The normed and Banach envelopes of WeakL 1

The space WeakL ¹ consists of all Lebesgue measurable functions on [0,1] such thatq(f)=supcλ{t:|f(t)|>c} c>0 is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f:q(f)≤1} of the quasi-normq, and letN be the null space of ρ. The normed envelope of WeakL ¹, which we denote byW, is the space (WeakL ¹/N, ρ). The Banach envelope of WeakL ¹, , is the completion ofW. We show that is isometrically lattice isomorphic to a sublattice ofW. It is also shown that all rearrangement invariant Banach function spaces are isometrically lattice isomorphic to a sublattice ofW.     

GENERAL INFORMATION

We characterize the convex envelope of a given function f as the unique solution of a convex programming problem. It allows us to build a sequence of convex and polygonal function ^un that converges uniformly to the convex envelope of f.     

Envelope of holomorphy of a tube domain over a complex Lie group

According to S. Bochner [6, 7]: IfD =B +iℝ ⁿ is a tube domain in ℂ ⁿ , where B is a domain in ℝ ⁿ , and if [(B)\tilde]\tilde B is the convex envelope of B, then any holomorphic function on D extends to the tube domain [(D)\tilde] = [(B)\tilde] + i\mathbbRⁿ \tilde D = \tilde B + i\mathbb{R}^n , which is a univalent envelope of holomorphy of D. We give a generalization of this result to (nonunivalent) tube domains over a complex Lie group which admit a closed sub-group as a real form. Application: If (V, φ) is a tube domain over ℂ ⁿ and if B is the convex envelope of ϕ(V)∩ℝ ⁿ in ℝ ⁿ , then [(V)\tilde] = B + i\mathbbRⁿ \tilde V = B + i\mathbb{R}^n is an envelope of holomorphy of (V, φ).     

Analysis of Bounds for Multilinear Functions

We analyze four bounding schemes for multilinear functions and theoretically compare their tightness. We prove that one of the four schemes provides the convex envelope and that two schemes provide the concave envelope for the product of p variables over .     

Almost tight upper bounds for lower envelopes in higher dimensions

We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d≥3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(n ^d−1+∈), for any ∈>0; the constant of proportionality depends on ∈, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(n ^d-2λ _q (n)) for some constantq depending on the shape and degree of the surfaces (where λ _q (n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n ^2+∈), and give several applications of the new bounds. Work on this paper has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

On Special Classes of Rectilinear Congruences

Let Σ be an oriented rectilinear congruence in three-dimensional Euclidean space E³ and f a one-to-one mapping between the points of the middle surface and the points of the middle envelope of it. In this paper we investigate some properties of Σ, when f is equiareal or isometric.     

On the Embedding of (k,p)-Arcs is Maximal Arcs

Our main result is that a (k,p)-arc in PG (2,q),q = p ^h , with k qp - q + p - can be extended to a maximal arc. Combining this result with the recent Ball, Blokhuis, Mazzocca theorem about the non-existence of maximal arcs for p > 2, it gives an upper bound for the size of a (k,p)-arc. The method can be regarded as a generalization of B. Segre's method for proving similar embeddability theorems for k-arcs (that is when n= 2). It is based on associating an algebraic envelope containing the short lines to the (k,p)-arc. However, the construction of the envelope is independent of Segre's method using the generalization of Menelaus' theorem.     

Convex Envelopes of Monomials of Odd Degree

Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems (NLP), particularly within the context of spatial Branch-and-Bound methods for global optimization. This paper proposes a nonlinear continuous and differentiable convex envelope for monomial terms of odd degree, x ^2k+1, where k N and the range of x includes zero. We prove that this envelope is the tightest possible. We also derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches.     

Enveloppes polynômiales d'ensembles compacts invariants

We consider the following problem: let V^? be a finite dimensional vector space, and U be a compact group of ?‐linear automorphisms of V^?. The polynomial envelope of a compact set Q ? V^? is defined as where ??(V^?) denotes the space of holomorphic polynomial functions on V^?. The problem is to determine the polynomial envelope of a compact set which is U‐invariant. We solve the problem when U is the isotropy subgroup at the origin of the automorphism group of a bounded symmetric domain of tube type. The case of a domain of type II has been solved by C. Sacré [1992], and, for a domain of type IV, it has been solved by L. Bou Attour [1993]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Lines of Minima and Teichmüller Geodesics

