Geometric Permutations Research Articles

Introduction to Discrete Mathematics and Fourier Analysis. A Practical Guide for third Year Undergraduate Students.

This book covers a quick introduction to high school mathematics, arithmetic and geometric series, infinite geometric series, mathematical induction, binomial theorem, factorials, permutations and combinations, determinants, systems of linear equations, boolean algebra, orthogonal functions, Legendre, Hermite, and Laguerre polynomials and orthogonal properties. For example, mathematical induction is a proof based on logical assumptions of equations that involve integers of one number that shows the proof for the next number. It is based on the base step and the inductive step. The base step is to prove the integers based on a formula. The inductive step involves statement of one integer number that shows the proof of the statement for the next integer. The determinant is a scalar and it is found by multiplying the two elements of the principal diagonal and subtracting them from the multiplication of the elements of the other diagonal.

An experimental study of forbidden patterns in geometric permutations by combinatorial lifting

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in $\mathbb{R}^3$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be ``lifted'' to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations. In particular, we prove that all triples of permutations of size $5$ are realizable, and give a complete list of triples of size $6$ that are not.

Oxidation-assisted pulsating three-stream non-Newtonian slurry atomization for energy production

Past work involving validated “cold-flow” CFD modeling of self-generating and self-sustaining pulsating near-sonic non-Newtonian slurry atomization elucidated acoustic signatures, atomization mechanisms, and the effects of numerics and geometric permutations. The numerical method has now been incorporated with exothermic oxidation reaction kinetics relations along with radiation, i.e. no longer cold-flow. These models provide substantially increased model rigor and allow for new pulsing thermal measures which help assess injector thermal stresses. Twelve models have been run for extended periods of time in order to investigate the effects of dramatic changes in gas feed rate and prefilming (retraction) length. Given the new metrics and models, multiple statistically optimized designs are potentially available depending on the objective function(s) and their relative weightings in the overall value proposition to the project. In the case in which all metrics have equal value to the project and are simultaneously considered in a statistical model, the optimum design involves a mid-level of retraction and a mid-level gas feed rate. If, however, more relative weighting is placed on the importance of droplet size minimization and injector thermal management in lieu of feed passage pressure drop minimization, the optimum design involves a similar retraction but a very high level of gas feed rate.

Geometric permutations of non-overlapping unit balls revisited

Given four congruent balls A,B,C,D in Rδ that have disjoint interior and admit a line that intersects them in the order ABCD, we show that the distance between the centers of consecutive balls is smaller than the distance between the centers of A and D. This allows us to give a new short proof that n interior-disjoint congruent balls admit at most three geometric permutations, two if n⩾7. We also make a conjecture that would imply that n⩾4 such balls admit at most two geometric permutations, and show that if the conjecture is false, then there is a counter-example that is algebraically highly degenerate.

Improved Bounds for Geometric Permutations

We show that the number of geometric permutations of an arbitrary collection of n pairwise disjoint convex sets in ${\mathbb R}^d$, for $d\geq 3$, is $O(n^{2d-3}\log n)$, improving Wenger's 20-year-old bound of $O(n^{2d-2})$.

On k-convex polygons

We introduce a notion of k-convexity and explore polygons in the plane that have this property. Polygons which are k-convex can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of 2-convex polygons, a particularly interesting class, and show how to recognize them in O ( n log n ) time. A description of their shape is given as well, which leads to Erdős–Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of 2-convex objects considered.

Quasi-steady capillarity-driven flows in slender containers with interior edges

AbstractIn the absence of significant body forces the passive manipulation of fluid interfacial flows is naturally achieved by control of the specific geometry and wetting properties of the system. Numerous ‘microfluidic’ systems on Earth and ‘macrofluidic’ systems aboard spacecraft routinely exploit such methods and the term ‘capillary fluidics’ is used to describe both length-scale limits. In this work a collection of analytic solutions is offered for passive and weakly forced flows where a bulk capillary liquid is slowly drained or supplied by a faster capillary flow along at least one interior edge of the container. The solutions are enabled by an assumed known pressure (or known height) dynamical boundary condition. Following a series of assumptions this boundary condition can be in part determined a priori from the container dimensions and further quantitative experimental evidence, but not proof, is provided in support of its expanded use herein. In general, a small parameter arises in the scaling of the problems permitting a decoupling of the edge flow from the global bulk meniscus flow. The quasi-steady asymptotic system of equations that results may then be easily solved in closed form for a useful variety of geometries including uniform and tapered sections possessing at least one critically wetted interior edge. Draining, filling, bubble displacement and other imbibing flows are studied. Cursory terrestrial and drop tower experiments agree well with the solutions. The solutions are valued for the facility they provide in computing designs for selected capillary fluidics problems by way of passive transport rates and meniscus displacement. Because geometric permutations of any given design are myriad, such analytic tools are capable of efficiently identifying and comparing critical design criteria (i.e. shape and size) and the impact of various wetting conditions resulting from the fluid properties and surface conditions. Sample optimizations are performed to demonstrate the utility of the method.

