On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

Empty pseudo-triangles in point sets

We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between Θ(n²) and Θ(n³) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between Θ(n³) and Θ(n⁶). If we count only star-shaped pseudo-triangles, the bounds are Θ(n²) and Θ(n⁵). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n³) time. In the general case, the running times are O(n⁶) for the maximization problems and O(nlogn) for the minimization problems.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u^? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

Normally generated line bundles on general curves, II

Let C be a general curve of genus g≥3. Here we prove that there is a normally generated L∈Pic^d(C) such that h⁰(C,L)=r+1≥4 (i.e. a very ample line bundle which embeds C in P^r as a projectively normal curve) if and only if (r+1)h¹≤g≤r(r−1)/2+2h¹, where h¹?g+r−d=h¹(C,L).     

A singular boundary value problem for odd-order differential equations

The odd-order differential equation (−1)ⁿx⁽²ⁿ⁺¹⁾=f(t,x,…,x⁽²ⁿ⁾) together with the Lidstone boundary conditions x^(2j)(0)=x^(2j)(T)=0, 0?j?n−1, and the next condition x⁽²ⁿ⁾(0)=0 is discussed. Here f satisfying the local Carathéodory conditions can have singularities at the value zero of all its phase variables. Existence result for the above problem is proved by the general existence principle for singular boundary value problems.     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

Noncrossing Hamiltonian paths in geometric graphs

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: determine the largest number h(n) such that when we remove any set of h(n) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that . We also establish several results related to special classes of geometric graphs. Let h₁(n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h₁(n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that . Let h₂(n) denote the largest number such that when we remove an arbitrary star with at most h₂(n) edges from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We show that h₂(n)=⌈n/2⌉-1. Further we prove that when we remove any matching from a complete geometric graph the resulting graph will have a noncrossing Hamiltonian path.     

A fast algorithm for multiplying min-sum permutations

The present article considers the problem for determining, for given two permutations over indices from 1 to n, the permutation whose distribution matrix is identical to the min-sum product of the distribution matrices of the given permutations. This problem has several applications in computing the similarity between strings. The fastest known algorithm to date for solving this problem executes in O(n^1.5) time, or very recently, in O(nlogn) time. The present article independently proposes another O(nlogn)-time algorithm for the same problem, which can also be used to partially solve the problem efficiently with respect to time in the sense that, for given indices g and i with 1≤g<i≤n+1, the proposed algorithm outputs the values R(h) for all indices h with g≤h<i in O(n+(i−g)log(i−g)) time, where R is the solution of the problem.     

Ordering connected graphs having small degree distances

The concept of degree distance of a connected graph G is a variation of the well-known Wiener index, in which the degrees of vertices are also involved. It is defined by D^′(G)=∑_x∈V(G)d(x)∑_y∈V(G)d(x,y), where d(x) and d(x,y) are the degree of x and the distance between x and y, respectively. In this paper it is proved that connected graphs of order n≥4 having the smallest degree distances are K_1,n−1,BS(n−3,1) and K_1,n−1+e (in this order), where BS(n−3,1) denotes the bistar consisting of vertex disjoint stars K_1,n−3 and K_1,1 with central vertices joined by an edge.     

Degree sum conditions for oriented forests in digraphs

Let F be an oriented forest with n vertices and m arcs and D be a digraph without loops and multiple arcs. In this note we prove that D contains a subdigraph isomorphic to F if D has at least n vertices and min{d⁺(u)+d⁺(v),d⁻(u)+d⁻(v),d⁺(u)+d⁻(v)}≥2m−1 for every pair of vertices u,v∈V(D) with uv∉A(D). This is a common generalization of two results of Babu and Diwan, one on the existence of forests in graphs under a degree sum condition and the other on the existence of oriented forests in digraphs under a minimum degree condition.     

Cells in any simple polygon formed by a planar point set

Let P be a finite point set in general position in the plane. We consider empty convex subsets of P such that the union of the subsets constitute a simple polygon S whose dual graph is a path, and every point in P is on the boundary of S. Denote the minimum number of the subsets in the simple polygons S's formed by P by f_p(P), and define the maximum value of f_p(P) by F_p(n) over all P with n points. We show that ⌈(4n-17)/15⌉?F_p(n)?⌊n/2⌋.     

Multidimensional Farey partitions

The linear action of SL(n, ?₊) induces lattice partitions on the (n − 1)-dimensional simplex †n−1. The notion of Farey partition raises naturally from a matricial interpretation of the arithmetical Farey sequence of order r. Such sequence is unique and, consequently, the Farey partition of order r on A 1 is unique. In higher dimension no generalized Farey partition is unique. Nevertheless in dimension 3 the number of triangles in the various generalized Farey partitions is always the same which fails to be true in dimension n > 3. Concerning Diophantine approximations, it turns out that the vertices of an n-dimensional Farey partition of order r are the radial projections of the lattice points in ?₊ⁿ ∩ [0, r]ⁿ whose coordinates are relatively prime. Moreover, we obtain sequences of multidimensional Farey partitions which converge pointwisely.     

On edges crossing few other edges in simple topological complete graphs

We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graphT=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple if any two edges meet in at most one common point.Let h=h(n) be the smallest integer such that every simple topological complete graph on n vertices contains an edge crossing at most h other edges. We show that Ω(n^3/2)≤h(n)≤O(n²/log^1/4n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn².     

S(n)-θ-closed spaces

A P-space is called P-θ-closed if it is θ-closed in every P-space in which it is embedded. S(n)-θ-closed spaces are characterized in terms of special covers and filters, and a categorical property of S(n)-θ-closed spaces is established. Examples are given to distinguish S(n)-θ-closed and S(n)-closed spaces and to show that the property of being S(n)-θ-closed is not productive.     

Hamiltonicity in (0–1)-polyhedra

The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ? $\text{R}$ⁿ is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ? $\text{R}$ⁿ has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ? $\text{R}$^d having (0–1)-valued vertices such that $\text{G(P) ? G(P′)}$. Some combinatorial consequences of these results are also discussed.     

Inversion formulas for complex radon transform on projective varieties and boundary value problems for systems of linear PDEs

Let G ? ?P ⁿ be a linearly convex compact set with smooth boundary, D = ?P ⁿ \ G, and let D* ? (?P ⁿ )* be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety V of dimension d we construct an explicit inversion formula for the complex Radon transform R _V : H ^d,d?1(V ∩ D) → H ^1,0(D*) and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on D*.