A sequence of two ferrotoroidal phase transitions in nickel-bromine boracite Ni3B7O13Br

The phenomenological theory of a sequence of two second-order phase transitions in Ni-Br boracite is presented. Two different components of the toroidal moment vector T _i are the order parameters of these transitions. Expressions are derived for the temperature dependences of the spontaneous values of T _i , polarization P _i , and magnetization M _i and the dielectric χ_ij=dP _i /dE _j , magnetic k _ij =dM _i /H _j , and magnetoelectric α _ij=dP _i /dH _j =dM _j /dE _i susceptibilities. Some of these susceptibilities display sharp temperature peaks in the vicinity of phase transitions.     

A phenomenological theory of possible sequences of ferrotoroidal phase transitions in boracites

A phenomenological theory of the sequence of two second-order phase transitions with close temperatures is considered; such transitions occur in the Ni-Br boracite. The thermodynamic potential is written as a function of polarization P _i, magnetization M _i, and toroidal moment T _i vectors and fields E _i and H _i; T _i is treated as an order parameter. It is assumed that only one coefficient of T _i ² passes through zero as T decreases. The possibility of a sequence of two proper ferrotoroidal phase transitions along the T ₁ and T ₂ components is demonstrated. Spontaneous T _i, P _i, and M _i vector values and equations for susceptibility tensors (dielectric χ _ij = dP _i/dE _j, magnetic k _ij = dM _i/dH _j, and magnetoelectric α_ij = dP _i/dH _j = dM _j/dE _i) were obtained for three phases. Some of these values have well-defined anomalies in the vicinity of transitions. All possible sequences of ferrotoroidal phase transitions in boracites are considered. Depending on two potential coefficient values, these sequences may consist of one, two, or three such transitions.     

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field F_ijand the current density four-vector Jⁱ, are invariant under the change of the metric g_ijby g_ij = g_ij+ l_il_jif l_iis a principal null direction of F_ijand that an analogous result holds in the case of the massless Klein-Gordon equation if l_iis null and orthogonal to the gradient of the field and in the case of the null dust equations if l_iis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (g_ij, F_ij) be an exact solution of the Einstein-Maxwell equations and let l_ibe a principal null direction of F_ij, then (g_ij+ l_il_j, F_ij) is also an exact solution of the Einstein-Maxwell equations if and only if (l_il_j, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (g_ij, F_ij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.     

Application of CW local correction approach to SLW modeling of radiative transfer in non-isothermal gaseous media

The solution of the RTE equation in non-isothermal media by the SLW method is handled with the help of the local correction factor instead of the traditionally used so-called reference approach. Two methods of calculation of the correction factor are shown: one is based on the cumulative wavenumber (CW) distribution function developed with the HITEMP spectroscopic data; the other method uses the relationship between the reordered wavenumber and cumulative wavenumber. It is shown how this method is applied to different forms of the partition of absorption cross-section into gray gases. Further, it is shown theoretically that this method approaches the exact solution for non-isothermal media in the limit as the number of gray gases tends to infinity.     

Moments of a Single Entry of Circular Orthogonal Ensembles and Weingarten Calculus

Consider a symmetric unitary random matrix V = (v _ij )_{1 ≤ i, j ≤ N} from a circular orthogonal ensemble. In this paper, we study moments of a single entry v _ij . For a diagonal entry v _ii , we give the explicit values of the moments, and for an off-diagonal entry v _ij , we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size N. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.     

Pressure dependence of elastic constants of orthorhombic Li2Ge7O15 above and below the ferroelectric transition

The pressure derivatives of the elastic constants c_ij of orthorhombic Li₂Ge₇O₁₅ have been determined at 293 K by the method of pressure-induced shifts of resonance frequencies of thick plates at ca. 15 MHz in the range between 0 and 1500 bar. Approaching the transition at ca. 630 bar, all P_ij = dc_ij/dp (i, j = 1, 2, 3; p pressure) develop strongly negative values. At higher pressures a similar behaviour with reversed sign is observed. The pressure derivatives of the pure “shear resistances” c₄₄, c₅₅, and c₆₆ depend only slightly on pressure even in the vicinity of the transition. The main interactions driving the transition are of the totally symmetric type. The values dK^?1/dp (K volume compressibility) deviate strongly from the quasi-invariant value of ca. 5 observed in almost all stable crystals (dK^?1/dp = ? 1750 at 620 bar and 1380 at 700 bar). The anomalous piezoelastic behaviour reflects the anomalous thermoelastic behaviour: negative P_ij in the low pressure (high temperature) phase correspond to positive T_ij = d log c_ij/dT (T temperature) and vice versa.     

