Planar sections of three-dimensional cylinders

Is it true that every interior point of a three-dimensional convex body lies on its planar section with an inscribed regular hexagon and the center of a centrally symmetric convex body lies on a planar section with an inscribed regular octagon? In this paper, we prove these propositions for cylinders of a special type.     

Outer Billiards outside a Regular Octagon: Periodicity of Almost All Orbits and Existence of an Aperiodic Orbit

The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.     

在某些正则区域上的多元B形式曲面

本文首先通过在多面体区域上抬高维数的技巧给出了多元Ｂ形式中曲面的一般性定义．由此我们构造了平行四边形域上、正六边形域上和正八边形成上Ｂ形式的同次曲面格式，并给出了其基函数的递推公式和求导公式．同时我们也给出了正六边形域上插值角点的Ｂ形式同次曲面的表示式．     

On automorphisms of a generalized octagon of order (2, 4)

Possible orders and fixed-point subgraphs of automorphisms of a generalized octagon of order (2, 4) are found. The vertex-symmetric generalized octagon of order (2, 4) is proved to be classical.     

Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon S, and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group.     

A restriction on the parameters of a suboctagon

If a generalized octagon with parameters (s, t) for t > s has a proper suboctagon with parameters (s₁, t₁), then st ? s₁²t₁.     

Characterizations of the Suzuki tower near polygons

In recent work, we constructed a new near octagon \(\mathcal {G}\) from certain involutions of the finite simple group \(G_2(4)\) and showed a correspondence between the Suzuki tower of finite simple groups, \(L_3(2)< U_3(3)< J_2< G_2(4) < Suz\), and the tower of near polygons, \(\mathrm {H}(2,1) \subset \mathrm {H}(2)^D \subset \mathsf {HJ} \subset \mathcal {G}\). Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall–Janko near octagon \(\mathsf {HJ}\) is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters (2, 4; 0, 3), but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, \(\mathrm {H}(2)^D\). We also give a complete classification of near hexagons of order (2, 2) and use it to prove the uniqueness result for \(\mathrm {H}(2)^D\).     

Zeta functions of line, middle, total graphs of a graph and their coverings

We consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G.     

Random sorting networks

A sorting network is a shortest path from 12?n to n?21 in the Cayley graph of Sn generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n→∞ the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.     

On von Neumann regular rings,III

This note is a natural sequel to [8] and [9]. Further characteristic properties of arbitrary von Neumann regular rings and strongly regular rings are given in terms of annihilators and simple modules. A prime ring with certain annihilator conditions is shown to be primitive (this is related to the following problem ofKaplansky: Are prime regular rings primitive?). Necessary and sufficient conditions for leftq-rings to be regular are also considered: For example, a leftq-ring is regular iff every simple rightA-module is flat. A sufficient condition is given for a leftqc-ring to be a uniserial, strongly left and strongly rightqc, left and rightq-ring. One of the main results ofJain, Mohamed andSingh onq-rings [5, Theorem 2.13] is generalised. Finally, it is shown that a prime left continuous ring either has zero socle or is primitive, left self-injective regular.     

Axiomatizations of Euclidean geometry in terms of points,equilateral triangles or squares, and incidence

Dimension-free Euclidean geometry over Euclidean ordered fields can be axiomatized in a two-sortedfirst-order language, with points and regular n-gons (with n = 3 or 4) as variables, and with a binary predicate standing for the incidence of a point and a regular n-gon as the only primitive notions.     

Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc $\mathcal {D}$ whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in $\mathcal {D}$ , which is invariant under the action of a lattice subgroup ?? of U(1,1), the group of isometries of ${\mathcal{D}}$ . In our case ?? generates a tiling of $\mathcal {D}$ with regular octagons. This problem was introduced as an example of spontaneous pattern formation in a model of image feature detection by the visual cortex where the features are assumed to be represented in the space of structure tensors. Under ??generic?? assumptions the bifurcation problem reduces to an ODE which is invariant by an irreducible representation of the group of automorphisms $\mathcal {G}$ of the compact Riemann surface $\mathcal {D}/\varGamma $ . The irreducible representations of $\mathcal {G}$ have dimensions one, two, three and four. The bifurcation diagrams for the representations of dimensions less than four have already been described and correspond to well-known group actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, thus completing the classification. In one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.     

q-Coverings,codes, and line graphs

In this paper we consider the relationship between q-coverings of a regular graph and perfect 1-codes in line graphs. An infinite class of perfect 1-codes in the line graphs L(I_k) is constructed.     

GQ(4, 2)-extensions, strongly regular case

A strongly regular locallyGQ(4, 2)-graph is a graph with parameters either (126, 45, 12, 8) or (190, 45, 12, 10). The existence and the uniqueness of the corresponding locallyGQ(4, 2)-graph in the first case are well known. We prove that theGQ(4, 2)-hyperoval on ten vertices either is the Petersen graph, or is the Möbius 5-prism, or consists of two (2, 3)-subgraphs connected by three edges. We obtain homogeneousGQ(4, 2)-solutions with a strongly regular point graph; in particular, this implies the negative answer to the question of F. Buekenhout concerning the existence of a locallyGQ(4, 2)-graph with the parameters (190, 45, 12, 10).     

The classification of self-adjoint boundary conditions: Separated, coupled, and mixed

There are three basic types of self-adjoint regular and singular boundary conditions: separated, coupled, and mixed. For even order problems with real coefficients, one regular endpoint and arbitrary deficiency index d, we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square-integrable solutions for some real value of the spectral parameter. In the separated case our construction yields non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available.     

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.     

Some New Upper Bounds for the Size of Partial Ovoids in Slim Generalized Polygons and Generalized Hexagon of Order (s, s3)

From an elementary observation, we derive some upper bounds for the number of mutually opposite points in the classical generalized polygons having 3 points on each line. In particular, it follows that the Ree-Tits generalized octagon O(2) of order (2, 4) has no ovoids. Also, we deduce from another observation a similar upper bound in any generalized hexagon of order (s, s ³).     

A 51-dimensional embedding of the Ree–Tits generalized octagon

We construct an embedding of the Ree–Tits generalized octagon defined over a field K in a 51-dimensional projective space over K arising from a 52-dimensional Lie algebra J of type . This construction derives from a quadratic map (related to a ‘standard’ duality of ) from the 26-dimensional module (see K. Coolsaet, Adv Geometry, to appear) into J. (This embedding is full if and only if K is a perfect field.) We provide explicit formulas for the coordinates of the points of the octagon in this embedding, in terms of their Van Maldeghem coordinates. We apply these results to compute the dimensions of subspaces generated by various special subsets of points of the octagon: the sets of points at a fixed distance from a given point or a given line and the Suzuki suboctagons. The results depend on whether K is the field of 2 elements, or not.      

Linear operators with compact supports, probability measures and Milyutin maps

The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for 0-dimensional spaces in terms of regular extension operators having compact supports. Milyutin maps are also considered and it is established that some topological properties, like paracompactness, metrizability and κ-metrizability, are preserved under Milyutin maps.