Vector Dependence Research Articles

Differential equations with linear algebra

1. First-Order Differential Equations. Differential Equations and Mathematical Model. Integrals as General and Particular Solutions. Direction Fields and Solution Curves. Separable Equations and Applications. Linear First-Order Equations. Substitution Methods and Exact Equations. 2. Mathematical Models and Numerical Methods. Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method. The Runge-Kutta Method. 3. Linear Systems and Matrices. Introduction to Linear Systems. Matrices and Gaussian Elimination. Reduced Row-Echelon Matrices. Matrix Operations. Inverses of Matrices. Determinants. Linear Equations and Curve Fitting. 4. Vector Spaces. The Vector Space R 3. The Vector Space Rn and Subspaces. Linear Combinations and Independence of Vectors. Bases and Dimension for Vector Spaces. General Vector Spaces. 5. Linear Equations of Higher Order. Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Undetermined Coefficients and Variation of Parameters. Forced Oscillations and Resonance. 6. Eigenvalues and Eigenvectors. Introduction to Eigenvalues. Diagonalization of Matrices. Applications Involving Powers of Matrices. 7. Linear Systems of Differential Equations. First-Order Systems and Applications. Matrices and Linear Systems. The Eigenvalue Method for Linear Systems. Second-Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Numerical Methods for Systems. 8. Matrix Exponential Methods. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems. Spectral Decomposition Methods. 9. Nonlinear Systems and Phenomena. Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. 10. Laplace Transform Methods. Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. 11. Power Series Methods. Introduction and Review of Power Series. Power Series Solutions. Frobenius Series Solutions. Bessel's Equation. References for Further Study. Appendix A: The Existence and Uniqueness of Solutions. Appendix B: Theory of Determinants. Answers to Selected Problems. Index.

Stress-strain behavior and geometrical properties of packings of elongated particles

We present a numerical analysis of the effect of particle elongation on the quasistatic behavior of sheared granular media by means of the contact dynamics method. The particle shapes are rounded-cap rectangles characterized by their elongation. The macroscopic and microstructural properties of several packings subjected to biaxial compression are analyzed as a function of particle elongation. We find that the shear strength is an increasing linear function of elongation. Performing an additive decomposition of the stress tensor based on a harmonic approximation of the angular dependence of branch vectors, contact normals, and forces, we show that the increasing mobilization of friction force and the associated anisotropy are key effects of particle elongation. These effects are correlated with partial nematic ordering of the particles which tend to be oriented perpendicular to the major principal stress direction and form side-to-side contacts. However, the force transmission is found to be mainly guided by cap-to-side contacts, which represent the largest fraction of contacts for the most elongated particles. Another interesting finding is that, in contrast to shear strength, the solid fraction first increases with particle elongation but declines as the particles become more elongated. It is also remarkable that the coordination number does not follow this trend so that the packings of more elongated particles are looser but more strongly connected.

Aggregation Behavior of Triple Helical Polysaccharide with Low Molecular Weight in Diluted Aqueous Solution

It has been proven in our previous work that the lentinan triple helical chains with high weight-average molecular weight (M(w) = 1.71 x 10(6)) formed easily a self-entangle one, and no ordered aggregates were detected. In the present work, we used the ultrasonic method to degrade the lentinan triple helical chains to obtain a sample with a mean value of M(w) approximately 5.0 x 10(5) g/mol. Subsequently, its dilute aqueous solution properties were studied by dynamic light scattering (DLS). The relaxation time distributions exhibited two modes (fast and slow) with different relaxation time scales. The fast mode was attributed to the relaxation of an individual triple helical lentinan, whereas the slow mode indicated the formation of large aggregates. On the basis of the scattering wave vector dependencies for the scattering intensity and for the amplitudes and characteristic times associated with the relaxation modes, the molecular parameters were calculated by combing static LS with DLS. The values of the radius of gyration for individuals (R(g)(indi)) and aggregates (R(g)(agg)) were 48.2 and 75.4 nm, and those of the hydrodynamic radius for individuals (R(h)(indi)) and aggregates (R(h)(agg)) were 14.9 and 98.4 nm, respectively. Furthermore, the structure-sensitive dimensionless parameter of individuals (rho(indi) = 3.23) and aggregates (rho(agg) = 0.766) indicated that the individual triple helical chains were stiff, whereas the aggregates existed as compact clusters. The aggregates consisted of short triple helical chains by packing close to form "faggot-like" assembly, and the ordered aggregates (near 10%) coexisted with the predominant triple helical chain in the aqueous solution. Atomic force microscopy provided straightforward evidence on the shape of the triple helical chains and their aggregates in water.