Note on conservation laws based upon traceless energy-momentum tensors in flat space-time

In this note we prove the following theorem. If in a flat space-time with metric g_ij(x) treferred to general coordinates xⁱ a vector ξⁱ(x) satisfies (Tⁱ_jξ^j)_;i=0 (semicolon denotes covariant differentiation) for all energy-momentum tensors of the set $\text{{}\text{T}{}^{\mathrm{ij}}\text{T}{}^{\mathrm{i}}{}_{\mathrm{j;i}}\text{=0;g}{}_{\mathrm{ij}}\text{T}{}^{\mathrm{ij}}\text{=0}$; T^ij = T^ji; T^iju_iu_j > 0 (where u_i is a time-like vector)}, then the vector ξⁱ defines a conformal motion. This theorem, which may be considered as a converse (in flat space-time) to a well-known result of Trautman, is a generalization of a result obtained by J. T. ?opuszański and J. Szczucka-Soko?owska [Reports on Mathematical Physics 11 (1977), 153] in which they assumed the vector ξⁱ was a polynomial in Minkowski coordinates.     

Robust output synchronization of second-order systems

A technique to synchronize a network of dynamical systems described by second-order ordinary differential equations is presented. Each system can be driven by a coupling control signal, which is synthesized such that, at steady-state, the outputs of two given systems, say y _i and y _j , i ≠ j, satisfy a specified ratio, that is, y _i /y _j = α _i /α _j , α _i ≠ 0 ≠ α _j . Among others, this includes the cases where the outputs are synchronized in-phase or anti-phase. The proposed synchronization technique is robust; this means that a small synchronization error is preserved at steady-state, even if the systems were perturbed by external disturbances. Some level of parameter uncertainty can also be tolerated. The coupling control signals are synthesized based on a classical controller and a robust observer that estimates the generalized velocities and provides an estimation of the perturbation terms. Some experimental results, showing the performance of the proposed synchronization technique, are included.     

Spectral methods and cluster structure in correlation-based networks

We investigate how in complex systems the eigenpairs of the matrices derived from the correlations of multichannel observations reflect the cluster structure of the underlying networks. For this we use daily return data from the NYSE and focus specifically on the spectral properties of weight W_ij=|C|_ij−δ_ij and diffusion matrices D_ij=W_ij/s_j−δ_ij, where C_ij is the correlation matrix and s_i=∑_jW_ij the strength of node j. The eigenvalues (and corresponding eigenvectors) of the weight matrix are ranked in descending order. As in the earlier observations, the first eigenvector stands for a measure of the market correlations. Its components are, to first approximation, equal to the strengths of the nodes and there is a second order, roughly linear, correction. The high ranking eigenvectors, excluding the highest ranking one, are usually assigned to market sectors and industrial branches. Our study shows that both for weight and diffusion matrices the eigenpair analysis is not capable of easily deducing the cluster structure of the network without a priori knowledge. In addition we have studied the clustering of stocks using the asset graph approach with and without spectrum based noise filtering. It turns out that asset graphs are quite insensitive to noise and there is no sharp percolation transition as a function of the ratio of bonds included, thus no natural threshold value for that ratio seems to exist. We suggest that these observations can be of use for other correlation based networks as well.     

