Remarks on the Wilson-Zimmermann expansion and some properties of them-point distribution

This paper contains a few simple remarks on a paper by S. Schlieder and E. Seiler. For the special class of local fields treated by these authors we arrive at the same necessary condition for the existence of the Wilson-Zimmermann expansion (considered both as an operator expansion and as an expansion in bilinear forms) of the product ofn real scalar fields under the assumption that the singularities occurring asx _j x _j+1;j=1, 2, ...,n–1, do not influence each other as long as these limits are simultaneously taken.This paper is part of a thesis presented to the University of Manitoba in partial fulfillment of the degree of Doctor of Philosophy.     

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.     

Reconstruction of singularities for solutions of Schrödinger's equation

We determine the behavior in time of singularities of solutions to some Schrödinger equations onR ⁿ . We assume the Hamiltonians are of the formH ₀+V, where \(H_0 = 1/2\Delta + 1/2 \sum\limits_{k = 1}^n { \omega _k^2 x_k^2 } \) , and whereV is bounded and smooth with decaying derivatives. When all ω _k =0, the kernelk(t,x,y) of exp (?itH) is smooth inx for every fixed (t,y). When all ω₁ are equal but non-zero, the initial singularity “reconstructs” at times \(t = \frac{{m\pi }}{{\omega _1 }}\) and positionsx=(?1) ^m y, just as ifV=0;k is otherwise regular. In the general case, the singular support is shown to be contained in the union of the hyperplanes \(\{ x|x_{js} = ( - 1)^l js_{y_{js} } \} \) , when ω _j t/π=l _j forj=j ₁,...,j _r .     

Unusual Pr antiferromagnetic ordering in high-Tc cuprates

The anomalous Pr antiferromagnetic order with high Néel temperature T_N(Pr) are reported throughout the whole Pr_1+xBa_2−xCu₃O_7−y or 1212-type Cu(Ba_2−xPr_x)PrCu₂O_7−y system (−0.2<x<1; −0.4<y<1), where three distinct crystal structures were observed: orthorhombic 1212-chain O(I) (space group Pmmm), tetragonal T (P4/mmm), and orthorhombic O(II) (Cmmm). Systematic variation of T_N(Pr) in this system as well as in other 1212 and 2212 cuprates M_nA₂PrCu₂O₇ (n=1, 2, M=Hg, Tl, Pb/Cu, Bi, Nb, Cu; A=Sr, Ba, Ba/Pr) was discussed through the correlation of T_N(Pr) with Pr–O bond length. The importance of quasi-two-dimensional Pr–O–Pr superexchange magnetic coupling through strong wave function overlap between the overextended Pr-4f orbital with eight O-2p_π orbitals in the adjacent CuO₂ bi-layer is discussed. No superconductivity was observed in the present study.     

Effect of elevated concentrations of strontium and iron on the structural and dielectric characteristics of Ba(1−x−y)Sr(x)Ti Fe(y)O3 prepared through sol–gel technique

Nano-composite Ba_1−xSr_(x)TiO₃ (BST), where x=0.01–0.50 and doped with different concentrations of iron Ba_(1−x−y)Sr_(x)TiFe _(y)O₃ (BSTF), where x=0.01 and y=0.01–0.05 powders were prepared by sol–gel method.     

