Antipodal Pairs Research Articles

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.

On antipodal and adjoint pairs of points for two convex bodies

The numbers of antipodal and of adjoint pairs of points are estimated for a given pair of disjoint convex bodies inEd.

Measuring the Hubble constant and our Virgo-infall velocity independently

A sample of spiral galaxies with B(sub T) less than 14.5 located in two local volumes, one in the direction of, but behind, the Virgo Cluster (behind-Virgo volume (BV)) and the other in the opposite direction (anti-Virgo volume (AV)), were used via a Tully-Fisher (TF) relation to derive the following two parameters: H(sub AB), the mean Hubble ratio between AV and BV, and delta v(sub parallel), the peculiar velocity of the Local Group in the direction of the Virgo Cluster (VC) with respect to a uniformly expanding reference system defined by our AV and BV sub-samples. The two sampled volumes, separated by a velocity interval of 5600 km/s, form an antipodal pair. This particular geometry not only allows us to derive the two parameters independently but also reduces the dynamical effect of the Local Supercluster on H(sub AB) without increasing the Malmquist bias. By limiting our sample to spiral galaxies having large velocity widths W(sub R), we effectively reduce the TF scatter and Malmquist bias in our sample. The TF zero point and dispersion were then determined by further correcting for the small residual Malmquist bias. An additional sample of fainter galaxies was used to test for a non-Gaussian tail to the TF disperison. We found no evidence for such a tail and formally give an upper limit of about 18% for the fractional contribution of an unseen tail. The average intrinsic TF dispersion for the dominant Gaussian component is sigma(sub TF)(sup 0) approximately 0.33 mag for W(sub R) approximately equal to or greater than 180 km/s. Our numerical results are delta v(sub parallel) approximately equals 414 +/- 82 km/s and H(sub AB) approximately equals (84.0 +/- 2.4)(1 + epsilon) km/s Mpc, where (1 + epsilon) accounts for any systematic error between the calibrators and the sample galaxies. Various dynamical models were tested to explore the effect on H(sub AB) of the uncertainties in the local velocity field. Constrained by our observed delta v(sub parallel) as well as other observational quantities, we found that the rms deviation from unity of H(sub AB)/H(sub 0) (where H(sub 0) is the Hubble constant for each model) is 5%, making H(sub AB) a good indicator for H(sub 0). Taking this variation as an additional error, our formal estimate for the Hubble constant is H(sub 0) approximately equals (84 +/- 5)(1 + epsilon) km/s Mpc.

Lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convex polytope

The greatest lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convexd-polytope withn vertices are determined.

On the number of antipodal or strictly antipodal pairs of points in finite subsets ofR d , II

The paper is a continuation of [MM], namely containing several statements related to the concept of antipodal and strictly antipodal p~rs of points in a subset X of R d, which has cardinality n. The points zi, z i E X are called antipodal if each of them is eontsJned in one of two different parallel supporting hyperplanes of the convex hull of X. If such hyperplanes contain no further point of X, then zi, zj are even strictly antipodal. We shall prove some lower bounds on the number of strictly antipodal pairs for given d and n. Furthermore, this concept leads us to a statement on the quotient of the lengths of longest and shortest edges of special d-simpliees, and finally a generalization (concerning strictly antipodal segments) is proved.

Do gamma-ray burst sources repeat?

Following the discovery by Quashnock and Lamb (1993) of an apparent excess of $\gamma$-ray burst pairs with small angular separations, we reanalyze the angular distribution of the bursts in the BATSE catalogue. We find that in addition to an excess of close pairs, there is also a comparable excess of antipodal bursts, i.e pairs of bursts separated by about 180 degrees in the sky. Both excesses have only modest statistical significance. We reject the hypothesis put forward by Quashnock and Lamb that burst sources are repeaters, since it is obvious that this hypothesis does not predict an excess of antipodal coincidences. Lacking any physical model of bursts that can explain the antipodal pairs, we suggest that the two excesses seen in the data are either due to an unusual statistical fluctuation or caused by some unknown selection effect.

Hotspots and sunspots: surface tracers of deep mantle convection in the Earth and Sun

