Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time \tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1) is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most \ell in time \tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i,j] of the product matrix C a witness of rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for C[i,j], with high probability. The algorithm runs in \tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).Comment: 14 pages, 3 figure

Magnetization steps in Zn_(1-x)Mn_xO: Four largest exchange constants and single-ion anisotropy

Magnetization steps (MST's) from Mn pairs in several single crystals of Zn_(1-x)Mn_xO (0.0056<=x<=0.030, and in one powder (x=0.029), were observed. The largest two exchange constants, J1/kB=-18.2+/-0.5K and J1'/kB=-24.3+/-0.6K, were obtained from large peaks in the differential susceptibility, dM/dH, measured in pulsed magnetic fields, H, up to 500 kOe. These two largest J's are associated with the two inequivalent classes of nearest neighbors (NN's) in the wurtzite structure. The 29% difference between J1 and J1' is substantially larger than 13% in CdS:Mn, and 15% in CdSe:Mn. The pulsed-field data also indicate that, despite the direct contact between the samples and a superfluid-helium bath, substantial departures from thermal equilibrium occurred during the 7.4 ms pulse. The third- and fourth-largest J's were determined from the magnetization M at 20 mK, measured in dc magnetic fields H up to 90 kOe. Both field orientations H||c and H||[10-10] were studied. (The [10-10] direction is perpendicular to the c-axis, [0001].) By definition, neighbors which are not NN's are distant neighbors (DN's). The largest DN exchange constant (third-largest overall), has the value J/kB=-0.543+/-0.005K, and is associated with the DN at r=c. Because this is not the closest DN, this result implies that the J's do not decrease monotonically with the distance r. The second-largest DN exchange constant (fourth-largest overall), has the value J/kB=-0.080 K. It is associated with one of the two classes of neighbors that have a coordination number z=12, but the evidence is insufficient for a definite unique choice. The dependence of M on the direction of H gives D/kB=-0.039+/-0.008K, in fair agreement with -0.031 K from earlier EPR work.Comment: 12 pages, 10 figures. Submitted to PR

Magnetization steps in a diluted Heisenberg antiferromagnetic chain: Theory and experiments on TMMC:Cd

A theory for the equilibrium low-temperature magnetization M of a diluted Heisenberg antiferromagnetic chain is presented. The magnetization curve, M versus B, is calculated using the exact contributions of finite chains with 1 to 5 spins, and the "rise and ramp approximation" for longer chains. Some non-equilibrium effects that occur in a rapidly changing B, are also considered. Specific non-equilibrium models based on earlier treatments of the phonon bottleneck, and of spin flips associated with cross relaxation and with level crossings, are discussed. Magnetization data on powders of TMMC diluted with cadmium [i.e., (CH_3)_4NMn_xCd_(1-x)Cl_3, with 0.16<=x<=0.50 were measured at 0.55 K in 18 T superconducting magnets. The field B_1 at the first MST from pairs is used to determine the NN exchange constant, J, which changes from -5.9 K to -6.5 K as x increases from 0.16 to 0.50. The magnetization curves obtained in the superconducting magnets are compared with simulations based on the equilibrium theory. Data for the differential susceptibility, dM/dB, were taken in pulsed magnetic fields (7.4 ms duration) up to 50 T, with the powder samples in a 1.5 K liquid-helium bath. Non-equilibrium effects, which became more severe as x decreased, were observed. The non-equilibrium effects are tentatively interpreted using the "Inadequate Heat Flow Scenario," or to cross-relaxation, and crossings of energy levels, including those of excited states.Comment: 16 pages, 14 figure

Processing of strong flux trapping high T(subc) oxide superconductors: Center director's discretionary fund

