Antipodal Pairs Research Articles

Exact Thermal Eigenstates of Nonintegrable Spin Chains at Infinite Temperature.

The eigenstate thermalization hypothesis (ETH) plays a major role in explaining thermalization of isolated quantum many-body systems. However, there has been no proof of the ETH in realistic systems due to the difficulty in the theoretical treatment of thermal energy eigenstates of nonintegrable systems. Here, we write down analytically thermal eigenstates of nonintegrable spin chains. We consider a class of theoretically tractable volume-law states, which we call entangled antipodal pair (EAP) states. These states are thermal, in the most fundamental sense that they are indistinguishable from the Gibbs state with respect to all local observables, with infinite temperature. We then identify Hamiltonians having the EAP state as an eigenstate and rigorously show that some of these Hamiltonians are nonintegrable. Furthermore, a thermal pure state at an arbitrary temperature is obtained by the imaginary time evolution of an EAP state. Our results offer a potential avenue for providing a provable example of the ETH.

When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry

A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy’s Theorem—simultaneously in both (i) Euclidean geometry and (ii) the relativistic model of hyperbolic geometry (also known as the Klein model)—motivates the study of four cyclic antipodal pairs of a circle, ordered arbitrarily counterclockwise. The translation of results from Euclidean geometry into hyperbolic geometry is obtained by means of hyperbolic trigonometry, called gyrotrigonometry, to which Einstein addition gives rise. Identities that extend the Pythagorean identity in both Euclidean and hyperbolic geometry are obtained.

On the enumeration of a certain type of hyperplane arrangements

In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discriminantal arrangement. Later we observe that the restriction we impose on the type of hyperplane arrangements is a mild one and that this conditional restriction is quite generic. Moreover the restriction is defined in terms of a normal system being concurrency free which is a generic condition. We also discuss two examples of normal systems which are not concurrency free in the last section and enumerate the number of isomorphism classes.

Effective Capture of Nongraspable Objects for Space Robots Using Geometric Cage Pairs

Capture and removal of space debris are challenging in robotic on-orbit servicing activities. A large portion of space debris does not possess any graspable features, which makes the conventional grippers inapplicable. To handle such nongraspable objects, a space robotic capture system is presented. A dual-arm space robot simulator that has the advantages of miniaturization and scalability is designed for ground tests. Inspired by the robotic caging, we propose a novel capture method that uses a series of hollow-shaped end-effector pairs to cage the antipodal pairs of the nongraspable objects. To apply the caging-pair method steadily, space robots need exerting a squeezing action on objects, which can be characterized by the motion and force manipulation of two robotic arms in the assigned directions. Based on the velocity and force manipulability transmission ratios, a caging compatibility index is proposed to describe the capturing ability in this manner. Via the optimization of the desired caging compatibility index, an effective algorithm is proposed to plan the near-optimal joint configurations for the pregrasping cages. Finally, both simulation studies and experimental tests are conducted to evaluate the performance of the proposed capture method.

Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

In this paper we study singular points of the Wigner caustic and affine lambda -equidistants of planar curves based on shapes of these curves. We generalize the Blaschke–Süss theorem on the existence of antipodal pairs of a convex curve.

Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability

We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid E(a, b, c): x2/a+ y2/b+ z2/c = 1, a < b < c. The equations of motion are transported to S2 ×S2 via a conformal map that combines confocal quadric coordinates for the ellipsoid and sphero-conical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.

Paracatadioptric camera calibration based on properties of polar line of infinity point with respect to circle and line

Two linear calibration methods based on space-line projection properties for a paracatadioptric camera are presented. Considering the central catadioptric system, a straight line is projected into a circle on the viewing spherical surface for the first projection. The tangent lines in a group at antipode point pairs with respect to the circle are parallel, with the infinity point being the intersection point; therefore, the infinity line can be obtained from two groups of antipode point pairs. Further, the direction of the polar line of an infinity point with respect to the circle is orthogonal to the direction of its infinity point. Hence, on the imaging plane, images of the circular points or orthogonal vanishing points are used to determine the intrinsic parameters. On the basis of the properties of the antipodal point pairs and a least-squares fitting, a corresponding optimization algorithm for line image fitting is proposed. Experimental results demonstrate the robustness of the two calibration methods, that is, for images of the circular points and orthogonal vanishing points.

