The proof of,and generalisations to,a conjecture by Baker and Essam

The weak and strong k-weights of a combinatorial geometry are defined and relationships between these constants and other properties of geometries are obtained. In particular, it is shown that a geometry has cohesion n if and only if the weak k-weights satisfy

$\text{σ}{}_{\mathrm{0?A?S}}\text{k(A) |A|}{}^{\mathrm{l}}\text{=0}\text{for}\text{1≤l<n:}\text{≠0}\text{for}\text{l=n}$

. Similar results are obtained for the connectivity of a geometry, and a separate discussion is given of the cohesion and connectivity of linear graphs.     

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.     

Intersections of t-reguli,rational curves,and orthogonal Latin squares

The celebrated net-embedding theorem of R.H. Bruck asserts that a net with a large number of parallel classes, i.e., a net with small deficiency, can be embedded in an affine plane. The only known class of examples of unembeddable nets of small deficiency has been constructed by geometry, using partial spreads of PG(3, q). In this note we attempt to generalize that discussion to PG(n, q), with particular emphasis on the case n=5.     

Problems in algebra inspired by universal algebraic geometry

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant K _Θ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H ₁ and H ₂ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) K _Θ(H ₁) and K _Θ(H ₂) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ⁰ be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ⁰) for different varieties Θ and also the groups of autoequivalences of Θ⁰. The problem is to describe these groups for different Θ.     

Affine geometry designs, polarities, and Hamada's conjecture

In a recent paper, two of the authors used polarities in PG(2d−1,p) (p?2 prime, d?2) to construct non-geometric designs having the same parameters and the same p-rank as the geometric design PGd(2d,p) having as blocks the d-subspaces in the projective space PG(2d,p), hence providing the first known infinite family of examples where projective geometry designs are not characterized by their p-rank, as it is the case in all known proven cases of Hamada's conjecture. In this paper, the construction based on polarities is extended to produce designs having the same parameters, intersection numbers, and 2-rank as the geometric design AGd+1(2d+1,2) of the (d+1)-subspaces in the binary affine geometry AG(2d+1,2). These designs generalize one of the four non-geometric self-orthogonal 3-(32,8,7) designs of 2-rank 16 (V.D. Tonchev, 1986 [12]), and provide the only known infinite family of examples where affine geometry designs are not characterized by their rank.     

Geometries on σ-Hermitian matrices

Ring geometry and the geometry of matrices naturally meet at the ring R := K ^n×n of (n×n)-matrices with entries in a field K (not necessarily commutative). Our aim is to strengthen the interaction between these disciplines. Below we sketch some results from either side, even though not in their most general form, but in a way that is tailored to our needs.     

Geometries simpliciales unimodulaires

Theorem: Let n, k be integers such that 2 ? k ? n ? 2. A complete k-simplicial geometry of order n is unimodular if and only if k = 2 or k = n ? 2.     

Geometry of compact complex manifolds with maximal torus action

We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S ¹) ^k . We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ? T _? = (?*) ^k . On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kähler form ω _F . As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.     

Dielectric breakdown in solids modeled by DBM and DLA

Using numerical simulation, two stochastic models of electrical treeing in solid dielectrics are compared. These are the diffusion-limited aggregation (DLA) model and the dielectric breakdown model (DBM or η-model). On a linear two-dimensional geometry, the relationship between both models, when the size of the structures is of the order of the experimental samples (the electrode gap is 100 times the length of the discharge channel), is explored by statistical methods. Although there is a one-to-one correspondence between DBM with η=1 and the DLA model when the structure size is very large, the case of rather smaller structures is not well known. From a fractal analysis, employing the method of the correlation function C(r), it follows that average fractal dimension of electrical trees, generated with the DLA or with the DBM (η=1), collapse (up to the numerical uncertainty), on a single curve that “universally” accounts for finite size effects. Even more, from this analysis we conclude that the two curves obtained for DLA and DBM (η=1) cannot be distinguished if one takes into account the error bars. This means that finite size effects in the fractal analysis of DLA and DBM (η=1) are quite the same (despite the differences in the algorithms respectively used to generate the electrical trees). To our knowledge no comparison has ever been made between the similarities and differences of the DBM and DLA approach on a geometry other than the open-planar geometry.     