Resolution Dependence of Singular Vectors Computed for Typhoon SINLAKU

To investigate the resolution dependence of singular vector (SV) for Typhoon SINLAKU in the recurvature stage, SVs were computed using the Japan Meteorological Agency (JMA) tangent linear and adjoint (hereafter TL/AD) model at three different resolutions of TL63L601, TL95L60 and TL159L60.The results show that depending on the horizontal resolution, SVs have their own peak energy height and spectral shifts from the initial to final time due to the difference of detectable influences from mid-latitude troughs and tropical cyclone (TC) surrounding flows. On the other hand, common features of wind fields are found among the three SV structures. As for the SV using the TL/AD model at a TL159L60 resolution, the 48 hour optimization time interval (OTI) provides a larger influence from mid-latitude troughs with higher peak energy height than in the case with 24 hour OTI.Those results suggest that in the present case of a recurving TC using dry SV with a dry total energy (TE) norm, a certain degree of resolution in the TL/AD model is required to effectively detect sensitive areas around the TC because the resolution can change the energy balance among the mechanisms contributing to the SV structure.1 We use T as an abbreviation for triangular truncation with Gaussian grid, TL for triangular truncation with linear Gaussian grid and L for vertical layers. Therefore, TL63L60 denotes spectral triangular truncation at 63 wave numbers with linear Gaussian grid and 60 vertical layers.

Microstructural regimes of colloidal rod suspensions, gels, and glasses

We review the diverse range of materials made up of rod-shaped colloids. A common feature of such suspensions is the strong and efficient contribution of rods to the material's solid-like rheological properties such as elastic modulus and yield stress. Colloidal rod suspensions span from biomaterials such as f-actin and fd virus to inorganic materials such as boehmite and hematite, and to commercial fibers such as cellulose. We argue that, depending on the strength of pair potential interactions, such rod suspensions form microstructures that vary between the two limits of heterogeneous fractal clusters and homogeneous fiber networks. The volume fraction range for transition between these two limiting cases is strongly aspect ratio dependent. The two limiting microstructures can be distinguished by differences in the scattering vector dependence of their structure factors, as long as the range of scattering vector probed is sufficient to span regimes both above and below qL ≈ 1. Here q is the scattering vector and L is the rod length. Theories of the Brownian dynamics of fractal clusters and fiber networks show that the two types of microstructure can lead to the arrested dynamics of gelation and the glass transition, respectively. The volume fraction and aspect ratio dependences of the dynamical slowing down of these two cases differ significantly; we suggest that probing these differences is a convenient way to distinguish between gels and glasses of colloidal rods. This distinction is important to clarify because these microstructures are determinants of rheological properties such as elasticity and yielding. By combining these structural and dynamical ideas about rods, we classify a set of literature measurements on more than fifteen different colloidal materials and thereby distinguish between regimes of gelation and vitrification. We conclude by suggesting directions for future research in the arrested dynamics, the non-linear rheology, and the absolute lower limit of gelation in colloidal rod suspensions.

Dynamics of apoB100-Containing Lipoproteins Determined by Incoherent Elastic Neutron Scattering

