Covering shadows with a smaller volume

For n?2 a construction is given for convex bodies K and L in Rn such that the orthogonal projection Lu onto the subspace u^⊥ contains a translate of Ku for every direction u, while the volumes of K and L satisfy Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection Lξ onto a k-dimensional subspace ξ contains a translate of Kξ, while the mth intrinsic volumes of K and L satisfy Vm(K)>Vm(L) for all m>k.For each k=1,…,n, we then define the collection Cn,k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms).It is subsequently shown that, if L∈Cn,k, and if the orthogonal projection Lξ contains a translate of Kξ for every k-dimensional subspace ξ of Rn, then Vn(K)?Vn(L).The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chain

Cn,1⊂Cn,2⊂?⊂Cn,n−1⊂Cn,n,     

Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fⁿ there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.     

On a class of degenerate extremal graph problems

Given a class ? of (so called “forbidden”) graphs, ex (n, ?) denotes the maximum number of edges a graphG ⁿ of ordern can have without containing subgraphs from ?. If ? contains bipartite graphs, then ex (n, ?)=O(n ^2?c ) for somec>0, and the above problem is calleddegenerate. One important degenerate extremal problem is the case whenC _2k , a cycle of 2k vertices, is forbidden. According to a theorem of P. Erd?s, generalized by A. J. Bondy and M. Simonovits [3₂, ex (n, {C _2k })=O(n ^1+1/k ). In this paper we shall generalize this result and investigate some related questions.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

A new family of expansive graphs

An affine graph is a pair (G,σ) where G is a graph and σ is an automorphism assigning to each vertex of G one of its neighbors. On one hand, we obtain a structural decomposition of any affine graph (G,σ) in terms of the orbits of σ. On the other hand, we establish a relation between certain colorings of a graph G and the intersection graph of its cliques K(G). By using the results we construct new examples of expansive graphs. The expansive graphs were introduced by Neumann-Lara in 1981 as a stronger notion of the K-divergent graphs. A graph G is K-divergent if the sequence |V(Kⁿ(G))| tends to infinity with n, where Kⁿ⁺¹(G) is defined by Kⁿ⁺¹(G)=K(Kⁿ(G)) for n?1. In particular, our constructions show that for any k?2, the complement of the Cartesian product C^k, where C is the cycle of length 2k+1, is K-divergent.     

Fractal boundaries are not typical

Let M be a C¹n-dimensional compact submanifold of Rn. The boundary of M, ∂M, is itself a C¹ compact (n−1)-dimensional submanifold of Rn. A carefully chosen set of deformations of ∂M defines a complete subspace consisting of boundaries of compact n-dimensional submanifolds of Rn, thus the Baire Category Theorem applies to the subspace. For the typical boundary element ∂W in this space, it is the case that ∂W is simultaneously nowhere-differentiable and of Hausdorff dimension n−1.     

On the zeros of a minimal realization

In an earlier work, the authors have introduced a coordinate-free, module-theoretic definition of zeros for the transfer function G(s) of a linear multivariable system (A,B,C). The first contribution of this paper is the construction of an explicit k[z]-module isomorphism from that zero module, Z(G), to V¹/R¹, where V¹ is the supremal (A,B)-invariant subspace contained in kerC and R¹ is the supremal (A,B)-controllable subspace contained in kerC, and where (A,B,C) constitutes a minimal realization of G(s). The isomorphism is developed from an exact commutative diagram of k-vector spaces. The second contribution is the introduction of a zero-signal generator and the establishment of a relation between this generator and the classic notion of blocked signal transmissions.     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

Plane Sections of Convex Bodies of Maximal Volume

Let K = {K ₀ ,... ,K _k } be a family of convex bodies in R ⁿ , 1≤ k≤ n-1 . We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k -dimensional plane A _k (subset, dbl equals) R ⁿ , called a common maximal k-transversal of K , such that, for each i∈ {0,... ,k} and each x∈ R ⁿ , where V _k is the k -dimensional Lebesgue measure in A _k and A _k +x . Given a family K = {K _i } _i=0 ^l of convex bodies in R ⁿ , l < k , the set C _k ( K ) of all common maximal k -transversals of K is not only nonempty but has to be ``large' both from the measure theoretic and the topological point of view. It is shown that C <sub>k</sub> ( K ) cannot be included in a ν -dimensional C <sup>1</sup> submanifold (or more generally in an ( H <sup>ν</sup> , ν) -rectifiable, H <sup>ν</sup> -measurable subset) of the affine Grassmannian AGr <sub>n,k</sub> of all affine k -dimensional planes of R <sup>n</sup> , of O(n+1) -invariant ν -dimensional (Hausdorff) measure less than some positive constant c <sub>n,k,l</sub> , where ν = (k-l)(n-k) . As usual, the ``affine' Grassmannian AGr _n,k is viewed as a subspace of the Grassmannian Gr _n+1,k+1 of all linear (k+1) -dimensional subspaces of R ⁿ⁺¹ . On the topological side we show that there exists a nonzero cohomology class θ∈ H ^n-k (G _n+1,k+1 ;Z ₂ ) such that the class θ ^l+1 is concentrated in an arbitrarily small neighborhood of C _k ( K ) . As an immediate consequence we deduce that the Lyusternik—Shnirel'man category of the space C _k ( K ) relative to Gr _n+1,k+1 is ≥ k-l . Finally, we show that there exists a link between these two results by showing that a cohomologically ``big' subspace of Gr _n+1,k+1 has to be large also in a measure theoretic sense. Received May 22, 1998, and in revised form March 27, 2000. Online publication September 22, 2000.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Harmonic analysis of fractal measures

