• Volume 110, Issue 4

November 2000,   pages  347-472

• A Quantum Spin System with Random Interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups $\{\mathcal{T}_t(𝜔)\}_{𝜔\in𝛺}$ of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space 𝛺, is used to prove the almost sure independence of the Arveson spectrum $\mathrm{Sp}(\mathcal{T}(𝜔))$ of $\mathcal{T}_t(𝜔)$. As a consequence, for any family of $(\mathcal{T}(𝜔), 𝛽)$-KMS states {ρ(𝜔)}, the spectrum of the generator of the group of unitaries which implement $\mathcal{T}(𝜔)$ in the GNS representation is also almost surely independent of 𝜔.

• Weighted Approximation of Continuous Functions by Sequences of Linear Positive Operators

In this work we obtain, under suitable conditions, theorems of Korovkin type for spaces with different weight, composed of continuous functions defined on unbounded regions. These results can be seen as an extension of theorems by Gadjiev in [4] and [5].

• Hyperfinite Representation of Distributions

Hyperfinite representation of distributions is studied following the method introduced by Kinoshita [2, 3], although we use a different approach much in the vein of [4]. Products and Fourier transforms of representatives of distributions are also analysed.

• Sampling and II-Sampling Expansions

Using the hyperfinite representation of functions and generalized functions this paper develops a rigorous version of the so-called `delta method' approach to sampling theory. This yields a slightly more general version of the classical WKS sampling theorem for band-limited functions.

• Formula for a Solution of 𝑢𝑡 + 𝐻(𝑢, D𝑢) = 𝑔

We study the continuous as well as the discontinuous solutions of Hamilton–Jacobi equation 𝑢𝑡 + 𝐻(𝑢, D𝑢) = 𝑔 in $\mathbb{R}^n × \mathbb{R}_+$ with 𝑢(𝑥, 0)=𝑢0(𝑥). The Hamiltonian 𝐻(𝑠, 𝑝) is assumed to be convex and positively homogeneous of degree one in 𝑝 for each 𝑠 in $\mathbb{R}$. If 𝐻 is non increasing in 𝑠, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.

• Completely Monotone Multisequences, Symmetric Probabilities and a Normal Limit Theorem

Let G𝑛, 𝑘 be the set of all partial completely monotone multisequences of order 𝑛 and degree 𝑘, i.e., multisequences 𝑐𝑛(𝛽1, 𝛽2,$\ldots$ ,𝛽k), 𝛽1, 𝛽2,$\ldots$ ,𝛽𝑘 = 0, 1, 2,$\ldots$ ,𝛽1 + 𝛽2 + \$cdots$ + 𝛽𝑘 ≤ n, 𝑐𝑛(0,0,$\ldots$ ,0) = 1 and $(-1)^{𝛽_0}𝛥^{𝛽_0}$ 𝑐𝑛(𝛽1, 𝛽2,$\ldots$ ,𝛽𝑘)≥ 0 whenever 𝛽0 ≤ 𝑛-(𝛽1 + 𝛽2 +$\cdots$ +𝛽𝑘) where 𝛥 𝑐𝑛(𝛽1, 𝛽2,$\ldots$ ,𝛽)=𝑐𝑛(𝛽1+1, 𝛽2,$\ldots$ ,𝛽𝑘)+ 𝑐𝑛(𝛽1,𝛽2+1,$\ldots$ ,𝛽𝑘)+$\cdots$ + 𝑐𝑛(𝛽1, 𝛽2,$\ldots$ ,𝛽𝑘+1)-𝑐𝑛(𝛽1,𝛽2,$\ldots$ ,𝛽𝑘)$. Further, let$\prod_{n,k}$be the set of all symmetric probabilities on${0, 1, 2,\ldots ,k}^{n}$. We establish a one-to-one correspondence between the sets G𝑛, 𝑘 and$\prod_{n, k}$and use it to formulate and answer interesting questions about both. Assigning to G𝑛, 𝑘 the uniform probability measure, we show that, as 𝑛 → ∞ , any fixed section {𝑐𝑛(𝛽1, 𝛽2,$\ldots$,𝛽𝑘), 1 ≤$\sum 𝛽𝑖≤ 𝑚}, properly centered and normalized, is asymptotically multivariate normal. That is, $\left\{\sqrt{\left(\binom{n+k}{k}\right)}(𝑐𝑛(𝛽1, 𝛽2,\ldots ,𝛽𝑘)-c_0(𝛽1, 𝛽2,\ldots ,𝛽𝑘), 1≤ 𝛽_1+𝛽2+\cdots +𝛽_k≤ m\right\}$ converges weakly to MVN[0,𝛴𝑚]; the centering constants 𝑐0(𝛽1, 𝛽2,$\ldots$ ,𝛽𝑘) and the asymptotic covariances depend on the moments of the Dirichlet $(1, 1,\ldots ,1; 1)$ distribution on the standard simplex in 𝑅𝑘.

• An Asymptotic Derivation of Weakly Nonlinear Ray Theory

Using a method of expansion similar to Chapman–Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an error 𝑂(𝜖2) where 𝜖 is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet–Bruhat's theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.

• Steady-State Response of a Micropolar Generalized Thermoelastic Half-Space to the Moving Mechanical/Thermal Loads

Microrotation effect of a load applied normal to the boundary and moving at a constant velocity along one of the co-ordinate axis in a generalized thermoelastic half-space is studied. The analytical expressions of the displacement component, force stress, couple stress and temperature field for two different theories i.e. Lord-Shulman (L-S) and Green-Lindsay (G-L) for supersonic, subsonic and transonic velocities in case of mechanical and thermal sources applied, are obtained by the use of Fourier transform technique. The integral transforms have been inverted by using a numerical technique and the numerical results are illustrated graphically for magnesium crystal-like material.

• Subject Index

• Author Index

• # Proceedings – Mathematical Sciences

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