Approximate solution of u'=-A2u using a relation between exp(-tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u(t) = -A(2)u. Using a representation of the semigroup exp(-A(2)t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w(t) = W-yy, with initial values v solving the initial value problem for v(y) = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2(nd) order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4(th) order equation for u to that of the 2(nd) order equation for v, followed by the solution of the heat equation in one space variable.

Relation between part position and kinematic freedom: An expository survey

This paper makes explicit the relation between relative part position and kinematic freedom of the parts which is implicitly available in the literature. An extensive set of representative papers in the areas of assembly and kinematic modelling is reviewed to specifically identify how the ideas in the two areas are related and influencing the development of each other. The papers are categorised by the approaches followed in the specification, representation, and solution of the part relations. It is observed that the extent of the part geometry is not respected in modelling schemes and as a result, the causal flow of events (proximity–contact–mobility) during the assembling process is not realised in the existing modelling paradigms, which are focusing on either the relative positioning problem or the relative motion problem. Though an assembly is a static description of part configuration, achievement of this configuration requires availability of relative motion for bringing parts together during the assembly process. On the other hand, the kinematic freedom of a part depends on the nature of contacting regions with other parts in its static configuration. These two problems are thus related through the contact geometry. The chronology of the approaches that significantly contributed to the development of the subject is also included in the paper.

Relation between Local Structure and Glass Forming Ability of Liquid Alloys

An attempt has been made to describe the glass forming ability (GFA) of liquid alloys, using the concepts of the short range order (SRO) and middle range order (MRO) characterizing the liquid structure.A new approach to obtain good GFA of liquid alloys is based on the following four main factors: (1) formation of new SRO and competitive correlation with two or more kinds of SROs for crystallization, (2) stabilization of dense random packing by interaction between different types of SRO, (3) formation of stable cluster (SC) or middle range order (MRO) by harmonious coupling of SROs, and (4) difference between SRO characterizing the liquid structure and the near-neighbor environment in the corresponding equilibrium crystalline phases. The atomic volume mismatch estimated from the cube of the atomic radius was found to be a close relation with the minimum solute concentration for glass formation. This empirical guideline enables us to provide the optimum solute concentration for good GFA in some ternary alloys. Model structures, denoted by Bernal type and the Chemical Order type, were again tested in the novel description for the glass structure as a function of solute concentration. We illustrated the related energetics of the completion between crystal embryo and different types of SRO. Recent systematic measurements also provide that thermal diffusivity of alloys in the liquid state may be a good indicator of their GFA.

XXVI. The relation between thermo-optic and piezo-optic phenomena in crystals

It is pointed out that the change in refractive index with temperature of a crystal is different from what is calculated from the accompanying change in volume and the piezo-optic coefficients. The difference, which is a pure temperature effect, is explained as being due to the change in polarizability of the atoms produced by a change in the amplitude of vibration. The polarizability (α) can be expanded as a Taylor series in the changes of the distance (r) between the atoms and it is found that while the piezo-optic coefficient depends only on ∂α/∂r, the pure temperature effect is a function of ∂ 2 a/∂r 2. Making use of the experimental data, the values of a and its first two derivatives can be determined. These values are foundto be of the same order as those deduced from the intensities of Rayleigh and Raman scattering of light. The theory predicts that dn/dT should vary as the coefficient of cubical expansion at different temperatures and this is verified to be true. Finally, calculations are made of the thermo- and piezo-optic coefficients, considering the electrostatic interaction between the atoms. These do not adequately explain the observed facts, since no provision is made for the distortion of electron atmospheres around the atoms and the consequent changes in polarizability.

On the relation between K-means and PLSA

Non-negative matrix factorization [5](NMF) is a well known tool for unsupervised machine learning. It can be viewed as a generalization of the K-means clustering, Expectation Maximization based clustering and aspect modeling by Probabilistic Latent Semantic Analysis (PLSA). Specifically PLSA is related to NMF with KL-divergence objective function. Further it is shown that K-means clustering is a special case of NMF with matrix L2 norm based error function. In this paper our objective is to analyze the relation between K-means clustering and PLSA by examining the KL-divergence function and matrix L2 norm based error function.