For two measured laminations ν⁺ and ν⁻ that fill up a hyperbolizable surface S and for , let be the unique hyperbolic surface that minimizes the length function e ^t l(ν⁺) + e ^-t l(ν⁻) on Teichmüller space. We characterize the curves that are short in and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface on the Teichmüller geodesic whose horizontal and vertical foliations are respectively, e ^t ν⁺ and e ^t ν⁻. By deriving additional information about the twists of ν⁺ and ν⁻ around the short curves, we estimate the Teichmüller distance between and . We deduce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is bounded independently of t. Received: May 2006, Revision: November 2006, Accepted: February 2007     

Unit-distance graphs,graphs on the integer lattice and a Ramsey type result

Summary Let (R ², 1) denote the graph withR ² as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic number(R ², 1) is still open; however,(R ², 1) is known to be between 4 and 7. By a theorem of de Bruijn and Erdös, it is enough to consider only finite subgraphs of (R ², 1). By a recent theorem of Chilakamarri, it is enough to consider certain graphs on the integer lattice. More precisely, forr > 0, let (Z ²,r, ) denote a graph with vertex setZ ² and two vertices adjacent if and only if their Euclidean distance is in the closed interval [r – ,r + ]. A simple graph is faithfully -recurring inZ ² if there exists a real numberd > 0 such that, for arbitrarily larger, G is isomorphic to a subgraph of (Z ²,r, ) in which every pair of vertices are at least distancedr apart. Chilakamarri has shown that, ifG is a finite simple graph, thenG is isomorphic to a subgraph of (R ², 1) if and only ifG is faithfully -recurring inZ ². In this paper we prove that(Z ²,r, ) 5 for integersr 1. We also prove a Ramsey type result which states that for any integerr > 1, and any coloring ofZ ² either there exists a monochromatic pair of vertices with their distance in the closed interval [r – ,r + ] or there exists a set of three vertices closest to each other with three distinct colors.     

On the security of stepwise triangular systems

In 2003 and 2004, Kasahara and Sakai suggested the two schemes RSE(2)PKC and RSSE(2)PKC, respectively. Both are examples of public key schemes based on ultivariate uadratic equations. In this article, we first introduce Step-wise Triangular Schemes (STS) as a new class of ultivariate uadratic public key schemes. These schemes have m equations, n variables, L steps or layers, r the number of equations and new variables per step and q the size of the underlying finite field . Then, we derive two very efficient cryptanalytic attacks. The first attack is an inversion attack which computes the message/signature for given ciphertext/message in O(mn ³ Lq ^r + n ² Lrq ^r ), the second is a structural attack which recovers an equivalent version of the secret key in O(mn ³ Lq ^r + mn ⁴) operations. As the legitimate user also has a workload growing with q ^r to recover a message/compute a signature, q ^r has to be small for efficient schemes and the attacks presented in this article are therefore efficient. After developing our theory, we demonstrate that both RSE(2)PKC and RSSE(2)PKC are special instances of STS and hence, fall to the attacks developed in our article. In particular, we give the solution for the crypto challenge proposed by Kasahara and Sakai. Finally, we demonstrate that STS cannot be the basis for a secure ultivariate uadratic public key scheme by discussing all possible variations and pointing out their vulnerabilities.     

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso ⁰(M ⁿ ) of the isometry group Iso(M ⁿ ) of M ⁿ admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M ⁿ ) acting polarly on M ⁿ , we prove that Φ(G) acts polarly on , and we obtain that f(M ⁿ ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.      

The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L ^q -theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L ^q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L ^p -regularity of in the respective weighted L ^q -spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L ^p -regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L _p -regularity. Preprint (1999)] with the fact that for L ^q -spaces -boundedness is implied by weighted estimates.     

L 2-lower semicontinuity of functionals of quadratic type

Summary A representation formula for the L ²-lower semicontinuous envelope of a quadratic integral of Calculus of Variations is given. Some particular cases are explicited in the details.This paper has been supported by G.N.A.F.A. (Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni).     

Sobolev embedding in ℂn and the\overline \partial - equation

We prove a Sobolev embedding theorem for functions that are in a Sobolev space while their is in L^t, fort large enough. This allows us to deduceL ^p or Lipschitz estimates with loss from classical Sobolev estimates for the solution of in weakly pseudo-convex domains.