We consider affine systems inR ⁿ constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ _bx:=R^–1x+b; the corresponding measure μ onR ⁿ is a probability measure fixed by the self-similarity $\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} } $ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR ⁿ such that the exponentials {e^iλ·x}Λ form anorthogonal basis forL ²(μ); and (ii) the existence of a certaindual pair of representations of theC ^*-algebraO _N wheren is the cardinality of the setB. (For eachN, theC ^*-algebraO _N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL ²(μ) fail to span a dense subspace. Instead we show that theC ^*-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse ^iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC ^*-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.     

Centerpole sets for colorings of abelian groups

A subset C?G of a group G is called k-centerpole if for each k-coloring of G there is an infinite monochromatic subset G, which is symmetric with respect to a point c??C in the sense that S=cS ^?1 c. By c _k (G) we denote the smallest cardinality c _k (G) of a k-centerpole subset in G. We prove that c _k (G)=c _k (? ^m ) if G is an abelian group of free rank m??k. Also we prove that c ₁(? ⁿ⁺¹)=1, c ₂(? ⁿ⁺²)=3, c ₃(? ⁿ⁺³)=6, 8??c ₄(? ⁿ⁺⁴)??c ₄(?⁴)=12 for all n????, and ${\frac{1}{2}(k^{2}+3k-4)\le c_{k}(\mathbb{Z}^{n})\le2^{k}-1-\max_{s\le k-2}\binom {k-1}{s-1}}$ for all n??k??4.     

On the number of divisors of binomial coefficients

This paper deals with the problems of the upper and lower orders of growth of the ratios of the divisor functions of “adjacent” binomial coefficients, i.e., of the numbers of combinations of the form C _n ^k and C _n ^k+1 or C _n ^k and C _n+1 ^k . The suprema and infima of the corresponding ratios are obtained.     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

Two comments on Dvoretzky’s sphericity theorem

For any two positive integersk, l and anyɛ>0 there exists anN(k, l, ɛ) so that given anyl convex bodiesC ₁, …,C _l symmetric about the origin inE ⁿ withn≧N there exists a subspaceE ^k so that eachC _i intersectsE ^k, or has a projection intoE ^k, in a set which is nearly spherical (asphericity <ɛ). The measure of the totality ofE ^k which intersect a given body inE ⁿ in a nearly ellipsoidal set is considered and an affine invariant measure is introduced for that purpose.     

Pairs of B-Splines with Small Support on the Four-Directional Mesh Generating a Partition of Unity

Let τ be the four-directional mesh of the plane and let Σ₁ (respectively Λ₁) be the unit square (respectively the lozenge) formed by four (respectively eight) triangles of τ. We study spaces of piecewise polynomial functions in C ^k (R ²) with supports Σ₁ or Λ₁ having sufficiently high degree n, which are invariant with respect to the group of symmetries of Σ₁ or Λ₁ and whose integer translates form a partition of unity. Such splines are called complete Σ₁ and Λ₁-splines. We first give a general study of spaces of linearly independent complete Σ₁ and Λ₁-splines of class C ^k and degree n. Then, for any fixed k≥0, we prove the existence of complete Σ₁ and Λ₁-splines of class C ^k and minimal degree, but they are not unique. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of these splines.     

Deletion-restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A₀. Let A^′=A?{A₀} be the deletion of A₀ from A and A^″={A∩A₀|A∩A₀≠∅} be the restriction of A to A₀. Let M=V?_?A∈AA be the complement of A in V. If A is an arrangement of complex affine hyperplanes, then there is a split short exact sequence, 0→Hk(M^′)→Hk(M)→Hk+1−codimR(A₀)(M^″)→0. In this paper, we determine conditions for when the triple (A,A^′,A^″) of arrangements of affine subspaces yields the above split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion-restriction subspace arrangements.     

Distance to Ck hypersurfaces

It is shown by elementary means that a C^k hypersurface M of positive reach in $\text{R}$^{n + 1} has the property that the signed distance function to it is C^k, k ? 1. This extends and complements work of Federer, Gilbarg and Trudinger, and Serrin.     

On the motivic class of the stack of bundles

Let G be a split connected semisimple group over a field. We give a conjectural formula for the motivic class of the stack of G-bundles over a curve C, in terms of special values of the motivic zeta function of C. The formula is true if C=P¹ or G=SLn. If k=C, upon applying the Poincaré or called the Serre characteristic by some authors the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If k=Fq, upon applying the counting measure, it reduces to the fact that the Tamagawa number of G over the function field of C is |π₁(G)|.