Line Transversals of Convex Polyhedra in $\mathbb{R}^3$

We establish a bound of $O(n^2k^{1+\varepsilon})$, for any $\varepsilon>0$, on the combinatorial complexity of the set $\mathcal{T}$ of line transversals of a collection $\mathcal{P}$ of k convex polyhedra in $\mathbb{R}^3$ with a total of n facets, and we present a randomized algorithm which computes the boundary of $\mathcal{T}$ in comparable expected time. Thus, when $k\ll n$, the new bounds on the complexity (and construction cost) of $\mathcal{T}$ improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set $\mathcal{T}_{\ell_0}$ of line transversals which emanate from a fixed line $\ell_0$, establish an almost tight bound of $O(nk^{1+\varepsilon})$ on the complexity of $\mathcal{T}_{\ell_0}$, and provide a randomized algorithm which computes $\mathcal{T}_{\ell_0}$ in comparable expected time. Slightly improved combinatorial bounds for the complexity of $\mathcal{T}_{\ell_0}$ and comparable improvements in the cost of constructing this set are established for two special cases, both assuming that the polyhedra of $\mathcal{P}$ are pairwise disjoint: the case where $\ell_0$ is disjoint from the polyhedra of $\mathcal{P}$, and the case where the polyhedra of $\mathcal{P}$ are unbounded in a direction parallel to $\ell_0$. Our result is related to the problem of bounding the number of geometric permutations of a collection $\mathcal{C}$ of k pairwise-disjoint convex sets in $\mathbb{R}^3$, namely, the number of distinct orders in which the line transversals of $\mathcal{C}$ visit its members. We obtain a new partial result on this problem.

The combinatorial encoding of disjoint convex sets in the plane

We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2n-2 combinatorially distinct ways. The encoding generalizes the authors' encoding of point configurations by "allowable sequences" of permutations. Since it applies as well to a collection of compact connected sets with a specified pseudoline arrangement $$ \mathcal{A} $$ of separators and double tangents, the result extends the Edelsbrunner-Sharir theorem to the case of geometric permutations induced by pseudoline transversals compatible with $$ \mathcal{A} $$ .

Suballowable sequences and geometric permutations

We define suballowable sequences of permutations as a generalization of allowable sequences. We give a characterization of allowable sequences in the class of suballowable sequences, prove a Helly-type result on sets of permutations which form suballowable sequences, and show how suballowable sequences are related to problems of geometric realizability. We discuss configurations of points and geometric permutations in the plane. In particular, we find a characterization of pairwise realizability of planar geometric permutations, give two necessary conditions for realizability of planar geometric permutations, and show that these conditions are not sufficient.

Abundant Polymorphism in a System with Multiple Hydrogen-Bonding Opportunities: Oxalyl Dihydrazide

To date, only one crystal structure has been reported in the literature for oxalyl dihydrazide [H(2)N.NH.CO.CO.NH.NH(2)]. In the present paper, we report the discovery of four new polymorphs of oxalyl dihydrazide, obtained by crystallization from solution under different conditions, including the use of different crystallization solvents. All polymorphs have the trans-trans-trans conformation of the N-N-C-C-N-N backbone, but the positions of the hydrogen atoms of the NH(2) groups relative to this backbone differ between the different polymorphs through variation of the torsion angle around each NH-NH(2) bond. The different polymorphs display a range of different hydrogen-bonding arrangements, constructed from different types of hydrogen-bonded array. The existence of several different potential hydrogen-bond donor and hydrogen-bond acceptor groups in the oxalyl dihydrazide molecule, together with the fact that the N-H bonds of the NH(2) groups adopt different orientations with respect to the molecular plane, leads to several possible geometric permutations for hydrogen-bonding arrangements in the solid state. It would not be surprising if even more polymorphs of oxalyl dihydrazide are discovered in the future.