Establishing a Relation between the Mass and the Spin of Stellar-Mass Black Holes

Stellar mass black holes (SMBHs), forming by the core collapse of very massive, rapidly rotating stars, are expected to exhibit a high density accretion disk around them developed from the spinning mantle of the collapsing star. A wide class of such disks, due to their high density and temperature, are effective emitters of neutrinos and hence called neutrino cooled disks. Tracking the physics relating the observed (neutrino) luminosity to the mass, spin of black holes (BHs) and the accretion rate ((M) over dot) of such disks, here we establish a correlation between the spin and mass of SMBHs at their formation stage. Our work shows that spinning BHs are more massive than nonspinning BHs for a given (M) over dot. However, slowly spinning BHs can turn out to be more massive than spinning BHs if (M) over dot at their formation stage was higher compared to faster spinning BHs.

Relation Between the Diffusivity, Viscosity, and Ionic Radius of LiCl in Water, Methanol, and Ethylene Glycol: A Molecular Dynamics Simulation

A molecular dynamics (MD) investigation of LiCl in water, methanol, and ethylene glycol (EG) at 298 K is reported. Several; structural and dynamical properties of the ions as well as the solvent such as self-diffusivity, radial distribution functions, void and neck distributions, velocity autocorrelation functions, and mean residence times of solvent in the first solvation shell have been computed. The results show that the reciprocal relationship between the self-diffusivity of the ions and the viscosity is valid in almost all solvents with the exception of water. From an analysis of radial distribution functions and coordination numbers the nature of hydrogen bonding within the solvent and its influence on the void and neck distribution becomes evident. It is seen that the solvent solvent interaction is important in EG while solute solvent interactions dominate in water and methanol. From Voronoi tessellation, it is seen that the voids and necks within methanol are larger as compared to those within water or EG. On the basis of the void and neck distributions obtained from MD simulations and literature experimental data of limiting ion conductivity for various ions of different sizes we show that there is a relation between the void and neck radius on e one hand and dependence of conductivity on the ionic radius on the other. It is shown that the presence of large diameter voids and necks in methanol is responsible for maximum in limiting ion conductivity (lambda(0)) of TMA(+), while in water in EG, the maximum is seen for Rb+. In the case of monovalent anions, maximum in lambda(0) as a function ionic radius is seen for Br- in water EG but for the larger ClO4- ion in methanol. The relation between the void and neck distribution and the variation in lambda(0) with ionic radius arises via the Levitation effect which is discussed. These studies show the importance of the solvent structure and the associated void structure.

The relation between polarisation and electric field for trissarcosine calcium chloride

Measurement of the relation between polarisation and electric field for ferroelectric trissarcosine calcium chloride (TSCC) was made in the pressure range up to 6 kbar. The pressure dependence of the spontaneous polarisation and the coercive field were obtained, and the existence of a new pressure-induced phase and the paraelectric- ferroelectric-new phase triple point were found.

Relation between high-tc superconductivity and structure - a study of la3-xba3+xcu6o14+delta and yba2cu3o7-delta systems

Superconductivity is found in tetragonal La3−x Ba3+x Cu6O14+δ and La, Ba)6−x Sr x Cu6O14+δ even though they do not possess Cu-O chains or the K2NiF4 structure. Resistivity measurements confirm the occurrence of a transformation from chain-superconductivity to sheet-superconductivity in YBa2Cu3O7−δ as δ is varied in the range 0.0–0.5.

Relation between Chandrasekhar's X-function and the eigenvalues of integral operators with difference kernels

Abstract is not available.

On the paradoxical relation between the melting temperature and forbidden energy gap of nanoparticles

We comment on the paradox that seems to exist about a correlation between the size-dependent melting temperature and the forbidden energy gap of nanoparticles. By analyzing the reported expressions for the melting temperature and the band gap of nanoparticles, we conclude that there exists a relation between these two physical quantities. However, the variations of these two quantities with size for semiconductors are different from that of metals. (C) 2010 American Institute of Physics.[doi:10.1063/1.3466920].

Relation between Tc and hole concentration in superconducting cuprates

The superconducting transition temperature, Tc, of several series of cuprates shows a nonlinear dependence on the hole concentration, nh, determined by chemical titrations. The tc becomes maximum when nh is in the 0.12-0.15 range in cuprates containing a single Cu-O layer and around 0.2 in cuprates containing two Cu-O layers.