Apolipoprotein B100 (apoB100)-containing lipoproteins (very low density lipoprotein (VLDL) and low density lipoprotein (LDL)) are the principal fat and cholesterol carriers in blood. During metabolic conversion from VLDL to LDL, the particle size decreases (from ∼ 80 nm to 20 nm) and lipid composition is changed, however, the amphiphilic apoB100 molecule remains bound to its lipoprotein particle and most likely compensates for structural changes due to its inherent conformational flexibility and dynamics .Here, we report on motions in the time range of 100 ps to 1 ns in human-LDL, human VLDL and yolk-VLDL, which were recorded by elastic neutron-scattering temperature scans from 20K to 310 K using hydrated lipoprotein powders. The mean square displacement values were calculated from the scattering vector dependence of the elastic intensity I(Q). The effective force constants , which are a measure for the resilience of the particle, were derived from the slopes in the vs. T scans. In the low-temperature harmonic regime we found no substantial differences between lipoprotein fractions ( ∼1 N/m). Nevertheless, lipoproteins are softer compared to hydrated myoglobin powder (2 N/m) or purple membranes (1.7 N/m) [1]. Significant differences were observed with increasing temperatures. Both, human and yolk VLDL show two breaks in the scan with a steep increase in above 270K, whereas LDL shows a smooth behavior above a dynamic transition around 220K. Accordingly, at physiological temperatures VLDL-fractions are highly soft and mobile ( ∼0.08 N/m) as compared to LDL ( ∼ 0.2 N/m). Sucrose, added as cryoprotectant, significantly modified the dynamics of VLDL, as it confers extreme stability to VLDL over the whole temperature range and substantially suppresses dynamic transitions.[1] G. Zaccai, Science 288 (2000), 1604-1607

Magnetic octupole order in Ce0.7La0.3B6: A polarized neutron diffraction study

Abstract Recently, in phase IV of Ce x La 1 - x B 6 , weak but distinct superlattice reflections from the order parameter of phase IV have been detected by our unpolarized neutron scattering experiment [K. Kuwahara, K. Iwasa, M. Kohgi, N. Aso, M. Sera, F. Iga, J. Phys. Soc. Japan 76 (2007) 093702]. The scattering vector dependence of the intensity of superlattice reflections is quite unusual; the intensity is stronger for high scattering vectors. This result strongly indicates that the order parameter of phase IV is the magnetic octupole. However, the possibility that the observed superlattice reflections are due to lattice distortions could not be completely ruled out only on the basis of the unpolarized neutron scattering experiment. To confirm that the superlattice reflections are magnetic, therefore, we have performed a single crystal polarized neutron diffraction experiment on Ce 0.7 La 0.3 B 6 . The obtained result has clearly shown that the time reversal symmetry is broken by the order parameter of phase IV. This is further evidence for the magnetic octupole order in Ce x La 1 - x B 6 .

Spin Dynamics in Pyrochlore Heisenberg Antiferromagnets

We study the low temperature dynamics of the classical Heisenberg antiferromagnet with nearest neighbor interaction on the frustrated pyrochlore lattice. We present extensive results for the wave vector and frequency dependence of the dynamical structure factor, obtained from simulations of the precessional dynamics. We also construct a solvable stochastic model for dynamics with conserved magnetization, which accurately reproduces most features of the precessional results. Spin correlations relax at a rate independent of the wave vector and proportional to the temperature.

Quasiparticles at the Mott Transition inV2O3: Wave Vector Dependence and Surface Attenuation

We present an angle resolved photoemission study of V2O3, a prototype system for the observation of Mott transitions in correlated materials. We show that the spectral features corresponding to the quasiparticle peak in the metallic phase present a marked wave vector dependence, with a stronger intensity along the GammaZ direction. The analysis of their intensity for different probing depths shows the existence of a characteristic length scale for the attenuation of coherent electronic excitations at the surface. This length scale, which is larger than the thickness of the surface region as normally defined for noncorrelated electronic states, is found to increase when approaching the Mott transition. These results are in agreement with the behavior of quasiparticles at surfaces as predicted by Borghi et al.

Three-point susceptibilitiesχn(k;t)andχns(k;t): Mode-coupling approximation

Recently, it was argued that a three-point susceptibility equal to the density derivative of the intermediate scattering function, chi{n}(k;t)=dF(k;t)dn , enters into an expression for the divergent part of an integrated four-point dynamic density correlation function of a colloidal suspension [Berthier, J. Chem. Phys. 126, 184503 (2007)]. We show that, within the mode-coupling theory, the equation of motion for chi{n}(k;t) is essentially identical as the equation of motion for the q-->0 limit of the three-point susceptibility chi{q}(k;t) introduced by Biroli [Phys. Rev. Lett. 97, 195701 (2006)]. We present a numerical solution of the equation of motion for chi{n}(k;t) . We also derive and numerically solve an equation of motion for the density derivative of the self-intermediate scattering function, chi_{n};{s}(k;t)=dF;{s}(k;t)dn . We contrast the wave vector dependence of chi{n}(k;t) and chi{n};{s}(k;t) .