On the complement of the set of n-collinear points in PG(3, n)

Let ${S = (\mathcal{P}, \mathcal{L}, \mathcal{H})}$ be the finite planar space obtained from the 3-dimensional projective space PG(3, n) of order n by deleting a set of n-collinear points. Then, for every point ${p\in S}$ , the quotient geometry S/p is either a projective plane or a punctured projective plane, and every line of S has size n or n + 1. In this paper, we prove that a finite planar space with lines of size n + 1 ? s and n + 1, (s ≥ 1), and such that for every point ${p\in S}$ , the quotient geometry S/p is either a projective plane of order n or a punctured projective plane of order n, is obtained from PG(3, n) by deleting either a point, or a line or a set of n-collinear points.     

Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

To characterize Hermitian varieties in projective space PG(d, q) of d dimensions over the Galois field GF(q), it is necessary to find those subsets K for which there exists a fixed integer n satisfying (i) 3 ? n ? q ? 1, (ii) every line meets K in 1, n or q + 1 points. K is called singular or non-singular as there does or does not exist a point P for which every line through P meets K in 1 or q + 1 points. For q odd, a non-singular K is a non-singular Hermitian variety (M. Tallini Scafati “Caratterizzazione grafica delle forme hermitiane di un S_{r, q}” Rend. Mat. Appl.26 (1967), 273–303). For q even, q > 4 and d = 3, a non-singular K is a Hermitian surface or “looks like” the projection of a non-singular quadric in PG(4, q) (J.W.P. Hirschfeld and J.A. Thas “Sets of type (1, n, q + 1) in PG(d, q)” to appear). The case q = 4 is quite exceptional, since the complements of these sets K form a projective geometry code, a (21, 11) code for d = 2 and an (85, 24) code for d = 3. The full list of these sets is given.     

Spectral radius, index estimates for Schrödinger operators and geometric applications

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation ^′(vz^′)+Avz=0, where A, v are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schrödinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.     

Continuous dependence on initial geometry for a class of abstract equations in Hilbert space

Conditions are given under which members of a class of uniformly bounded solutions to the Cauchy problem associated with equations of the form Mu_tt ? Nu = F, in Hilbert space, depend continuously on perturbations of the initial geometry; our results generalize a similar theorem of Knops and Payne for classical solutions to initial-boundary value problems in linear elastodynamics.     

Complex hyperbolic geometry and Hilbert spaces with complete Pick kernels

Suppose H is a finite dimensional reproducing kernel Hilbert space of functions on X. If H has the complete Pick property then there is an isometric map, Φ, from X, with the metric induced by H, into complex hyperbolic space, ${\mathbb{CH}}^n$, with its pseudohyperbolic metric. We investigate the relationships between the geometry of $\mathrm{\Phi }(X)$ and the function theory of H and its multiplier algebra.     

Minkowski's successive minima and the zeros of a convexity-function

For a convex bodyK?E ^d letV _i (K),i=0, 1,...,d denote its normalized quermassintegrals. We show that there are various inequalities between the zeros of the polynomial W(?μ K) = ∑ _i=0 ^d V_i(K)(?μ)ⁱ and Minkowski's successive minima in geometry of numbers.     

Quadratic spaces over finite fields and codes

Let V be a finite-dimensional quadratic space over a finite field GF(?) of characteristic different from 2. It is shown that, even if V is singular, the geometry of V is completely determined by the number of points on the unit sphere, the “sphere of the nonsquares,” and the “0-sphere.” For ? = 3, this implies that two codes over GF(3) with the same weight enumerator are isometric.     

Solutions of the Klein-Gordon equation on manifolds with variable geometry including dimensional reduction

We develop the recent proposal to use dimensional reduction from the four-dimensional space-time (D = 1 + 3) to the variant with a smaller number of space dimensions D = 1 + d, d < 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation with a few toy examples (“educational toys”) of a space-time with a variable spatial geometry including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique for transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: (1) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). (2) The specific spectrum of scalar excitations resembles the spectrum of real particles; it reflects the geometry of the transition region and represents its “fingerprints.” (3) The parity violation due to the asymmetric character of the construction of our models could be related to the CP symmetry violation.