Coupled problems in transient fluid and structural dynamics in nuclear engineering: Part 2: A discrete model for squeeze flow between two adjacent LMFBR subassemlies

The paper deals with a detailed investigation of the interaction between sub-assemblies and coolant dynamics within an LMFBR core under transient pressure loading. The one-dimensional discrete model approximates the axial squeeze flow by a combined Lagrangian particle Euler cell approach. The coupling between the fluid and the structure is achieved by a step by step scheme. An energy balance criterion is used to improve its computational efficiency. An extension for two-dimensional squeeze flow, where axial and transverse flow around the subassembly may occur, completes the mathematical model.     

On the complexity of locating linear facilities in the plane

We consider the computational complexity of linear facility location problems in the plane, i.e., given n demand points, one wishes to find r lines so as to minimize a certain objective-function reflecting the need of the points to be close to the lines. It is shown that it is NP-hard to find r lines so as to minimize any isotone function of the distances between given points and their respective nearest lines. The proofs establish NP-hardness in the strong sense. The results also apply to the situation where the demand is represented by r lines and the facilities by n single points.     

Simple connectedness of hyperplane complements in some geometries related to buildings

For a class of parapolar spaces that includes the geometries E6,4, E7,7, and E8,1 with lines of size at least three, the metasymplectic spaces with lines of size at least four, and the polar line Grassmannians with lines of size at least four except D4,2(3), we show that the subgraph of the point-collinearity graph induced on the complement of a hyperplane is simply connected. We also show that these parapolar spaces have Veldkamp lines.     

On the structure of 3-nets embedded in a projective plane

We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular otherwise. It is completely irregular when no class of the 3-net consists of concurrent lines. We are interested in embeddings of 3-nets which are irregular but the lines of one class are concurrent. For an irregular embedding of a 3-net of order n?5 we prove that, if all lines from two classes are tangent to the same irreducible conic, then all lines from the third class are concurrent. We also prove the converse provided that the order n of the 3-net is smaller than p. In the complex plane, apart from a sporadic example of order n=5 due to Stipins [7], each known irregularly embedded 3-net has the property that all its lines are tangent to a plane cubic curve. Actually, the procedure of constructing irregular 3-nets with this property works over any field. In positive characteristic, we present some more examples for n?5 and give a complete classification for n=4.     

Shifting the balance of health care into the 21st century

Some of the present or likely future key shifts in the balance of health care in Europe are set out; shifts in when and where and how care is delivered, what is delivered and who is cared for. An illustrative assessment is given of ways in which ORMS can help in coping with the uncertainty, complexity and change that underlies many of the observed shifts in the balance of health care.     

Symmetric Gobos in a finite projective plane

A gobo G in any incidence structure K is a (perhaps degenerate) tactical configuration having the property that no three points in G are collinear and no three lines in G are concurrent. General results are obtained where K is a finite projective plane of order n and G has k points and k lines such that each point (line) lies on r lines (points) of G. Particular attention is called to the contrast between the case r = 1 and the case r ≠ 1 when k = n.     

Multiperfect numbers on lines of the Pascal triangle

A number n is said to be multiperfect (or multiply perfect) if n divides its sum of divisors σ(n). In this paper, we study the multiperfect numbers on straight lines through the Pascal triangle. Except for the lines parallel to the edges, we show that all other lines through the Pascal triangle contain at most finitely many multiperfect numbers. We also study the distribution of the numbers σ(n)/n whenever the positive integer n ranges through the binomial coefficients on a fixed line through the Pascal triangle.     

A property of the colored complete graph

If the lines of the complete graph K_n are colored so that no point is on more than $\text{1}\text{7}\text{(n?1)}$ lines of the same color or so that each point lies on more than $\text{1}\text{7}\text{(5n+8)}$ lines of different colors, then K_n contains a cycle of length n with adjacent lines having different colors.     