Motional Coherence in Fluid Phospholipid Membranes

We report a high energy-resolution neutron backscattering study, combined with in situ diffraction, to investigate slow molecular motions on nanosecond time scales in the fluid phase of phospholipid bilayers of 1,2-dimyristoyl-sn-glycero-3-phoshatidylcholine. A cooperative structural relaxation process was observed. From the in-plane scattering vector dependence of the relaxation rates in hydrogenated and deuterated samples, combined with results from a 0.1 micros long all-atom molecular dynamics simulation, it is concluded that correlated dynamics in lipid membranes occurs over several lipid distances, spanning a time interval from pico- to nanoseconds.

Independence of vectors in codes over rings

We study codes over Frobenius rings. We describe Frobenius rings via an isomorphism to the product of local Frobenius rings and use this decomposition to describe an analog of linear independence. Special attention is given to codes over principal ideal rings and a basis for codes over principal ideal rings is defined. We prove that a basis exists for any code over a principal ideal ring and that any two basis have the same number of vectors.

Dispersion Relation of an OH-Stretching Vibration from Inelastic X-Ray Scattering

We show that recent advances now allow us to measure the wave vector dependence of OH-stretching frequencies at energies around 400 meV by inelastic x-ray scattering using ID28@ESRF. We found a large, unexpected dispersion when we measured the dispersion relations of the hydrogen stretching frequencies of diaspore, alpha-AlOOH, where the hydrogen atoms participate in a hydrogen bond of intermediate strength. We can account for this behavior with density functional perturbation theory calculations and a simple model based on H-H interactions.

The wave surveyor technique for fast plasma wave detection in multi-spacecraft data

Abstract. Multi-satellite missions like Cluster allow to study the full spatio-temporal variability of plasma processes in near-Earth space, and both the frequency and the wave vector dependence of dispersion relations can be reconstructed. Existing wave analysis methods include high-resolution beamformers like the wave telescope or k-filtering technique, and the phase differencing approach that combines the correlations measured at pairs of sensors of the spacecraft array. In this paper, we make use of the eigendecomposition of the cross spectral density matrix to construct a direct wave identification method that we choose to call the wave surveyor technique. The analysis scheme extracts only the dominant wave mode but is much faster to apply than existing techniques, hence it is expected to ease survey-type detection of waves in large data sets. The wave surveyor technique is demonstrated by means of synthetic data, and is also applied to Cluster magnetometer measurements.

Competition between charge and spin order in thet−U−Vextended Hubbard model on the triangular lattice

Several new classes of compounds can be modeled in first approximation by electrons on the triangular lattice that interact through on-site repulsion $U$ as well as nearest-neighbor repulsion $V$. This extended Hubbard model on a triangular lattice has been studied mostly in the strong coupling limit for only a few types of instabilities. Using the extended two-particle self-consistent approach (ETPSC), that is valid at weak to intermediate coupling, we present an unbiased study of the density and interaction dependent crossover diagram for spin- and charge-density wave instabilities of the normal state at arbitrary wave vector. When $U$ dominates over $V$ and electron filling is large, instabilities are chiefly in the spin sector and are controlled mostly by Fermi surface properties. Increasing $V$ eventually leads to charge instabilities. In the latter case, it is mostly the wave vector dependence of the vertex that determines the wave vector of the instability rather than Fermi surface properties. At small filling, nontrivial instabilities appear only beyond the weak coupling limit. There again, charge-density wave instabilities are favored over a wide range of dopings by large $V$ at wave vectors corresponding to $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ superlattice in real space. Commensurate fillings do not play a special role for this instability. Increasing $U$ leads to competition with ferromagnetism. At negative values of $U$ or $V$, neglecting superconducting fluctuations, one finds that charge instabilities are favored. In general, the crossover diagram presents a rich variety of instabilities. We also show that thermal charge-density wave fluctuations in the renormalized-classical regime can open a pseudogap in the single-particle spectral weight, just as spin or superconducting fluctuations.

