertex for each i (reading subscripts modulo ℓ). An Euler cycle is symmetrical if there exists an automorphism of the graph such that ei → ei + 2 for each i. The cyclic group generated by this automorphism has one orbit on edges if ℓ is odd, or two orbits of length ℓ/2 if ℓ is even: that is to say, the group is regular or bi-regular on edges, respectively. Symmetrical Euler cycles arise naturally from arc-transitive embeddings of graphs in surfaces since, for each face of the embedded graph, the sequence of edges on the boundary of the face forms a symmetrical Euler cycle for the induced subgraph on this edge-set. We first classify all finite connected graphs which admit a cyclic subgroup of automorphisms that is regular or bi-regular on edges, and identify more than a dozen infinite families of examples. We then prove that exactly six of these families consist of graphs with symmetrical Euler cycles. These are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.

Three-Weight Codes over Rings and Strongly Walk Regular Graphs

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $${\mathbb {Z}}_{p^m},$$ for p a prime, and more generally over chain rings of depth m, and with a residue field of size q, a prime power. In the case of $$p=m=2,$$ strong necessary conditions ( diophantine equations) on the weight distribution are derived, leading to a partial classification in modest length. Infinite families of examples are built from Kerdock and generalized Teichmüller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear.

Four-Valent Oriented Graphs of Biquasiprimitive Type

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$ where $\Gamma$ is 4-valent, connected and $G$-oriented ($G$-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs $(\Gamma, G) \in\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \cong T^k$ for a simple group $T$ and $k\in \{1,2,4,8\}$. This provides a crucial step towards a general description of the long-studied family $\mathcal{OG}(4)$ in terms of a normal quotient reduction. We also give several methods for constructing pairs $(\Gamma, G)$ of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup $N$.

Reconstruction of Rooted Directed Trees

Let T be a rooted directed tree on n vertices, rooted at v. The rooted subtree frequency vector (RSTF-vector) of T with root v, denoted by rstf(T, v) is a vector of length n whose entry at position k is the number of subtrees of T that contain v and have exactly k vertices. In this paper we present an algorithm for reconstructing rooted directed trees from their rooted subtree frequencies (up to isomorphism). We show that there are examples of nonisomorphic pairs of rooted directed trees that are RSTF-equivalent, s.t. they share the same rooted subtree frequency vectors. We have found all such pairs (groups) for small sizes by using exhaustive computer search. We show that infinitely many nonisomorphic RSTF-equivalent pairs of trees exist by constructing infinite families of examples.

An infinite family of circulant graphs with perfect state transfer in discrete quantum walks

We study perfect state transfer in Kendon’s model of discrete quantum walks. In particular, we give a characterization of perfect state transfer purely in terms of the graph spectra, and construct an infinite family of 4-regular circulant graphs that admit perfect state transfer. Prior to our work, the only known infinite families of examples were variants of cycles and diamond chains.

On characteristic classes of exotic manifold bundles

Given a closed simply connected manifold M of dimension 2nge 6, we compare the ring of characteristic classes of smooth oriented bundles with fibre M to the analogous ring resulting from replacing M by the connected sum Msharp Sigma with an exotic sphere Sigma . We show that, after inverting the order of Sigma in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Sigma is necessary.

Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

We give two infinite families of examples of closed, orientable, irreducible 3-manifolds M M such that b 1 ( M ) = 1 b_1(M)=1 and π 1 ( M ) \pi _1(M) has weight 1, but M M is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl, and Wilton and provides the first examples of irreducible manifolds with b 1 = 1 b_1=1 that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.

Purely (non-)strongly real Beauville groups

We discuss Beauville groups whose corresponding Beauville surfaces are either always strongly real or never strongly real producing several infinite families of examples.

Minor Posets of Functions as Quotients of Partition Lattices

We study the structure of the partially ordered set of minors of an arbitrary function of several variables. We give an abstract characterization of such “minor posets” in terms of colorings of partition lattices, and we also present infinite families of examples as well as some constructions that can be used to build new minor posets.

Drinfeld orbifold algebras for symmetric groups

Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincaré–Birkhoff–Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces.

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

Algebraic quantum synchronizable codes

In this paper, we construct quantum synchronizable codes (QSCs) based on the sum and intersection of cyclic codes. Further, infinite families of QSCs are obtained from BCH and duadic codes. Moreover, we show that the work of Fujiwara~\cite{fujiwara1} can be generalized to repeated root cyclic codes (RRCCs) such that QSCs are always obtained, which is not the case with simple root cyclic codes. The usefulness of this extension is illustrated via examples of infinite families of QSCs from repeated root duadic codes. Finally, QSCs are constructed from the product of cyclic codes.

Intersections of Multiplicative Translates of 3-Adic Cantor Sets II: Two Infinite Families

ABSTRACTThis article studies the structure of finite intersections of general multiplicative translates for integers 1 ⩽ M1 < M2 < ⋅⋅⋅ < Mn, in which denotes the 3-adic Cantor set (of 3-adic integers whose expansions omit the digit 2), which has Hausdorff dimension log 32 ≈ 0.630929. This study was motivated by questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. The exceptional set is defined to be the set of all elements of whose forward orbits under this action intersect the 3-adic Cantor set infinitely many times. It is conjectured that it has Hausdorff dimension 0. An earlier article showed that upper bounds on the Hausdorff dimension of the exceptional set can be extracted from knowing Hausdorff dimensions of sets of the kind above, in cases where all Mi are powers of 2. These intersection sets were shown to be fractals whose points have 3-adic expansions describable by labeled paths in a finite automaton, whose Hausdorff dimension is exactly computable and is of the form log 3(β), where β is a real algebraic integer. It gave algorithms for determination of the automaton, and computed examples showing that the dependence of the automaton and the value β on the parameters (M1, …, Mn) is complicated. The present article studies two new infinite families of examples, illustrating interesting behavior of the automata and of the Hausdorff dimension of the associated fractals. One family has associated automata whose directed graph has a nested sequence of strongly connected components of arbitrarily large depth. The second family leads to an improved upper bound for the Hausdorff dimension of the exceptional set of log 3φ ≈ 0.438018, where φ denotes the Golden ratio.

COCHAIN SEQUENCES AND THE QUILLEN CATEGORY OF A COCLASS FAMILY

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of$p$-groups from different coclass families that have equivalent Quillen categories.

Contact structures, deformations and taut foliations

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibered spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.