The Maximal Number of Geometric Permutations for n Disjoint Translates of a Convex Set in ℝ Is Ω(n)

A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝd is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations in families of n disjoint translates of a convex set in ℝ3 is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝd is Ω(n).

Transversals to Line Segments in Three-Dimensional Space

We completely describe the structure of the connected components of transversals to a collection of n line segments in ℝ3. Generically, the set of transversals to four segments consists of zero or two lines. We catalog the non-generic cases and show that n≥ 3 arbitrary line segments in ℝ3 admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in ℝ3.

Geometric Permutations Induced by Line Transversals through a Fixed Point

A line transversal of a family S of n pairwise disjoint convex objects is a straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in Rd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K (n − 1, d − 1) and K (n, d − 1), where K (n, d) =d i=0 ( n−1 i ) = (nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d − 1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K (n, d − 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations. ∗ Work on this paper has been partially supported by a grant from the U.S.–Israeli Binational Science Foundation. The first author’s work on this paper has also been supported by NSF Grant CCR-9972568 and NSF ITR Grant CCR-0081964. Part of the work was performed while he was visiting the Mathematical Sciences Research Institute, Berkeley, California. Part of the work on this paper was performed while the second author was a Ph.D student at Tel-Aviv University, under the supervision of Micha Sharir, and later a postdoctoral fellow at the Mathematical Sciences Research Institute, Berkeley, California. 286 B. Aronov and S. Smorodinsky

Geometric permutations of disjoint unit spheres

We show that a set of n disjoint unit spheres in R d admits at most two distinct geometric permutations if n ⩾ 9 , and at most three if 3 ⩽ n ⩽ 8 . This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R 3 : if any subset of size at most 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.

Forbidden Families of Geometric Permutations in ℝd

A geometric permutation induced by a transversal line of a finite family ℱ of disjoint convex sets in ℝd is the order in which the transversal meets the members of the family. We prove that for each natural k, each family of k permutations is realizable (as a family of geometric permutations of some ℱ) in ℝd for d ≥ 2k – 1, but there is a family of k permutations which is non-realizable in ℝd for d ≤ 2k – 2.

Geometric permutations of higher dimensional spheres

In this paper, we prove that the maximum number of geometric permutations (induced by line transversals) of a set of n pairwise disjoint spheres with a bounded radius ratio in R d for d⩾3 is at most 2 ⌊ 2 M⌋+1 , where M is the ratio of the largest radius and the smallest radius of the spheres. Setting M to 1, this gives an upper bound of 4 on the maximum number of geometric permutations for congruent spheres in R d , matching a recently independently discovered result [Y. Zhou, S. Suri, in: Proc. of 12th Annual ACM-SIAM Symp. on Discrete Algorithms, 2001, pp. 234–243] on this case. Our result settles a conjecture in combinatorial geometry.

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

Abstract. Let F be a family of disjoint unit balls in R 3 . We prove that there is a Helly-number n 0 ≤ 46 , such that if every n 0 members of F ( | F | ≥ n 0 ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.

On neighbors in geometric permutations

We introduce a new notion of ‘neighbors’ in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in R d is O( n), and we prove this conjecture for d=2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O( N d−1 ) geometric permutations. Hence, we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.

Geometric permutations of balls with bounded size disparity

We study combinatorial bounds for geometric permutations of balls with bounded size disparity in d-space. Our main contribution is the following theorem: given a set S of n disjoint balls in R d , if n is sufficiently large and the radius ratio between the largest and smallest balls of S is γ, then the maximum number of geometric permutations of S is O( γ log γ ). When d=2, we are able to prove the tight bound of 2 on the number of geometric permutations for S, which is the best possible bound because it holds even when γ=1. Our theorem shows how the number of permutations varies as a function of the size disparity among balls, thus gracefully bridging the gap between two extreme bounds that were known before: the O(1) bound for congruent balls, and the Θ( n d−1 ) bound for arbitrary balls.