Antipodal Pairs Research Articles

Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux

Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between a pair of distinguished antipodes; (2) the number of geodesics between these antipodes is equal to twice the number of standard Young tableaux of a truncated shifted staircase shape.

Energy-Momentum Stability of Icosahedral Configurations of Point Vortices on a Sphere

We investigate the nonlinear stability of the icosahedral relative equilibrium configuration of point vortices on a sphere. The relative equilibrium problem is formulated as a problem of finding the nullspace of the configuration matrix that encodes the geometry of the icosahedron, as in Jamaloodeen and Newton (Proc. Royal Soc. A, Math. Phys. Eng. Sci. 462(2075):3277, 2006). The seven-dimensional nullspace of the configuration matrix, A, associated with the icosahedral geometry gives rise to a basis set of vortex strengths for which the icosahedron stays in relative formation, and we use these values to form the augmented Hamiltonian governing the stability. We choose the basis set made up of (i) one element with equal strength vortices on every vertex of the icosahedron (the uniform icosahedron); (ii) six elements made up of equal and opposite antipodal pairs. We start by proving nonlinear stability of the antipodal vortex pair (by direct methods). Following the methods laid out in Simo et al. (Arch. Ration. Mech. Anal. 115(1):15–59, 1991) and Pekarsky and Marsden (J. Math. Phys. 39(11):5894–5907, 1998) and more generally in Marsden and Ratiu (Introduction to Mechanics and Symmetry, 1999), we then combine our knowledge of the nullspace structure of A with the structure of the underlying Hamiltonian, and analyze the stability of the icosahedron using the energy-momentum method. Because the parameter space is large, we focus on the physically motivated and important case obtained by combining the basis elements into (i) the uniform icosahedron; (ii) a von Karman vortex street configuration of equal and opposite staggered, evenly spaced latitudinal rows equidistant from the equator (Chamoun et al. in Phys. Fluids 21:116603, 2009), and (iii) the North Pole–South Pole equal and opposite vortex pair. Stability boundaries in a three-parameter space are calculated for linear combinations of these grouped basis configurations.

Can a meso-type dinuclear complex be chiral?: dinuclear β-diketonato Ru(III) complexes

Dinuclear Ru(III) complexes, [Ru(III)(acac)(2)(dabe)Ru(III)(acac)(2)] (acacH = acetylacetone; dabeH(2) = 1, 2-diacetyl-1,2-dibenzoylethane) and [Ru(III)(acac)(2)(tbet)Ru(III)(acac)(2)] (tbetH(2) = 1,1,2,2-tetrabenzoylethane) were synthesized by reacting [Ru(acac)(2)(CH(3)CN)(2)]PF(6) with dabeH(2) and tbetH(2) respectively, in toluene. The X-ray structural analysis of a meso-type dinuclear Ru(III) complex, ΔΛ-[Ru(III)(acac)(2)(dabe)Ru(III)(acac)(2)], showed that the bridging part became chiral due to the orthogonal twisting of two non-symmetrical β-diketonato moieties. To confirm this conclusion, the complex was resolved chromatographically to provide a pair of optical antipodes. Such chirality in the bridging part was not generated for [Ru(III)(acac)(2)(tbet)Ru(III)(acac)(2)], because the β-diketonato moieties in tbet(2-) are symmetrical.

What can the detection of a single pair of circles-in-the-sky tell us about the geometry and topology of the Universe?

In a Universe with a detectable nontrivial spatial topology, the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations---the so-called circles-in-the-sky. Searches undertaken for nearly antipodal pairs of such circles in cosmic microwave background maps have so far been unsuccessful. Previously, we had shown that the negative outcome of such searches, if confirmed, should in principle be sufficient to exclude a detectable nontrivial spatial topology for most observers in very nearly flat ($0<\ensuremath{\mid}{\ensuremath{\Omega}}_{\mathrm{tot}}\ensuremath{-}1\ensuremath{\mid}\ensuremath{\lesssim}{10}^{\ensuremath{-}5}$) (curved) universes. More recently, however, we have shown that this picture is fundamentally changed if the universe turns out to be exactly flat. In this case, there are many potential pairs of circles with large deviations from antipodicity that have not yet been probed by existing searches. Here, we study under what conditions the detection of a single pair of circles-in-the-sky can be used to uniquely specify the topology and the geometry of the spatial section of the Universe. We show that from the detection of a single pair of matching circles one can infer whether the spatial geometry is flat or not, and if so we show how to determine the topology (apart from one case) of the Universe using this information. An important additional outcome of our results is that the dimensionality of the circles-in-the-sky parameter space that needs to be spanned in searches for matching pairs of circles is reduced from six to 5 degrees of freedom, with a significant reduction in the necessary computational time.

