An Alumanylyttrium Complex with an Absorption due to a Transition from the Al−Y Bond to an Unoccupied d-Orbital

The reaction between a dialkyl-substituted alumanyl anion and [Y(CH₂SiMe₃)₂(thf)₃][BPh₄] resulted in the formation of (dialkylalumanyl)yttrium complex 2 , which exhibits the first 2-center–2-electron (2 c-2 e) Al−Y bond. The ¹H and ¹³C NMR spectra of 2 together with an X-ray crystallographic analysis indicated a C_2v symmetrical structure. DFT calculations on 2 revealed that its LUMO consists of overlapping 3 p- and 4 d-orbitals of the Al and Y atoms, respectively, and that the HOMO–LUMO gap is narrow. The UV/Vis spectrum of 2 exhibited a visible absorption at 432 nm, which suggests that the strong σ-donating and π-accepting character of the three-coordinate dialkylalumanyl ligand generates a colored d⁰-complex that does not contain any π-electrons.     

Self-avoiding walk in five or more dimensions I. The critical behaviour

We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford5. We prove that the numberc _n ofn-step self-avoiding walks satisfiesc _n ~A ⁿ , where is the connective constant (i.e. =1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forc _n(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (^––Z)^1/2. The critical two-point function is shown to decay at least as fast as x^–2, and its Fourier transform is shown to be asymptotic to a multiple ofk ^–2 ask0 (i.e. =0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.Supported by the Nishina Memorial Foundation and NSF grant PHY-8896163.Supported by NSERC grant A9351     

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β=2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson’s model with β=2 and N particles, $\mbox {\boldmath $\mbox {\boldmath , is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function (z){\rm Ai}(z) is an entire function with zeros all located on the negative part of the real axis ℝ. We consider Dyson’s model with β=2 starting from the first N zeros of Ai(z){\rm Ai}(z) , 0>a ₁>⋅⋅⋅>a _N , N≥2. In order to properly control the effect of such initial confinement of particles in the negative region of ℝ, we put the drift term to each Brownian motion, which increases in time as a parabolic function: Y _j (t)=X _j (t)+t ²/4+{d ₁+∑ _ℓ=1 ^N (1/a _ℓ )}t,1≤j≤N, where d₁=Ai￠(0)/Ai(0)d_{1}={\rm Ai}'(0)/{\rm Ai}(0) . We show that, as the N→∞ limit of $\mbox {\boldmath $\mbox {\boldmath , we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of (z){\rm Ai}(z) on the negative ℝ is occupied by one particle, to the stationary state m_Ai\mu_{{\rm Ai}} . The stationary state m_Ai\mu_{{\rm Ai}} is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on ℝ and in which the Tracy-Widom distribution describes the rightmost particle position.     

Hydrogen-induced magnetic and structural transformations of GdCu

The magnetization of GdCu induced by hydrogen uptake was measured within the temperature range of 4.2 to 300 K, occurring phase changes were followed by X-ray diffraction measurements at ambient temperature. The prepared GdCu powder of CsCl-type structure readily absorbed hydrogen at ambient temperature, where hydrogen pressure was below 100 kPa. Hydrogenation changed the magnetism of GdCu in a complex manner from an antiferromagnetic-like type to a paramagnetic-like one. The changes in magnetic properties of GdCu by hydrogenation are governed by hydrogen-induced disproportionation. Within the composition range 0<[H]/[GdCu]<1, GdCu disproportionated according to 2GdCu+H₂→GdH₂+GdCu₂ . The magnetization was evaluated by the expression χ_total=(1-x)χ_GdCu+(x/2)(χ_GdH2+χ_GdCu2). GdCu hydride was not observed. Hydrogenation beyond [H]/[GdCu]>1 gave rise to the disproportionation of GdCu₂ causing the change in magnetization.     

A generalization of the weighted Strichartz estimates for wave equations and an application to self‐similar solutions

Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without a support restriction on the forcing term. The method of proof is based on expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self‐similar solutions to nonlinear wave equations are proved on up to five space dimensions. © 2006 Wiley Periodicals, Inc.     

On the sampling conditions for reconstruction of an acoustic field from a finite sound source

A simple and accurate method for the estimation of ultrasonic transducer fields is developed. In the method, the angular spectrum is employed to evaluate the three-dimensional propagation from a measured plane to an arbitrary parallel plane. The implementation uses a discrete convolution that is described in detail. Relative to conventional spatial-frequency representations, the implementation of the angular spectrum method in this paper has the advantage of being free from artifacts, enabling sample spacing to be greater than one half wavelength, using memory efficiently, and interpolating the measured data. The loosened sampling requirement and natural interpolation of the method permit efficient reconstruction of the full three-dimensional acoustic field from a coarse sound pressure measurement on single plane.     

Congruence classes of Frenet curves in complex quadrics

Congruent classes of Frenet curves of order 2 in the complex quadric are studied, obtaining that each congruence class is a level set of a family of certain smooth functions, that are generalizations of isoparametric functions on the unit sphere in the tangent space of the complex quadric.     

Resolutions of monomial almost complete intersections and generalizations

Let S=K[x₁,…,x_n] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.