The pattern of new appearances of hotspots on the Earth is investigated using available age data. The worldwide total of Cenozoic and Mesozoic hotspots, predicted by extrapolation from the observed number of continental flood basalts, is ∼ 40, in agreement with Crough's counts. There are found to be no true antipodal pairs. In one interpretation of the data, new appearances of hotspots occur first at high latitudes in both hemispheres and then migrate toward the equator; shortly before the migration cycle is finished, the next cycle begins. A relative deficiency of hotspots, however, is observed in very low equatorial and very high polar regions. In addition, hotspots occupy two large hemispherical groups in longitude that slowly drift in the same direction as the axial body rotation. An observed magnetic superchron seems to end when a new hotspot migration cycle in latitude begins; but consecutive superchrons show opposite polarity. Although the hotspot data are rather sparse, their suggested patterns resemble the well-known patterns of complexes of active regions on the Sun, as exemplified by proxy data, such as sunspots. Both the Earth and the Sun possess large convecting, rotating and magnetized mantles, in which the characteristic surface tracers are believed to reflect the patterns of convective phenomena very deep in the mantle (even though the tracers themselves—hotspots and sunspots—are certainly not analogs of each other). The present study supports existing theoretical ideas that the large-scale patterns of deep mantle convection in the Earth and Sun may fundamentally resemble each other, despite the enormous difference in their molecular viscosities. Magnetic polarity reversal histories also show close similarities in the two bodies, suggesting the operation of basically similar convective dynamos. However, the Sun has displayed no known analog of the Earth's magnetically disturbed intervals.

Effect of some stereoisomeric tricyclic antidepressants on45Ca uptake in synaptosomes from rat hippocampus

The present study has examined the inhibition of synaptosomal 45calcium uptake by trimipramine, oxaprotiline and doxepin, and their stereoisomers, in synaptosomes from the rat hippocampus. No significant difference in potency could be established for inhibition of net depolarization-induced 45calcium uptake for any pair of antipodes, and the IC50 values for calcium channel blockade were in the vicinity of 30 microM for this group of compounds. Trimipramine, doxepin and oxaprotiline also inhibited the 45calcium uptake mediated by Na(+)-Ca2+ exchange, with IC50 values of 71 microM, 110 microM, and 100 microM, respectively. The similar potency of doxepin isomers for inhibition of voltage-dependent calcium channels is in harmony with their reported similar potency in the clinic. A slight difference in potency is reported between the isomers of oxaprotiline in the behavioral despair test in rats, and the dextrorotatory isomer of trimipramine is reported to be a much more potent antidepressant than the levorotatory isomer: these order of potencies do not correspond perfectly with the similar potency of the antipodes against voltage-dependent calcium channels. The present study of stereoisomeric tricyclic antidepressants therefore fails to provide unequivocal support for the hypothesis that calcium channel blockade by tricyclic antidepressants is involved in their therapeutic effect.

Antipodal hotspot pairs on the Earth

The results of statistical analyses performed on three published hotspot distributions suggest that significantly more hotspots occur as nearly antipodal pairs than is anticipated from a random distribution, or from their association with geoid highs and divergent plate margins. The observed number of antipodal hotspot pairs depends on the maximum allowable deviation from exact antipodality. At a maximum deviation of ≤700 km, 26% to 37% of hotspots form antipodal pairs in the published lists examined here, significantly more than would be expected from the general hotspot distribution. Two possible mechanisms that might create such a distribution include: (1) symmetry in the generation of mantle plumes, and (2) melting related to antipodal focusing of seismic energy from large‐body impacts.

Stereochemical studies of chiral H-1 antagonists of histamine: the resolution, chiral analysis, and biological evaluation of four antipodal pairs.

The resolution of the H-1 antihistamines chloropheniramine, dimethindene, carbinoxamine, and mebrophenhydramine is described. The optical purity of antipodal products is investigated by chiral HPLC (use of alpha 1-acid glycoprotein and beta-cyclodextrin columns) and NMR (spectra of beta-cyclodextrin inclusion complexes). Configurational relationships among the group are reviewed and assignments are confirmed and extended by circular dichroism evidence. Affinity constants of antipodal pairs for guinea pig ileum and cerebellum sites, determined by gut bath and binding experiments respectively, are reported together with some in vivo tests in man for central effects. Results are discussed in terms of configurational requirements for activity and variations in antipodal potency ratios within the group.

Closed-shell fullerene and fulleroid carbon cylinders

Electronic closed shells are predicted for series of cylindrical carbon clusters incorporating antipodal pairs of heptagonal, octagonal or nonagonal rings of atoms. Although not technically fullerenes, these clusters can all be accommodated by extensions of the known leapfrog and cylinder rules for fullerene closed shells.

Computing the width of a set

For a set of points P in three-dimensional space, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n+I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is the number of vertices; in the worst case, I=O(n/sup 2/). For a convex polyhedra the time complexity becomes O(n+I). If P is a set of points in the plane, the complexity can be reduced to O(nlog n). For simple polygons, linear time suffices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On rosettes and almost rosettes

The closed plane curves of class C 2 which have curvature k(s) > 0 or k(s) ≥ 0 with a finite number of zeros are studied. The results concern the existence of normal lines which divide the perimeter into equal parts and the existence of some special kinds of pairs of points on these curves as orthodiameter pairs, antipodal pairs, etc. The paper also contains some generalizations of the theorems of Blaschke-Suss and Barbier.