A strong log-concavity property for measures on Boolean algebras

We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner, a new proof of a theorem of Liggett stating that ultra-log-concavity of sequences is preserved by convolutions, and some progress on a well-known log-concavity conjecture of J. Mason.

Automorphism groups of root system matroids

Given a root system R , the vector system R ̃ is obtained by taking a representative v in each antipodal pair { v , − v } . The matroid M ( R ) is formed by all independent subsets of R ̃ . The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root system matroids M ( R ) are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root system matroids.

Inflation on the brane with vanishing gravity

Many existing models of brane inflation suffer from a steep irreducible gravitational2 potential between the branes that causes inflation to end too early. Inspired by the fact that point masses in 2+1 D exert no gravitational force, we propose a novel unwarped and non-supersymmetric setup for inflation, consisting of 3-branes in two extra dimensions compactified on a sphere. The size of the sphere is stabilized by a combination of a bulk cosmological constant and a magnetic flux. Computing the 4D effective potential between probe branes in this background, we find a non-zero contribution only from exchange of level-1 KK modes of the graviton and radion. For antipodal pairs of branes the contribution from these modes is absent; we can thus eliminate entirely the troublesome gravitational contribution to the inflationary potential.

Magic square spectra

To study the eigenvalues of low order singular and non-singular magic squares we begin with some aspects of general square matrices. Additional properties follow for general semimagic squares (same row and column sums), with further properties for general magic squares (semimagic with same diagonal sums). Parameterizations of general magic squares for low orders are examined, including factorization of the linesum eigenvalue from the characteristic polynomial. For nth order natural magic squares with matrix elements 1 , … , n 2 we find examples of some remarkably singular cases. All cases of the regular (or associative, or symmetric) type (antipodal pair sum of 1 + n 2 ) with n - 1 zero eigenvalues have been found in the only complete sets of these squares (in fourth and fifth order). Both the Jordan form and singular value decomposition (SVD) have been useful in this study which examines examples up to 8th order. In fourth order these give examples illustrating a theorem by Mattingly that even order regular magic squares have a zero eigenvalue with odd algebraic multiplicity, m. We find 8 cases with m = 3 which have a non-diagonal Jordan form. The regular group of 48 squares is completed by 40 squares with m = 1 , which are diagonable. A surprise finding is that the eigenvalues of 16 fourth order pandiagonal magic squares alternate between m = 1 , diagonable, and m = 3 , non-diagonable, on rotation by π / 2 . Two 8th order natural magic squares, one regular and the other pandiagonal, are also examined, found to have m = 5 , and to be diagonable. Mattingly also proved that odd order regular magic squares have a zero eigenvalue with even multiplicity, m = 0 , 2 , 4 , . . . Analyzing results for natural fifth order magic squares from exact backtracking calculations we find 652 with m = 2 , and four with m = 4 . There are also 20, 604 singular seventh order natural ultramagic (simultaneously regular and pandiagonal) squares with m = 2 , demonstrating that the co-existence of regularity and pandiagonality permits singularity. The singular odd order examples studied are all non-diagonable.

Antipodal pairs, critical pairs, and Nash angular equilibria in convex cones

In this paper, we discuss three related geometric concepts that arise in connection with the angular analysis of a convex cone K⊂ℝ d . Although antipodal pairs and critical pairs in convex cones have been studied in our earlier works, several additional results are presented here for the first time. The concept of Nash angular equilibrium is new, and we set up the basic ingredients of the theory.

The LSB Theorem Implies the KKM Lemma

(2007). The LSB Theorem Implies the KKM Lemma. The American Mathematical Monthly: Vol. 114, No. 2, pp. 156-159.

