Separating convex sets in the plane

Given a setA inR ² and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane.We prove that, for any collectionF ofn disjoint disks inR ², there is a lineL that separates a disk inF from a subcollection ofF with at least (n–7)/4 disks. We produce configurationsH _n andG _n , withn and 2n disks, respectively, such that no pair of disks inH _n can be simultaneously separated from any set with more than one disk ofH _n , and no disk inG _n can be separated from any subset ofG _n with more thann disks.We also present a setJ _m with 3m line segments inR ², such that no segment inJ _m can be separated from a subset ofJ _m with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least n/3+1 elements ofF.     

Triangles inscribed in simple closed curves

We show that given any triangle T and any simple closed curve J ² there are infinitely many triangles similar to T whose vertices are contained in J. In fact, the set of such vertices is dense in J.     

On Modified Strong Dyadic Integral and Derivative

For functions f L(R ₊), we define a modified strong dyadic integral J(f) L(R ₊) and a modified strong dyadic derivative D(f) L(R ₊). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition f(x)dx = 0, we prove the equalities J(D(f)) = f and D(J(f)) = f. We find a countable set of eigenfunctions of the operators J and D. We prove that the linear span L of this set is dense in the dyadic Hardy space H(R ₊). For the functions f H(R ₊), we define a modified uniform dyadic integral J(f) L (R ₊).     

Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

Let be the tiling of R ³ with unit cubes whose vertices belong to the fundamental lattice L ₁ of points with integer coordinates. Denote by L _n the lattice consisting of all points x in R ³ such that nx belongs to L ₁. When the vertices of a polyhedron P in R ³ are restricted to lie in L ₁ then there is a formula which relates the volume of P to the numbers of all points of two lattices L ₁ and L _n lying in the interior and on the boundary of P. In the simplest case of the lattices L ₁ and L ₂ there are 27 points in each cube from whose relationships to the polyhedron P must be examined. In this note we present a new formula for the volume of lattice polyhedra in R ³ which involves only nine points in each cube of : one from L ₂ and eight belonging to L ₄. Another virtue of our formula is that it does not employ any additional parameters, such as the Euler characteristic.     

A Homological Lower Bound for Order Dimension of Lattices

We prove that if a finite lattice L has order dimension at most d, then the homology of the order complex of its proper part L vanishes in dimensions d – 1 and higher. If L can be embedded as a join-sublattice in N ^d , then L actually has the homotopy type of a simplicial complex with d vertices.     

Classification of finite commutative rings with planar,toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J _R , is a graph with vertex-set R J(R), such that two distinct vertices a and b in R J(R) are adjacent if and only if 1 ? ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(J _R ). In this paper, we characterize all finite commutative rings R such that the graphs L(J _R ) are planar, toroidal or projective.     

Bessel Potentials with Logarithmic Components and Sobolev-Type Embeddings

In a recent paper Edmunds, Gurka, and Opic [5] showed that Sobolev spaces of order k, based on the Zygmund spaces L ^n/k (log L) (R ⁿ ), are continuously embedded into L (R ⁿ ) if > 1/p, p n/k. In this paper we replace L ^n/k (log L) (R ⁿ ) by the Lebesgue space L ^n/k (R ⁿ ) and increase the smoothness of the functions involved by a "logarithmic" order > 1/p to obtain the continuous embedding into L (R ⁿ ). Both approaches turn out to be equivalent. We also derive results of Trudinger-type [16] on embeddings into Orlicz spaces in the limit case k = n/p as well as results of Brézis-Wainger-type [2] on almost Lipschitz continuity in the superlimiting case k = n/p + 1.     

A Local-Global Principle for Macaulay Posets

We consider the shadow minimization problem (SMP) for Cartesian powers P ⁿ of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L ⁿ . Namely, we show that under certain conditions the shadow of any initial segment of the order L ⁿ for n 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP.     