S -matrix calculation of radiation characteristics from dipole oscillation in two-dimensional photonic crystal slabs

The effects of the photonic band structure on the radiation from a dipole moment placed within photonic crystal (PC) slabs are investigated. Radiation above the slab surface and coupling with the slab’s propagating mode are calculated for frequencies above and below the light line of air cladding. The strongly enhanced emission from the surface and coupling with propagating modes and its polarization vector dependence are explained in terms of the photonic band of the PC slab structures. The emitted electromagnetic field is mainly extracted through coupling with the leaky modes of PC slabs.

Diffraction and electron energy loss to plasmons in silicon slabs

Dynamical diffraction patterns were calculated for $25\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ slabs of silicon with [001], [111], and [110] faces for a $120\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$ electron beam. The calculation used the mixed dynamical form factor in the dielectric formulation. Dielectric matrices with wave vector and frequency dependence were calculated within the local density approximation using the random phase approximation. The energy losses, $10--25\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, span the plasmon peak. Near the zone axes, the results show the preservation of elastic contrast and both excess and deficit Kikuchi lines.

Small-angle neutron scattering measurement of silicon nanoparticle size

We have determined the particle size distribution profiles of octane-terminated siliconnanoparticle suspensions, produced using the sonication of electrochemically etchedSi wafers. Small-angle neutron scattering data was analyzed separately in high(0.4 nm−1<q<3.0 nm−1) and low(q<0.4 nm−1) scattering vectorranges. Data in the high q range is consistent with the log-normal distribution of isolated spherical particles with median diameterd = 3 ± 0.2 nm.Particle sizes were also indirectly assessed from photoluminescenceand optical transmission spectroscopy using the size/bandgap relation:Eg = 3.44d−0.5, whereEg is ineV and d in nm. Both measurements were consistent with the particle size distribution profiles,estimated from ANS data fitting and TEM image analysis.A subpopulation of larger, irregular shape structures in the sizerange 10–50 nm was also indicated by neutron scattering in the lowq range and HRTEM images. However, further studies are warranted toexplain a relationship between the slope of scattering intensity versusscattering vector dependence in the intermediate scattering vector range(0.4 nm−1<q<1.0 nm−1) and the role of non-geometrical Si nanoparticle characteristics (mutual interaction forces,surface termination, etc).

From cooperative paramagnetism to Néel order inY2Ru2O7: Neutron scattering measurements

Spin correlations in the pyrochlore antiferromagnet Y_2Ru_2O_7 with Curie-Weiss temperature $\Theta_{CW}=-1100$ K and critical temperature T_N=77 K were examined through neutron scattering. For $T_N<T<\Theta_{CW}/3$ the data show spin relaxation with a rate $\hbar\Gamma= 1.17(9)k_BT$. For T<T_N spectral weight moves to higher energies with substantial changes up to $4\times k_BT_N$. For T<<T_N there is a $\Delta=11(1)$ meV energy gap and a pronounced spectral maximum at 19.7 meV. Throughout the temperature range examined the wave vector dependence of inelastic scattering exhibits a broad peak for $Qd\approx 3.8$ (d is Ru-Ru spacing) consistent with dipolar spin correlations.

Properties of creation and reduction according to the equinumber principle for adaptive vector quantisation

For adaptive vector quantisation, properties of creation and reduction according to the equinumber principle are presented. The equinumber principle is that, partition errors are mutually equivalent when the number of inputs in a partition space is mutually equal, and average distortion is asymptotically minimised. Then creation and reduction of adaptive vector quantisation have been introduced to avoid the initial dependence of reference vectors. In creation, output units are sequentially created according to the equinumber principle in the learning process to reach a predetermined number of units. In reduction, output units are sequentially deleted according to the equinumber principle to reach the prespecified number. A novel algorithm is unified by creation and reduction, in which reduction is carried out after creation is conducted. Experimental results show the properties of our approach. Furthermore our approach is applied to image data and the practicability is confirmed for image coding.