Magnetic suspension effect was first observed in samples of YBa2Cu3O7/AgO(Y-123/AgO) composites. Magnetization measurements of these samples show a much larger hysteresis which corresponds to a large critical current density. In addition to the Y-123AgO composites, recently similar suspension effects in other RE-123/AgO, where RE stands for rare-Earth elements, were also observed. Some samples exhibit even stronger flux pinning than that of the Y-123/AgO sample. An interesting observation was that in order to form the composite which exhibits strong flux trapping effect the sintering temperature depends on the particular RE-123 compound used. The paper presents the detailed processing conditions for the formation of these RE-123/AgO composites, as well as the magnetization and critical field data

"Quasi two-dimensional" spin distributions in II-VI magnetic semiconductor heterostructures: Clustering and dimensionality

Spin clustering in diluted magnetic semiconductors (DMS) arises from antiferromagnetic exchange between neighboring magnetic cations and is a strong function of reduced dimensionality. Epitaxially-grown single monolayers and abrupt interfaces of DMS are, however, never perfectly two-dimensional (2D) due to the unavoidable inter-monolayer mixing of atoms during growth. Thus the magnetization of DMS heterostructures, which is strongly modified by spin clustering, is intermediate between that of 2D and 3D spin distributions. We present an exact calculation of spin clustering applicable to arbitrary distributions of magnetic spins in the growth direction. The results reveal a surprising insensitivity of the magnetization to the form of the intermixing profile, and identify important limits on the maximum possible magnetization. High-field optical studies of heterostructures containing "quasi-2D" spin distributions are compared with calculation.Comment: 5 pages (RevTeX), 5 embedded EPS figs, published in PRB v61 p1736 (2000

Magnetic structures and reorientation transitions in noncentrosymmetric uniaxial antiferromagnets

A phenomenological theory of magnetic states in noncentrosymmetric tetragonal antiferromagnets is developed, which has to include homogeneous and inhomogeneous terms (Lifshitz-invariants) derived from Dzyaloshinskii-Moriya couplings. Magnetic properties of this class of antiferromagnets with low crystal symmetry are discussed in relation to its first known members, the recently detected compounds Ba2CuGe2O7 and K2V3O8. Crystallographic symmetry and magnetic ordering in these systems allow the simultaneous occurrence of chiral inhomogeneous magnetic structures and weak ferromagnetism. New types of incommensurate magnetic structures are possible, namely, chiral helices with rotation of staggered magnetization and oscillations of the total magnetization. Field-induced reorientation transitions into modulated states have been studied and corresponding phase diagrams are constructed. Structures of magnetic defects (domain-walls and vortices) are discussed. In particular, vortices, i.e. localized non-singular line defects, are stabilized by the inhomogeneous Dzyaloshinskii-Moriya interactions in uniaxial noncentrosymmetric antiferromagnets.Comment: 18 pages RevTeX4, 13 figure

Colossal Magnetoresistance in the Mn2+ Oxypnictides NdMnAsO1-xFx

Colossal magnetoresistance (CMR) is a rare phenomenon in which the electronic resistivity of a material can be decreased by orders of magnitude upon application of a magnetic field. Such an effect could be the basis of the next generation of magnetic memory devices. Here we report CMR in the antiferromagnetic oxypnictide NdMnAsO1-xFx as a result of competition between an antiferromagnetic insulating phase with strong electron correlations and a paramagnetic semiconductor upon application of a magnetic field. The discovery of CMR in antiferromagnetic Mn2+ oxypnictide materials could open up an array of materials for further investigation and optimisation for technological applications

Quantum analogue of the spin-flop transition for a spin pair

Quantum (step-like) magnetization curves are studies for a spin pair with antiferromagnetic coupling in the presence of a magnetic field parallel to the easy axis of the magnetic anisotropy. The consideration is done both analytically and numerically for a wide range of the anisotropy constants and spins up to $S \gtrsim 100$. Depending on the origin of the anisotropy (exchange or single-ion), the magnetization curve can demonstrate the jumps more than unity and the concentration of the unit jumps in a narrow range of the field. We also point the region of the problem parameters, where the behavior is quasiclassical for $S = 5$, and where system is substantially quantum in the limit $S \to \infty$.Comment: 5 pages, 5 figure