On lines, joints, and incidences in three dimensions

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R³ and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for m?n, and Θ(m2/3n2/3+m+n) for m?n. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R³. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9].     

Algebraic methods in discrete analogs of the Kakeya problem

We prove the joints conjecture, showing that for any N lines in R³, there are at most points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N² lines in R³ so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N³). Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.     

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in ?², there is a line ? such that in both line sets, for both halfplanes delimited by ?, there are $\sqrt{n}$ lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are $\sqrt{n/3}$ of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erd?s–Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to graph drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labeled lines that are universal for all n-vertex labeled planar graphs. In contrast, the main result by Pach and Toth (J. Graph Theory 46(1):39–47, 2004), has, as an easy consequence, that every set of n (unlabeled) lines is universal for all n-vertex (unlabeled) planar graphs.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Bilinear flocks

A flock in PG(3, q) is a set of q planes which do not contain the vertex of a cone and have the property that the intersections of the planes of the flock with the cone partition the points of the cone except for the vertex. In this paper, we examine flocks, called bilinear flocks, where the planes of the flock pass through at least one of two distinct lines, called supporting lines in PG(3, q). We classify and provide examples of cones that admit bilinear flocks whose supporting lines intersect in PG(3, q). We also examine bilinear flocks whose supporting lines are skew, providing an example and also showing that this situation can not occur under certain conditions.     

Graphs with Hamiltonian cycles having adjacent lines different colors

Let G be a complete graph K_p (or a complete bipartite graph K_m,m) with its lines colored so that no point is on more than k lines of the same color. If p ≥ 17k (or m ≥ 25k) then G has a cycle of every possible size with adjacent lines different colors.     

On the degree of points and lines in a restricted linear space

In this paper we show that, with the exception of a few easily characterized linear spaces, all restricted linear spaces of square order n have the maximal degree of the lines equal to n + 1, the degree of every point at least n + 1, and further we show that p ? n² + n + 1 ? q, where p is the number of points and q the number of lines.     

Localization method for lines of discontinuity of an approximately defined function of two variables

A function of two variables with lines of discontinuity of the first kind is considered. It is assumed that outside the discontinuity lines the function is smooth and has a bounded partial derivative. An approximation to the function in L ₂ and a perturbation level are known. The problem in question belongs to a class of nonlinear ill-posed problems, which are solved by constructing some regularizing algorithms. We propose a simple theoretical approach to solving the problem of localizing the discontinuity lines of a function that is noisy in the space L ₂. Some conditions on the exact function are imposed ??in the small.?? Methods of averaging are constructed, and error estimates of localizing the lines (in the small) are obtained.     

Numerical solution for degenerate scale problem arising from multiple rigid lines in plane elasticity

This paper studies the degenerate scale problem arising from multiple rigid lines in plane elasticity. In the first step, the problem should be formulated on a degenerate scale by distribution of body force densities along rigid lines. The condition of vanishing displacement along lines is also assumed. The coordinate transform with a reduced factor “h” is performed in the next step. The new obtained BIE is a particular non-homogenous BIE defined in the transformed coordinates with normal scale. In the normal scale, the integral operator is invertible. By using two fundamental solutions that are formulated in the normal scale, the new obtained BIE can be reduced to an equation for finding the factor “h”. Finally, the degenerate scale is obtained. It is proved from computed results that the degenerate scale only depends on the configuration of rigid lines, and does not depend on the initial normal scale used. In addition, the degenerate scale is invariant with respect to the rotation of rigid lines. Many examples are carried out.     

Orchards in elliptic curves over finite fields

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.     

Proof of Grünbaum's conjecture on the stretchability of certain arrangements of pseudolines

We prove Grünbaum's conjecture that every arrangement of eight pseudolines in the projective plane is stretchable, i.e., determines a cell complex isomorphic to one determined by an arrangement of lines. The proof uses our previous results on ordered duality in the projective plane and on periodic sequences of permutations of [1,n] associated to arrangements of n lines in the euclidean plane.