On Dantzig figures from graded lexicographic orders

We construct two families of Dantzig figures, which are d-dimensional polytopes with 2d facets and an antipodal vertex pair, from convex hulls of initial subsets for the graded lexicographic (grlex) and graded reverse lexicographic (grevlex) orders on Z≥0d. These two polytopes have the same number of vertices, O(d2), and the same number of edges, O(d3), but are not combinatorially equivalent. We provide an explicit description of the vertices and the facets for both families and describe their graphs along with analyzing their basic properties such as the radius, diameter, existence of Hamiltonian circuits, and chromatic number. Moreover, we also analyze the edge expansions of these graphs.

Half of an antipodal spherical design

We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$.

Approximation of convex bodies by polytopes with respect to minimal width and diameter

Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a polytope $P$ with at most $n$ vertices for which $\Lambda_n ({\mathcal K}^d) \leq \frac{w(P)}{w(C)}$. We give a lower estimate of $\Lambda_n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\big\lfloor {\frac{n}{2}} \big\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that $\Lambda_3 ({\mathcal K}^2) \geq {\frac 1 2}(3- \sqrt 3)$, and $\Lambda_n ({\mathcal K}^2) \geq \cos {\frac \pi {2 \lfloor {n/2} \rfloor}}$ for every $n \geq 4$. We also consider the dual question of estimating the smallest number $\Delta_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ there exists a polytope $P \supset C$ with at most $n$ facets for which $\frac{{\rm diam}(P)}{{\rm diam}(C)} \leq \Delta_n ({\mathcal K}^d)$. We give an upper bound of $\Delta_n ({\mathcal K}^d)$ for $n \geq 2d$. In particular, $\Delta_n ({\mathcal K}^2) \leq 1/ \cos {\frac \pi {2 \lfloor {n/2} \rfloor}}$ for $n \geq 4$.

On the 1-switch conjecture

Feder and Subi conjectured that for any 2-coloring of the edges of the n-dimensional cube, there is an antipodal pair of vertices connected by a path that changes color at most once. They proved that if the coloring is such that there are no properly edge colored 4-cycles, the conjecture is true, without a color change. We generalize their theorem by weakening the assumption on the coloring. Our method can be applied to a similar question on any graph, if the condition on the coloring is satisfied. We solve the corresponding problem on the toroidal grid of size 2a×2b.

Uniform multicommodity flows in the hypercube with random edge‐capacities

ABSTRACTWe give two results for multicommodity flows in the d‐dimensional hypercube with independent random edge‐capacities distributed like a random variable C where . Firstly, with high probability as , the network can support simultaneous multicommodity flows of volume close to between all antipodal vertex pairs. Secondly, with high probability, the network can support simultaneous multicommodity flows of volume close to between all vertex pairs. Both results are best possible. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 437–463, 2017

Backdoor use of philosophers in judicial decision-making? Antipodean reflections

ABSTRACTThe citation of philosophers in common law judicial decision-making has been rarely explored. Yet this issue divided two American commentators: Rao, in 1998, critiqued the United States Supreme Court’s (USSC) citation of philosophers, labelling it a ‘backdoor method for judicial policy making’; Brooks, in 2003, offered a trenchant rejoinder, arguing that philosophy is potentially an ‘ally in the pursuit of sensible legal reasoning’. As issues of the curial use of exogenous or extra-legal knowledge are not settled, there is value in reflecting on Australian judicial practice. This article proceeds in three stages. First, it locates Rao’s analysis of the USSC’s citation of philosophers within a larger picture that concerns issues of citation analysis. Second, it critiques Rao’s definitions of philosophy and adjudication in terms of three ‘antipodal pairs’ of characteristics – pragmatism versus abstraction, institutionalism versus intellectualism, and precedent versus permeability. Third, rejecting Rao’s thesis, this article conducts an exploratory survey to consider how the Australian High Court cites the named philosophers in Rao and Brooks’s surveys.

Characterization of Extremal Antipodal Polygons

Let $$S$$S be a set of $$2n$$2n points on a circle such that for each point $$p \in S$$p?S also its antipodal (mirrored with respect to the circle center) point $$p'$$p? belongs to $$S$$S. A polygon $$P$$P of size $$n$$n is called antipodal if it consists of precisely one point of each antipodal pair $$(p,p')$$(p,p?) of $$S$$S. We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of $$S$$S. Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization of antipodal polygons to higher dimensions we show that a similar characterization does not exist.

Compounds from the Chinese black ant (Polyrhachis dives) and NMR behavior of the isomers with formamide group

Two new dopamine derivatives divesamides A (1) and B (2), along with six known N-containing compounds were isolated from the Chinese black ant (Polyrhachis dives). Their structures were determined on the basis of spectroscopic methods. Compound 1 is a racemate, and chiral HPLC separation yielded a pair of antipodes. The absolute configuration of (+)-1 was assigned by a computational method. The double signals in the 1H and 13C NMR spectra of 2 that resulted from the presence of a formamide group were discussed. The T- and B-lymphocytes proliferation assay showed that 2 has moderate immunosuppressive activity toward T- and B-lymphocytes proliferation at a concentration of 20 μM.

Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous "parity proofs" of the Kochen-Specker theorem ruling out the existence of noncontextual hidden variables theories. The symmetries of the 120-cell are exploited to give a simple construction of its Kochen-Specker diagram, which is exhibited in the form of a "basis table" showing all the orthogonalities between its rays. The basis table consists of 675 bases (a basis being a set of four mutually orthogonal rays), but all the bases can be written down from the few listed in this paper using some simple rules. The basis table is shown to contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at the low end to 41 bases at the high end. Some explicit examples of these proofs are given, and their implications are discussed.

Seismic interferometry with antipodal station pairs

In this study, we analyze continuous data from all Global Seismographic Network stations between year 2000 and 2009 and demonstrate that several body wave phases (e.g., PP, PcPPKP, SKSP, and PPS) propagating between nearly antipodal station pairs can be clearly observed without array stacking using the noise/coda cross‐correlation method. Based on temporal correlations with global seismicity, we show that the observed body waves are clearly earthquake related. Moreover, based on single‐earthquake analysis, we show that the earthquake coda energy observed between ~10,000 and 30,000 s after a large earthquake contributes the majority of the signal. We refine our method based on these observations and show that the signal can be significantly improved by selecting only earthquake coda times. With our improved processing, the PKIKP phase, which does not benefit from the focusing effect near the antipode, can now also clearly be observed for long‐distance station pairs.

Arc permutations

Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, and the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircase shape. Finally, a bijection from non-unimodal arc permutations to Young tableaux of certain shapes, which preserves the descent set, is described and applied to deduce a conjectured character formula of Regev.

Axially chiral monomeric and dimeric square planar Pd(ii) complexes and their application to chiral tectonics

Mononuclear and dinuclear square planar palladium(II) complexes (denoted by [(hfac)Pd(II)(L-LH)] and [(hfac)Pd(II)(L-L)Pd(II)(hfac)], respectively) were synthesized. Here hfac(-), HL-L(-) and L-L(2-) denote hexafluoroacetylacetonato, monoprotonated and non-protonated bis-β-diketonato ligands, respectively. Three bis-β-diketones were used as HL-LH: 1,2-diacetyl-1,2-dibenzoylethane (denoted by dabeH2), 1,2-diacetyl-1,2-bis(3-methylbutanoyl)ethane (baetH2) and 1,2-diacetyl-1,2-propanoylethane (dpeH2). Both the monomeric and dimeric Pd(II) complexes were chiral due to the orthogonal twisting of the two non-symmetric diketonato moieties in HL-L(-) and L-L(2-), respectively. Optical resolution of [(hfac)Pd(II)(dabe)Pd(II)(hfac)] was achieved chromatographically on a chiral column to obtain a pair of optical antipodes which were stable against racemization. As for the other complexes, resolution was possible only after replacing hfac(-) with a bulky ligand such as dibenzoylmethanato (dbm(-)). Although a dinuclear complex with a symmetric bis-β-diketonato ligand, [(hfac)Pd(II)(taet)Pd(II)(hfac)] (taet(2-) = 1,1,2,2-tetraacetylethanato), was achiral, the replacement of the terminal ligands with non-symmetric β-diketonates yielded an axially chiral complex such as [(phacac)Pd(II)(taet)Pd(II)(phacac)], wherein phacac(-) denotes 1-phenyl-1,3-butanedionato. The UV and CD spectra of the Pd(II) complexes were analyzed with the help of the TDDFT calculations. The chiral monomeric species, [(dbm)Pd(II)(R- or S-baetH)], formed a heterometallic tetranuclear complex, [Fe(III){(dbm)Pd(II)(R- or S-baet)}3], in methanol solution.

Interaction of Global-scale Atmospheric Vortices: Modeling based on Hamiltonian Dynamic System of Antipodal Point Vortices on Rotating Sphere

It is shown for the first time that only an antipodal vortex pair (APV) is the elementary singular vortex object on the rotating sphere compatible with the hydrodynamic equations. The exact weak solution of the absolute vorticity equation on the rotating sphere is obtained in the form of Hamiltonian dynamic system for N interacting APVs. This is the first model describing interaction of Barrett vortices corresponding to atmospheric centers of action (ACA). In particular, new steady-state conditions for N=2 are obtained. These analytical conditions are used for the analysis of coupled cyclone-anticyclone ACAs over oceans in the Northern Hemisphere.