Reentrant transitions in the van Hemmen model of a spin glass

The van Hemmen model of a spin glass, which is an Ising model with random couplings J_ij between sites i and j equal to J₀ + J(ξ_iη_j + ξ_jη_i), where (ξ_i, η_i) are independent, identically distributed random variables, is studied in the pair approximation of the cluster variation method. For the family of probability distributions (1 − p)δ(ξ_i − a) + pδ(ξ_i) + (1 − p)δ(ξ_i + a), where p is varied, phase diagrams are constructed. They are qualitatively different from the mean-field phase diagrams and display a competition between tendencies towards spin-glass and towards ferromagnetic ordering, which results in reentrant transitions. It is argued that the observed effects are not accidental but are borne by the competition of bonds of the underlying lattice system.     

The electromagnetic energy and momentum of finite charged bodies moving at constant velocity

It is shown that a purely electromagnetic, divergence-free tensorS ^ij , can be defined for any electrically charged body which is held in equilibrium by some cohesive force and moving at some constant velocity. This tensor appears to represent the electromagnetic energy-momentum of the body; the integral (1/c) ∝S ^ij dS _j (dS _j is the differential element of any spacelike hypersurface) iscM ₀ μ ⁱ the electromagnetic four-momentum of the system (M ₀ is the electromagnetic rest mass of the system,U _i is the four-velocity). The divergence-free property ofS ^ij depends only on Maxwell's equation and the condition of uniform motion. It is suggested that whatever the nature of the cohesive forces within such a system the total stress-energy tensor will, in effect, break up into two parts which are separately divergence-free: the purely electromagnetic tensor,S ^ij , and a tensor representing the energy-momentum of the cohesive forces. Just as it makes sense to speak of the electromagnetic mass of a system at rest without regard to the cohesive forces, it makes sense to talk about the electromagnetic momentum of the system, when it is moving at constant velocity, without reference to the cohesive forces.     

Representations of CCR Algebras in Krein Spaces of Entire Functions

Representations of CCR algebras in spaces of entire functions are classified on the basis of isomorphisms between the Heisenberg CCR algebra $\mathcal{A}_H$ and star algebras of holomorphic operators. To each representation of such algebras, satisfying a regularity and a reality condition, one can associate isomorphisms and inner products so that they become Krein star representations of $\mathcal{A}_H$ , with the gauge transformations implemented by a continuous U(1) group of Krein space isometries. Conversely, any holomorphic Krein representation of $\mathcal{A}_H$ , having the gauge transformations implemented as before and no null subrepresentation, are shown to be contained in a direct sum of the above representations. The analysis is extended to CCR algebras with [a _i , a _j ^*]=δ _{i j} η _i , η _i =±1, i=1,...,M, the infinite-dimensional case included, under a spectral condition for the implementers of the gauge transformations.     

Growing random networks with fitness

Three models of growing random networks with fitness-dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness η, with rate (k−1)+η. For η>0 we find the connectivity distribution is power law with exponent γ=〈η〉+2. In the second model (B), the network is built by connecting nodes to nodes of connectivity k, random additive fitness η and random multiplicative fitness ζ with rate ζ(k−1)+η. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness (α,β), i incoming links and j outgoing links gains a new incoming link with rate α(i+1), and a new outgoing link with rate β(j+1). The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections.     

The spectral-line moment-based (SLMB) modeling of the wide band and global blackbody-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform gaseous media

The spectral-line moment-based (SLMB) modeling is proposed for the calculation of radiative properties of gases on any spectral width. The associated mathematical formulation is obtained by applying several concepts of the k-distribution methods such as the reordering of the wavenumber scale by monotonic variations of the absorption coefficient, together with the application of the k-moment method's principles. This approach gives both a general formula for the BTF and a simple and readily applicable approximation for the blackbody-weighted cumulated k-distribution function of the absorption coefficient. The model is applied for the computation of wide band BTFs and cumulative k-distributions for uniform columns of CO₂ and H₂O in the temperature range (300-2400 K) at atmospheric pressure. Model parameters are deduced from line-by-line (LBL) spectra calculated using the HITEMP database. Comparisons with LBL reference data as well as with contemporary modeling approaches (SLW, FSK, SNB) are performed and discussed.     