Nitrile biotransformations for the practical synthesis of highly enantiopure azido carboxylic acids and amides, ‘click’ to functionalized chiral triazoles and chiral β-amino acids

Under very mild conditions, biotransformations of racemic azido nitriles using Rhodococcus erythropolis AJ270, a nitrile hydratase/amidase-containing microbial whole-cell catalyst, afforded highly enantiopure, ( R)-α-arylmethyl- and (+)-α-cyclohexylmethyl-β-azidopropanoic acids and their ( S)- and (−)-carboxamide derivatives in excellent yields. The resulting functionalized chiral organoazides were converted in a straightforward fashion to a pair of antipodes of α-benzyl-β-amino acids ( R)- 13 and ( S)- 13. Azido carboxamide ( S)- 11a and azido carboxylic acid ( R)- 12a underwent ‘click’ reactions with diethyl acetylenedicarboxylate and phenylacetylene to produce functionalized chiral triazoles 14 and 15, respectively. The easy preparation of the starting nitrile substrates, highly efficient and enantioselective biotransformation reactions, and versatile utility of the resulting functionalized azido carboxylic acids and amide derivatives, render this method very attractive and practical in organic synthesis.

Antipodal hotspots and bipolar catastrophes: Were oceanic large-body impacts the cause?

One aspect of the hotspot distribution that has received little attention is its antipodal character. Of 45 ‘primary’ hotspots found in most hotspot compilations 22 (49%) form antipodal pairs within observed hotspot drift limits (≤ 20 mm/yr). In addition, the available ages, or possible age ranges, for both hotspots of an antipodal pair tend to be similar (≤ 10 Myr difference) or overlap. Monte Carlo simulations indicate that the antipodal primary hotspots' locations and ages are not due to chance at the > 99% confidence level ( p < 0.01). All hotspot pairs include at least one oceanic hotspot, and these are consistently opposite those hotspots related to large igneous provinces (LIPs) and continental volcanism. A mechanism of formation is considered in which minor hotspot volcanism is induced at, and flood basalt volcanism is triggered by seismic energy focused antipodal to, oceanic large-body impact sites. Because continental impacts are expected to have lower seismic efficiencies, continents possibly acted as shields to the formation of antipodal hotspot pairs. Published numerical models indicate that large oceanic impacts (10-km-diameter bolide) generate megatsunami capable of altering coastal depositional environments on a global scale. Past impact-generated megatsunami, consequently, could have left widespread stratigraphic records, possibly misinterpreted as indicating large rapid changes in eustatic sea level, and widely disrupted continental and marine sediment reservoirs responsible for abrupt changes in the isotopic composition of seawater. Phanerozoic mass extinction events, therefore, might have resulted primarily from catastrophic megatsunami in a dominantly oceanic hemisphere and the near contemporaneous effusion of vast quantities of noxious gases from flood basalt eruptions in a dominantly continental one.

Ghosts of the Milky Way: a search for topology in new quasar catalogues

We revisit the possibility that we inhabit a compact multi-connected flat, or nearly-flat, Universe. Analysis of COBE data has shown that, for such a case, the size of the fundamental domain must be a substantial fraction of the horizon size. Nevertheless, there could be several copies of the Universe within the horizon. If the Milky Way was once a quasar we might detect its `ghost' images. Using new large quasar catalogues we repeat the search by Fagundes & Wichoski for antipodal quasar pairs. By applying linear theory to account for the peculiar velocity of the local group, we are able to narrow the search radius to 134 arcsec. We find seven candidate antipodal quasar pairs within this search radius. However, a similar number would be expected by chance. We argue that, even with larger quasar catalogues, and more accurate values of the cosmological parameters, it is unlikely to be possible to identify putative ghost pairs unambiguously, because of the uncertainty of the correction for peculiar motion of the Milky Way.

From circle packing to covering on a sphere with antipodal constraints

The problem treated here is: how must N antipodal pairs of equal circles (spherical caps) of given angular radius r be arranged on the surface of a sphere so that the area covered by the circles will be as large as possible? Conjectured solutions of this problem for N = 4,5,7 are given where r varies from the maximum antipodal packing radius to the minimum antipodal covering radius. These cases exhibit sequences of symmetry– and edge–number transitions in the contact polyhedra, which, in general, differ from those of the unconstrained problem where centrosymmetry is not enforced. New minimum antipodal covering arrangements are conjectured for 4, 5 and 7 pairs: they are a hexagonal bipyramid, a D 2h deltahedron and the cube+octahedron compound, respectively.

