Given a family of sets/vectors of the same cardinality/dimension you get the shadow by deleting one element/coordinate from a set/vector in all possible ways. You find the family with the smallest shadow by ordering all sets/vectors. The set case was solved by Kruskal (1963), and Katona (1966), and has many applications. We study two orderings which solve the 0, 1 vector case, and give the shadow size.

Let t(G, q) denote the smallest integer t such that the vertex set V of the graph G can be partitioned into t classes V(G)=[Cup ]ti=1Vi such that the number of edges in the induced subgraph 〈Vi〉 for 1[les ]i[les ]t, is divisible by q. Using an algebraic theorem due to Baker and Schmidt we prove that if q is a prime power then t(G, q) can be computed and a corresponding partition can be presented in a polynomial time.

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.

Let d be a fixed positive integer, and ε>0. We show that there is a natural number M such that if G is any graph with m[ges ]M edges and maximum degree at most d, then the harmonious chromatic number h(G) satisfies

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We introduce two related families of polynomials, easily computable by simple recursions into which any Kazhdan–Lusztig (and inverse Kazhdan–Lusztig) polynomial of any Coxeter group can be expanded linearly, and we give combinatorial interpretations to the coefficients in these expansions. This yields a combinatorial rule for computing the Kazhdan–Lusztig polynomials in terms of paths in a directed graph, and a completely combinatorial reformulation of the nonnegativity conjecture [15, p. 166].

It is proved that quadratic systems of autonomous differential equations in the plane admit forty-five topologically distinct phase configurations at infinity. The related conditions on the coefficients are given in affine invariant form.

Let f be analytic and univalent in Δ={z:[mid ]z[mid ]<1}. We write

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for 0[les ]r<1 and θ∈[0, 2π], and remark that L(r, θ) is the length of the image in f(Δ) of the ray joining 0 to reiθ in Δ. We shall be concerned with the growth of L(r, θ), as a function of r, for univalent functions in Δ and, more particularly, for functions in the subclass of close-to-convex univalent functions in Δ.

For a large class of Banach algebras we solve the Cohomology Comparison Problem for the second cohomology groups, in that we prove that for all amenable strongly regular Banach function algebras A and all Banach A-bimodules E the second comparison map between continuous and algebraic cohomology is an embedding. Accepting the continuum hypothesis we show that for algebras of this kind there exist Banach A-bimodules E such that i2 is not onto.

JB*-triples occur in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on C*-algebras. These spaces are equipped with a ternary product {·,·,·}, the Jordan triple product, and are essentially geometric objects in that the linear isometries between them are exactly the linear bijections preserving the Jordan triple product (cf. [23]).

Certain projections of the real polytopes {3, 3, 4}, {3, 4, 3}, {3, 3, 5} suggest highly symmetric coordinates for the self-reciprocal complex polygons 3{3}3, 4{3}4, 3{4}3, 5{3}5 and 3{5}3. Although there are a number of interesting complications, this suggestion is essentially correct and leads to elegant coordinates for all the sporadic complex polygons. Among the by-products of producing these coordinates we count most significant our new insights about 2{6}3 and our simple proof that the 600 vertices of the real polytope {5, 3, 3} are quite unrelated to the 600 vertices of either 5{6}2 or 5{4}3.

For reciprocation with respect to a sphere [sum ]x2=c in Euclidean n-space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere [sum ]uu=c. This is now applied, for the first time, to complex polytopes.

When a regular polytope Π has a palindromic Schläfli symbol, it is self-reciprocal in the sense that its reciprocal Π′, with respect to a suitable concentric sphere or antisphere, is congruent to Π. The present article reveals that Π and Π′ usually have together the same vertices as a third polytope Π+ and the same facet-hyperplanes as a fourth polytope Π− (where Π+ and Π− are again regular), so as to form a ‘compound’, Π+[2Π]Π−. When the geometry is real, Π+ is the convex hull of Π and Π′, while Π− is their common content or ‘core’. For instance, when Π is a regular p-gon {p}, the compound is

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The exceptions are of two kinds. In one, Π+ and Π− are not regular. The actual cases are when Π is an n-simplex {3, 3, ..., 3} with n[ges ]4 or the real 4-dimensional 24-cell {3, 4, 3}=2{3}2{4}2{3}2 or the complex 4-dimensional Witting polytope 3{3}3{3}3{3}3. The other kind of exception arises when the vertices of Π are the poles of its own facet-hyperplanes, so that Π, Π′, Π+ and Π− all coincide. Then Π is said to be strongly self-reciprocal.

The Bers–Greenberg theorem tells us that the Teichmüller space of a Riemann surface with branch points (orbifold) depends only on the genus and the number of special points, and not on the particular ramification values. On the other hand, the Maskit embedding provides a mapping from the Teichmüller space of an orbifold, into the product of one-dimensional Teichmüller spaces. In this paper we prove that there is a set of isomorphisms between one-dimensional Teichmüller spaces that, when restricted to the image of the Teichmüller space of an orbifold under the Maskit embedding, provides the Bers–Greenberg isomorphism.

The first examples of triangular AF algebras to be studied were the refinement algebra and the standard algebra. Both are analytic algebras with the property that the cocycle can be taken to be constant on the matrix units of the algebra. The latter property, called local constancy, is quite special and is still ill-understood. In the present paper we examine the class of nest algebras [Tscr ] in AF C*-algebras which share the distinctive features of the refinement algebra:

(1) [Tscr ] is a nest algebra in which the nest generates the diagonal,

(2) [Tscr ] admits a locally constant cocycle.

To each finitely generated nilpotent group Γ, we associate an integer d and show that the 2-dimensional isoperimetric function (the Dehn function) of Γ is bounded below by a polynomial function of degree d. In the case of free nilpotent groups of class c we find (if we exclude ℤ) that d=c+1.

Let X and Y be pointed spaces. A phantom map from X to Y is a map whose restriction to any finite skeleton of X is null-homotopic. Let Ph (X, Y) denote the set of homotopy classes of phantom maps from X to Y. As a pointed set it is isomorphic to the lim1 term of the tower of groups

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where Y(n) denotes the Postnikov approximation of Y through dimension n. The homomorphisms in this tower are induced by the projections ΩY(n)← ΩY(n+1)). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.

In this paper, we study some asymptotic aspects of the positive eigenfunctions of the combinatorial Laplacian associated to a homogeneous tree. The results are inspired by results of Dennis Sullivan concerning λ-harmonic functions on the hyperbolic spaces ℍn and contained in the paper [9].