Dielectric relaxation phenomena of rigid polar liquid molecules under giga hertz electric field

The dielectric relaxation phenomena of rigid polar liquid molecules chloral and ethyltrichloroacetate (j) in benzene, n-hexane and n-heptane (i) under 4.2, 9.8 and 24.6 GHz electric fields at 30°C are studied to show the possible existence of double relaxation times τ ₂ and τ ₁ for rotations of the whole and the flexible parts of molecules. The probability of showing double relaxation is more in aliphatic solvents indicating their nonrigidity. The symmetric and asymmetric distribution parameters γ and δ are obtained from X _ij ^′ /X _0ij and X _ij ^″ /X _0ij and w _j →0 where X _ij ^′ and X _ij ^″ are real and imaginary parts of the complex orientational susceptibility X _i ^* and X _0ij is the low frequency susceptibility which is real. X _ij ’s are involved with the measured dielectric relative permittivities ε _ij ^′ , ε _in ^″ , ε _0ij and ε _∞ij of solutions. The theoretical weighted contributions c ₁ and c ₂ towards dielectric dispersions by Fröhlich’s method are compared with the experimental ones obtained from the graphical variation of X _ij ^′ /X _0ij and X _ij ^″ /X _0ij with weight fractions w _j ’s at w _j → 0. The measured dipole moments μ ₂ and μ ₁ of the whole and the flexible part of a polar molecule in terms of the linear coefficients β’s of X _ij ^′ ’s with w _j ’s and the estimated τ ₂ and τ ₁ reveal their associations with aliphatic solvents. The theoretical dipole moments μ _theo’s from the available bond angles and bond moments of the substituent polar groups of the molecules with the estimated μ’s suggest the mesomeric, inductive and electromeric effects in them under GHz electric field.     

An Inhomogeneous Contact Process Model for Speciation

We propose the following model on $\mathbb{Z}^{+}$ for speciation and extinction. A species at site i gives birth to a new species at site j at rate λp(i,j) where i and j are nearest neighbors. A death at site i occurs at rate δ _i . We show that the existence of a phase transition in λ depends critically on the value of the limit of $\frac{p(n,n+1)}{\delta_{n}}$ .     

Spouge's Conjecture on Complete and Instantaneous Gelation

We investigate the stochastic counterpart of the Smoluchowski coagulation equation, namely the Marcus–Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluster in an arbitrarily small time, which is known as a complete and instantaneous gelation phenomenon. Indeed, Spouge (also Domilovskii et al. for a special case) conjectured that K(i, j)=(ij) , >1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function (i, j), increasing in both i and j such that _j=1 1/(j(i, j))< for all i, and K(i, j)ij(i, j) for all i, j, then complete and instantaneous gelation occurs. Evidently, this implies that any kernels K(i, j)ij(log(i+1)log(j+1)) , >1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim _i+j ij/K(i, j)=0 and _{i, j=1} 1/K(i, j)=, then gelation time T _g satisfies 0<T _g<. Moreover, the gelation is complete after T _g.     

Coagulation equations with gelation

Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernelK _ij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelationt _c , and give an exact postgelation solution for the modelK _ij =(ij ^ω ) (ω>1/2) andK _ij =a ^i+j (a>1). For the modelK _ij =i ^ω +j ^ω (ω<1, without gelation) initial solutions are given. It is argued that the kernelK _ij ～ij ^ω with ω?1?1/d (d is dimensionality) effectively models the sol-gel transformation in polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (Ω=1). For allΩ the exponents,t=Ω+3/2 andσ=Ω?1/2,γ=(3/2?Ω)/(Ω ? 1/2) andΒ=1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.     

A stochastic derivation of the Sivashinsky equation for the self-turbulent motion of a free particle

Within the framework of the Kershaw approach and of a hypothesis on spatial stochasticity, the relativistic equations of Lehr and Park, Guerra and Ruggiero, and Vigier for stochastic Nelson mechanics are obtained. In our model there is another set of equations of the hydrodynamical type for the drift velocityv _i(x _j,t) and stochastic velocityu _i(x _j,t) of a particle. Taking into account quadratic terms in l, the universal length, we obtain from these equations the Sivashinsky equations forv _i(x _j,t) in the caseu _i →0. In the limit l →0, these equations acquire the Newtonian form.