A generalization of addition formulae

Let R denote the set of reals, J a real interval and X a real linear space. We determine the functions g : X J, M : J R and H : J ² R satisfying the equationg(x+M(g(x))y)=H(g(x),g(y)),under the assumptions that g is continuous on rays, M is continuous, and H is symmetric. As a consequence we obtain characterizations of some groups and semigroups.     

Algorithms for graphs of bounded treewidth via orthogonal range searching

We show that, for any fixed constant k3, the sum of the distances between all pairs of vertices of an abstract graph with n vertices and treewidth at most k can be computed in O(nlog^k−1n) time.We also show that, for any fixed constant k2, the dilation of a geometric graph (i.e., a graph drawn in the plane with straight-line segments) with n vertices and treewidth at most k can be computed in O(nlog^k+1n) expected time. The dilation (or stretch-factor) of a geometric graph is defined as the largest ratio, taken over all pairs of vertices, between the distance measured along the graph and the Euclidean distance.The algorithms for both problems are based on the same principle: data structures for orthogonal range searching in bounded dimension provide a compact representation of distances in abstract graphs of bounded treewidth.     

Eine Übertragung des Satzes von Bonnet auf Regelflächen im einfach isotropen Raum

J ₃ ⁽¹⁾ . Bonnet's classical theorem about ruled surfaces in the three-dimensional Euclidean space does not hold inJ ₃ ⁽¹⁾ . To get an isotropic version of this theorem the terms geodetic line and isogonal-trajectory of the generators are replaced by new, the isotropic space adapted properties of curves on ruled surfaces.     

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N?Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and edges, with N=⌈B^′n⌉ for some constant B^′ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is . Our approach is based on random graphs; in fact, we show that the classical Erd?s–Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.     

Construction of Equilibrium Magnetic Vector Potentials

In [9] we introduced the notion of equilibrium surface current JdS_x on the closed C smooth surfaces in R ³ as a generalization of electric solenoid, and proved their existence. We here show an algorithm for JdS_x starting from a given surface current J dS_x on .     

A characterization of the interpolation spaces ofH 1 andL ∞ on the line

The Calderón-Mitjagin theorem characterizes all interpolation spaces of the pair of Lebesgue spaces (L ¹,L ) as the rearrangement-invariant spaces. The results of this paper show that the interpolation spaces ofH ¹(R) andL (R) consist of elements whose nontangential maximal functions lie in rearrangement-invariant spaces.Communicated by Jaak Peetre.     

Anti-Wick Symbols for Infinite Products in K-Homology

We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L ²(R), and compare our infinite tensor power construction with an extension of pseudodifferential operators on R . We show that the K-theory connecting maps coincide. We propose a natural definition of ellipticity for anti-Wick operators on R, compute the corresponding index, and draw some consequences concerning these operators.     

Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem

We consider a boundary value problem where f(x) ∈ L_p(R), p ∈ [1,∞] (L_∞(R) ≔ C(R) and 0 ≤ q(x) ∈ L^loc₁( R). Boundary value problem (0.1) is called correctly solvable in the given space L_p(R) if for any f(x) ∈ L_p(R) there is a unique solution y(x) ∞ L_p(R) and the following inequality holds with absolute constant c(p) ∈ (0,∞). We find criteria for correct solvability of the problem (0.1) in L_p(R).     

Variations on cops and robbers

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R≥1 edges at a time, establishing a general upper bound of , where α = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng, and Scott and Sudakov. We also show that in this case, the cop number of an n‐vertex graph can be as large as n^{1 ? 1/(R ? 2)} for finite R≥5, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is O(n(loglogn)²/logn). Our approach is based on expansion. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2

Let Ω be a bounded smooth domain inR ². Letf:R→R be a smooth non-linearity behaving like exp{s ²} ass→∞. LetF denote the primitive off. Consider the functionalJ:H ₀ ¹ (Ω)→R given by