The vertical spectrum of H2CO revisited: (SC)2-CI and CC calculations

The vertical electronic spectrum of formaldehyde has been studied by means of (SC)²-MR-SDCI and CCLR methods. Two basis sets of atomic natural orbitals (ANOs) complemented with a one-centre series of Rydberg orbitals were used. The first was taken from the CASPT2 study by Merchán, M., and Roos, B. O., 1995, Theoret. Chim. Acta, 92, 221, and may be described as C,O[4s3pld]/H[2slp] with a lslpld Rydberg series centred in the charge centroid of the ²B₂ state of the cation. The second was a larger basis set that may be described as C,O[6s5p3d2f]/H[4s3p2d] + 3s3p3d in the same centre. The (SC)² dressing may be applied efficiently to an MR-SDCI method and comparison with the dressed CAS-SDCI is satisfactory, in spite of the remarkable reduction in the CI space dimension. The consistency of the (SC)²-MR-SDCI results was tested also against the CCLR and CASPT2 results using the same basis sets and against the CCLR results using Dunning's aug- and daug-cc-pVQZ basis sets. The ³A₁(π → π *) state is correctly placed as the second excited triplet while the highly multi-configurational nature of the ¹A₁(π → π *) state is confirmed as well as its greatly mixed valence-Rydberg nature. This state is predicted as lying under the 10 eV level, on top of the (n_y → 3d) Rydberg states that are predicted in the 8.9–9.5eV region. The 5 ¹B₂(n^y → 4s) Rydberg state and the ¹B₂(σ_y → π*) also are predicted in this region. The triplet states also were calculated with the (SC)²-MR-SDCI method. The vertical ordering of the 2 ¹A₁(n_y → 3p_y) and 2 ¹B₂(n_y → 3p_z) states is discussed, as well as that of the ¹B₁(σ → π*) and the Rydberg ¹B₁(n_y → 3d_xy) states. This work shows the highly reliable values that may be reached applying the dressing method along with a large basis set. Such a procedure is made possible using an MR-SDCI selection of spaces instead of the CAS-SDCI that was used up to now in most (SC)² dressing applications.     

Johnson's SB distribution function as applied in the mathematical representation of particle size distributions. Part 1: Theoretical background and numerical simulation

An investigation was carried out of the transformation between the number, length, surface and volume size distributions expressed by Johnson's S_B distribution function – the bounded log-normal distribution function. As is well known, if any of the number, length, surface and volume distributions is log-normal, all the others will also be log-normal. Theoretical analysis suggests that the S_B function may have a similar property. This was confirmed by a computer-aided numerical simulation, in which emphasis was given to the transformation between successive order size distributions, i.e. ?_i(x) → ?_{i + 1}(x) or ?_i(x) → ?_{i ? 1}(x). The numerical results can be applied to the particle size distribution transformation because this transformation can generally be made step by step, for example, ?_i → ?_i?1 (x) → ?_{i ? 2}(x) → … → ?_j(x) for ?_i(x) → ?_j(x) ( i > j).     

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

The length and the relative length of tangent vectors of finsler spaces

We consider the length of a vector in a Finsler space with the fundamental function L(x,y). The length of a vector X is usually defined as the value L(x,X) of L. On the other hand, we have an essential tensor g_ij(x,y), called the fundamental tensor, and the concept of relative length |X_y| of X may be introduced by |X|_y^y = g_ij(x,y)XⁱX^j with re spect to a supporting element y. The question arises whether is L(x,X) the minimum of |X|_y or not? If there exists a supporting element y satisfying |X|_y < L(x,X), then a curve x(t) in the Finsler space will be measured shorter than the usual length, by integrating |dx/dt|_y with the field of such supporting element y(t) along the curve.     

The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period⩽2 for a nonlinear transformation

We prove that the one-site distribution of Gibbs states (for any finite spin setS) on the Bethe lattice is given by the points satisfying the equation π=T ²π, whereT=h·A·?, with?(x)=x ^(q?1/q,h(x)=(x∥x∥ _q ) ^q ,A=(a(r, s)∶r, s∈S), and $$a(r,s) = \exp (K[r,s] + (1/q)[N,r + s])$$ We also show that forA a symmetric, irreducible operator the nonlinear evolution on probability vectorsx(n+1)=Ax(n) ^p ∥Ax(n) ^p ∥₁ withp>0 has limit pointsξ of period?2. We show thatA positive definite implies limit points are fixed points that satisfy the equationAξ ^p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