First photosensitized enantiodifferentiating isomerization by optically active sensitizer immobilized in zeolite supercages.

Enantiodifferentiating photoisomerization of (Z)-cyclooctene sensitized by (R)- or (S)-1-methylheptyl benzoate immobilized in zeolite supercages afforded the respective enantiomer pair, (-)- and (+)-(E)-isomer (1E) in 5% enantiomeric excess, whilst racemic 1E was obtained upon homogeneous-phase photosensitization with the same antipodal sensitizer pair, thus demonstrating for the first time that chirally modified zeolites not only serve as supramolecular photosensitizing media but also enhance the original enantiodifferentiating ability of chiral photosensitizer.

Two noncolourable configurations in four dimensions illustrating the Kochen-Specker theorem

It is demonstrated that the 60 rays corresponding to antipodal pairs of vertices of the 600-cell, and the 300 rays corresponding to antipodal pairs of vertices of the 120-cell, can both be used to give noncolouring proofs of the Bell-Kochen-Specker theorem in four dimensions.

Note on packings in Grassmannian space G (3,1)

How must 2N non-overlapping equal circles forming antipodal pairs be packed on a sphere so that the angular diameter of the circles will be as great as possible? In this note, some unnoticed putative solutions to this problem are mentioned, and attention is called to the Danzerian rigidity of the graphs of locally optimal antipodal packings.

Diastereomeric selectivity of carbon-coated zirconia reversed-phase liquid chromatographic media.

The determination of enantiomeric excess, that is, the relative amount of any pair of optical antipodes, constitutes a integral part of the work of analytical and synthetic chemists involved in natural products research or pharmaceutical development. Mosher's reagent [alpha-methoxy-alpha-(trifluoromethyl)phenylacetyl chloride] has evolved into a major tool for the determination of absolute configuration by NMR. We report here on the separation of diastereomers formed by derivatizing enantiomers with Mosher's reagent. We have shown that reversed-phase liquid-solid adsorption chromatography on carbon surfaces frequently gives considerably superior resolution of diastereomeric pairs than does RPLC on conventional bonded phases. The improved resolution results from the very high sensitivity of solid carbon surfaces to the geometric organization of the solute rather than from differences in column efficiency. We compare the separation of pharmaceutically and biologically important stereoisomeric mixtures, including (+/-)-warfarin and (+/-)-amino acid esters, on both conventional bonded phases and carbon surfaces prepared by chemical vapor deposition of organic compounds on porous zirconia microparticles.

All convex polyhedra can be clamped with parallel jaw grippers

We study various classes of polyhedra that can be clamped using parallel jaw grippers. We show that all n-vertex convex polyhedra can be clamped regardless of the gripper size and present an O( n + k) time algorithm to compute all positions of a polyhedron that allow a valid clamp where k is the number of antipodal pairs of features. We also observe that all terrain polyhedra and orthogonal polyhedra can be clamped and a valid clamp can be found in linear time. Finally we show that all polyhedra can be clamped with some size of gripper.

Some constructions of spherical 5-designs

Spherical designs were introduced by Delsarte, Goethals, and Seidel in 1977. A spherical t-design in, R n is a finite set X ⊂ S n−1 with the property that for every polynomial p with degree ⩽ t, the average value of p on X equals the average value of p on S n−1 . This paper contains some existence and nonexistence results, mainly for spherical 5-designs in R 3. Delsarte, Goethals, and Seidel proved that if X is a spherical 5-design in R 3, then | X| ⩾ 12 and if | Xz. sfnc; = 12, then X consists of the vertices of a regular icosahedron. We show that such designs exist with cardinality 16, 18, 20, 22, 24, and every integer ⩾ 26. If X is a spherical 5-design in R n , then | X| ⩾ n( n + 1); if | X| = n( n + 1), then X has been called tight. Tight spherical 5-designs in R n are known to exist only for n = 2, 3, 7, 23 and possibly n = u 2 − 2 for odd u ⩾ 7. Any tight spherical 5-design in R n must consist of n(n + 1) 2 antipodal pairs of points. We show that for n ⩾ 3, there are no spherical 5-designs in R n consisting of n(n + 1) 2 + 1 antipodal pairs of points.