Enantiomeric Separation of Rotenone and Rotenolone on Chiral Stationary Phases

Abstract Analytical and preparative high-performance liquid chromatoqraphic procedures have been developed for separation of optical antipodes of rotenone and rotenolone on three chiral stationary phases. Rotenone enantiomers were resolved on (+)-poly (triphenylmethylmethacrylate)-bonded silica, whereas optical resolution of rotenolone enantiomers was accomplished by using a (R)-N-3,5-dinitrobenzoylphenylglycine silica column (DNBPG). In experiments where resolution of enantiomers was achieved, each antipodal pair in both rotenone and rotenolone series was sufficiently resolved, although simultaneous resolution of all four isomers of rotenone and rotenolone was not accomplished. In all cases, the elution of (-)-(6aβ, 12aβ)-enantiomers preceded that of their antipodes. Separation factors (α) obtained with the covalent DNBPG stationary phase were slightly higher than those observed with the ionic valiant. The method has been applied to the resolution of racemates of deguelin and tephrosin.

On a problem of Yuzvinsky on separating the n-cube

The following problem of Yuzvinsky is solved here: how many vertices of the n-cube must be removed from it in order that no connected component of the rest contains an antipodal pair of vertices? Some further results and problems are described as well.

Lösbarkeit Nichtlinearer Gleichungen Mit Orthogonalitätssätzen Vom Borsukschen Typ

In this paper we will prove two theorems which are similar to BORSUKs antipodal theorem on the n-dimensional sphere Sn. However, instead of antipodal pairs of points used in BORSUKs theorem, orthogonal pairs of points are regarded. By these “orthogonal versions” of BORSUKs theorem we get existence theorems about the solutions of non-odd equations F(x) = 0 in Rn and on fixed-point-equations in Hilbert spaces.

The symmetric all-furthest- neighbor problem

Given a set P of n points on the plane, a symmetric furthest-neighbor (SFN) pair of points p, q is one such that both p and q are furthest from each other among the points in P . A pair of points is antipodal if it admits parallel lines of support. In this paper it is shown that a SFN pair of P is both a set of extreme points of P and an antipodal pair of P . It is also shown that an asymmetric furthest-neighbor (ASFN) pair is not necessarily antipodal. Furthermore, if P is such that no two distances are equal, it is shown that as many as, and no more than, ⌊ n/2⌋ pairs of points are SFN pairs. A polygon is unimodal if for each vertex p k , k = 1,…, n the distance function defined by the euclidean distance between p k and the remaining vertices (traversed in order) contains only one local maximum. The fastest existing algorithms for computing all the ASFN or SFN pairs of either a set of points, a simple polygon, or a convex polygon, require 0( n log n) running time. It is shown that the above results lead to an 0( n) algorithm for computing all the SFN pairs of vertices of a unimodal polygon.

Philosophy and Democracy

prestige of political very high these days. It commands attention of economists and lawyers, two groups of academics most closely connected to shaping of public policy, as it has not done in a long time. And it claims attention of political leaders, bureaucrats, and judges, most especially judges, with a new and radical forcefulness. command and claim follow not so much from fact that philosophers are doing creative work, but from fact that they are doing creative work of a special sort-which raises again, after a long hiatus, possibility of finding objective truths, true meaning, right answers, the philosopher's stone, and so on. I want to accept this possibility (without saying very much about it) and then ask what it means for democratic politics. What standing of philosopher in a democratic society? This an old question; there are old tensions at work here: between truth and opinion, reason and will, value and preference, one and many. These antipodal pairs differ from one another, and none of them quite matches pair philosophy and democracy. But they do hang together; they point to a central problem. Philosophers claim a certain sort of authority for their conclusions; people claim a different sort of authority for their decisions. What relation between two? I shall begin with a quotation from Wittgenstein that might seem to resolve problem immediately. The philosopher, Wittgenstein wrote, is not a citizen of any community of ideas. That what makes him into a philosopher.' This more than an assertion of detachment in its usual sense, for citizens are surely capable, sometimes, of detached judgments even of their own ideologies, practices, and institutions. Wittgenstein asserting a more radical detachment. philosopher and must be an outsider; standing apart, not occasionally (in judgment) but systematically (in thought). I do not know whether philosopher

Antipodal Embeddings of Graphs

An antipodal graph D of diameter d has the property that each vertex υ has a unique (antipodal) vertex υ at distance d from υ in D. We show that any such D has circuits of length 2d passing through antipodal pairs of vertices. The identification of antipodal vertex-pairs in D produces a quotient graph G with a double cover projection morphism p : D→G. Using the two-fold quotient map of surfaces π : S2→RP2 where the real projective plane is obtained from the sphere, we study the relation between embeddings of a planar graph in S2 and embeddings of G in RP2. In particular, our main theorem establishes that every planar antipodal graph D has an embedding in S2 such that p is a restriction of the projection π.

Double covers of graphs

A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.