Eine scheinbare Abschwächung der Lokalitätsbedingung

For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following condition: For any two natural numbersn andj with 1≦j<n and for any configurationX(n, j):X ₁, ...,X _j?1,X _j ,X _j+1,X _j+2, ...X _n that is totally space-like in both orders: 1, ...,j?1,j, j+1,j+2, ...,n and 1, ...,j?,j+1,j,j + 2, ...,n there exist constants α(n,j)>2,C(X(n, j))>0,h(X(n, j))>0 such that with \(x_k = X_k \sqrt { - x^2 } \) : $$\begin{gathered} |\langle A(x_1 ) \ldots A(x_{j - 1} )[A(x_j ), A(x_{j + 1} )] A(x_{j + 2} ) \ldots A(x_n )\rangle |< \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,< C(X(n, j)) exp\{ - h(X(n, j))\sqrt { - x^2 } ^{\alpha (n, j)} \} \hfill \\ \end{gathered} $$ for ?x ²>1.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

The ground state of two-dimensional electrons in a nonuniform magnetic field

An exact solution to the Schrödinger equation for the ground state of two-dimensional Pauli electrons in a nonuniform transverse magnetic field H is presented for two cases. In the first case, the field H depends on a single variable, H = H(y), while in the second case, the field is axially symmetric, H = H(ρ), ρ²=x ²+y ². The electron density distributions n = n(y) and n = n(ρ) that correspond to a completely filled lower level are found. For quasiuniform fields of fixed sign, the functions n(y) and n(ρ) are locally related to the magnetic field: n(y) = H(y)/?₀ and n(ρ) = H(ρ)/?₀, where ?₀ = hc/|e| is a magnetic flux quantum. Magnetic fields are considered that are periodic, singular, and bounded in the plane xy. Finite electron objects in a nonuniform magnetic field are analyzed.     

Lie-theoretic generating relations of two variable Laguerre polynomials

Generating relations involving two variable laguerre polynomials L_n(x, y) are derived. The process involves the construction of a three-dimensional Lie algebra isomorphic to special linear algebra sl(2) with the help of Weisner's method by giving suitable interpretations to the index n of the polynomials L_n(x, y).     

EPR and optical absorption studies of V O ions in L-asparagine monohydrate

Electron paramagnetic resonance (EPR) studies of V O²⁺ ions in L-asparagine monohydrate single crystals are reported at room temperature. It is found that the V O²⁺ ion takes up an interstitial site. The angular variations of the EPR spectra in three mutually perpendicular planes are used to determine the principal g and A values and their direction cosines. The values of g and A parameters are: g_x=1.9011, g_y=2.1008, g_z=1.9891 and A_x=100, A_y=78, A_z=126 (×10⁻⁴) cm⁻¹. The optical absorption spectrum of V O²⁺ ions in L-asparagine monohydrate is also studied at room temperature. The band positions are calculated using the energy expressions and compared with the observed band positions to confirm the transitions. The best-fit values of the crystal field (Dq) and tetragonal (Ds and Dt) parameters are evaluated from the observed band positions.     

Degree Complexity of Birational Maps Related to Matrix Inversion

For a q × q matrix x = (x _{i, j} ) we let ${J(x)=(x_{i,j}^{-1})}For a q × q matrix x = (x _{i, j} ) we let J(x)=(x_i,j^-1){J(x)=(x_{i,j}^{-1})} be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x)=(x_i,j)^-1{I(x)=(x_{i,j})^{-1}} denote the matrix inverse, and we define K=I°J{K=I\circ J} to be the birational map obtained from the composition of these two involutions. We consider the iterates Kⁿ=K°?°K{K^n=K\circ\cdots\circ K} and determine the degree complexity of K, which is the exponential rate of degree growth d(K)=lim_n?￥( deg(Kⁿ) )^1/n{\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}} of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet.     

Existence of a phase-transition in a one-dimensional Ising ferromagnet

Existence of a phase-transition is proved for an infinite linear chain of spins μ _j =±1, with an interaction energy $$H = - \sum J(i - j)\mu _i \mu _j ,$$ whereJ(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log logn) [n ³ J(n)]^?1 both converge. In particular, as conjectured byKac andThompson, a transition exists forJ(n)=n ^?α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed.     

Finite-N fluctuation formulas for random matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic Σ _j ^N =1 (x _j ? 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½Σ _j ^N =1 (θ _j ?π) and ? Σ _j ^N =1 log 2 |sinθ _j/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.