Manifestation of geometric frustration on magnetic and thermodynamic properties of the pyrochlores Sm2X2O7 (X=Ti,Zr)

We present here magnetization, specific heat, and Raman studies on single-crystalline specimens of the first pyrochlore member Sm2Ti2O7 of the rare-earth titanate series. Its analogous compound Sm2Zr2O7 in the rare-earth zirconate series is also investigated in the polycrystalline form. The Sm spins in Sm2Ti2O7 remain unordered down to at least T=0.5 K. The absence of magnetic ordering is attributed to very small values of exchange (θcw∼−0.26 K) and dipolar interaction (μeff∼0.15 μB) between the Sm3+ spins in this pyrochlore. In contrast, the pyrochlore Sm2Zr2O7 is characterized by a relatively large value of Sm-Sm spin exchange (θcw∼−10 K); however, long-range ordering of the Sm3+ spins is not established at least down to T=0.67 K due to frustration of the Sm3+ spins on the pyrochlore lattice. The ground state of Sm3+ ions in both pyrochlores is a well-isolated Kramers doublet. The higher-lying crystal field excitations are observed in the low-frequency region of the Raman spectra of the two compounds recorded at T=10 K. At higher temperatures, the magnetic susceptibility of Sm2Ti2O7 shows a broad maximum at T=140 K, while that of Sm2Zr2O7 changes monotonically. Whereas Sm2Ti2O7 is a promising candidate for investigating spin fluctuations on a frustrated lattice, as indicated by our data, the properties of Sm2Zr2O7 seem to conform to a conventional scenario where geometrical frustration of the spin excludes their long-range ordering.

Topological properties of the one dimensional exponential random geometric graph

In this article we study the one-dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008.

A Geometric Algorithm for Learning Oblique Decision Trees

In this paper we present a novel algorithm for learning oblique decision trees. Most of the current decision tree algorithms rely on impurity measures to assess goodness of hyperplanes at each node. These impurity measures do not properly capture the geometric structures in the data. Motivated by this, our algorithm uses a strategy, based on some recent variants of SVM, to assess the hyperplanes in such a way that the geometric structure in the data is taken into account. We show through empirical studies that our method is effective.

Geometric quantum computation using fictitious spin-1/2 subspaces of strongly dipolar coupled nuclear spins

Geometric phases have been used in NMR to implement controlled phase shift gates for quantum-information processing, only in weakly coupled systems in which the individual spins can be identified as qubits. In this work, we implement controlled phase shift gates in strongly coupled systems by using nonadiabatic geometric phases, obtained by evolving the magnetization of fictitious spin-1/2 subspaces, over a closed loop on the Bloch sphere. The dynamical phase accumulated during the evolution of the subspaces is refocused by a spin echo pulse sequence and by setting the delay of transition selective pulses such that the evolution under the homonuclear coupling makes a complete 2 pi rotation. A detailed theoretical explanation of nonadiabatic geometric phases in NMR is given by using single transition operators. Controlled phase shift gates, two qubit Deutsch-Jozsa algorithm, and parity algorithm in a qubit-qutrit system have been implemented in various strongly dipolar coupled systems obtained by orienting the molecules in liquid crystal media.

Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs

Let n points be placed independently in d-dimensional space according to the density f(x) = A(d)e(-lambda parallel to x parallel to alpha), lambda, alpha > 0, x is an element of R-d, d >= 2. Let d(n) be the longest edge length of the nearest-neighbor graph on these points. We show that (lambda(-1) log n)(1-1/alpha) d(n) - b(n) converges weakly to the Gumbel distribution, where b(n) similar to ((d - 1)/lambda alpha) log log n. We also prove the following strong law for the normalized nearest-neighbor distance (d) over tilde (n) = (lambda(-1) log n)(1-1/alpha) d(n)/log log n: (d - 1)/alpha lambda <= lim inf(n ->infinity) (d) over tilde (n) <= lim sup(n ->infinity) (d) over tilde (n) <= d/alpha lambda almost surely. Thus, the exponential rate of decay alpha = 1 is critical, in the sense that, for alpha > 1, d(n) -> 0, whereas, for alpha <= 1, d(n) -> infinity almost surely as n -> infinity.

Geometric Characterizations for Patterson-Sullivan Measures of Geometrically Finite Kleinian Groups

The main results of this thesis show that a Patterson-Sullivan measure of a non-elementary geometrically finite Kleinian group can always be characterized using geometric covering and packing constructions. This means that if the standard covering and packing constructions are modified in a suitable way, one can use either one of them to construct a geometric measure which is identical to the Patterson-Sullivan measure. The main results generalize and modify results of D. Sullivan which show that one can sometimes use the standard covering construction to construct a suitable geometric measure and sometimes the standard packing construction. Sullivan has shown also that neither or both of the standard constructions can be used to construct the geometric measure in some situations. The main modifications of the standard constructions are based on certain geometric properties of limit sets of Kleinian groups studied first by P. Tukia. These geometric properties describe how closely the limit set of a given Kleinian group resembles euclidean planes or spheres of varying dimension on small scales. The main idea is to express these geometric properties in a quantitative form which can be incorporated into the gauge functions used in the modified covering and packing constructions. Certain estimation results for general conformal measures of Kleinian groups play a crucial role in the proofs of the main results. These estimation results are generalizations and modifications of similar results considered, among others, by B. Stratmann, D. Sullivan, P. Tukia and S. Velani. The modified constructions are in general defined without reference to Kleinian groups, so they or their variants may prove useful in some other contexts in addition to that of Kleinian groups.

Transformation formula for exponential sums involving fourier coefficients of modular forms

In 1984 Jutila [5] obtained a transformation formula for certain exponential sums involving the Fourier coefficients of a holomorphic cusp form for the full modular group SL(2, Z). With the help of the transformation formula he obtained good estimates for the distance between consecutive zeros on the critical line of the Dirichlet series associated with the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper we follow Jutila to obtain a transformation formula for exponential sums involving the Fourier coefficients of either holomorphic cusp forms or certain Maass forms for congruence subgroups of SL(2, Z) and prove similar estimates for the corresponding Dirichlet series.

Ground state geometric distortions Image distal substituent effects in determining the β-facial selectivity in 7-norbornenones

The X-ray structure of Image and MNDO optimized geometries of related 7-norbornenone derivatives show a clear tilt of the carbonyl bridge away from the C=C double bond. The preferred reduction from the more hindered face of the diester reveals the electron/electrostatic origin of π - facial selectivity in these systems. X-ray structure and MNDO calculations reveal the dominance of electronic effects in determining the π-facial selectivity in 4a.

Flat connections, geometric invariants and the symplectic nature of the fundamental group of surfaces

In this paper we associate a new geometric invariant to the space of fiat connections on a G (= SU(2))-bundle on a compact Riemann surface M and relate it tcr the symplectic structure on the space Hom(pi(1)(M), G)/G consisting of representations of the fundamental group pi(1)(M) Of M into G module the conjugate action of G on representations.