On eigenvectors,approximations and the Feynman propagator

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product $〈x_0|K^t|x_1〉$ of the eigenvector $|x_0〉$ of the position operator with eigenvalue $x_0$ and $K^t(|x_1〉)$, where $K^t$ is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.     

Multipoint Super-Halley Type Approximation Algorithms in Banach Spaces

We use a new multipoint iteration of at least R-order three to solve a nonlinear operator equation in a Banach space. Under standard Newton-Kantorovich assumptions, we establish two Kantorovich convergence theorems using a new system of recurrence relations. We also give some explicit error bounds for this iteration.     

Intermediate statistics as a consequence of deformed algebra

We present a formulation of deformed oscillator algebra which leads to intermediate statistics as a continuous interpolation between Bose-Einstein and Fermi-Dirac statistics. It is deduced that a generalized permutation or exchange symmetry leads to the introduction of the basic number and it is then established that this in turn leads to the deformed algebra of oscillators. We obtain the mean occupation number describing the particles obeying intermediate statistics which thus establishes the interpolating statistics and describe boson-like and fermion-like particles obeying intermediate statistics. We also obtain an expression for the mean occupation number in terms of an infinite continued fraction, thus clarifying successive approximations.     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let

     

On the 3-restricted edge connectivity of permutation graphs

An edge cut W of a connected graph G is a k-restricted edge cut if G−W is disconnected, and every component of G−W has at least k vertices. The k-restricted edge connectivity is defined as the minimum cardinality over all k-restricted edge cuts. A permutation graph is obtained by taking two disjoint copies of a graph and adding a perfect matching between the two copies. The k-restricted edge connectivity of a permutation graph is upper bounded by the so-called minimum k-edge degree. In this paper some sufficient conditions guaranteeing optimal k-restricted edge connectivity and super k-restricted edge connectivity for permutation graphs are presented for k=2,3.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

Combinatorial constructions of fault-tolerant routings with levelled minimum optical indices

The design of fault-tolerant routings with levelled minimum optical indices plays an important role in the context of optical networks. However, not much is known about the existence of optimal routings with levelled minimum optical indices besides the results established by Dinitz, Ling and Stinson via the partitionable Steiner quadruple systems approach. In this paper, we introduce a new concept of a large set of even levelled -design of order v and index 2, denoted by -LELD, which is equivalent to an optimal, levelled (v−2)-fault-tolerant routing with levelled minimum optical indices of the complete network with v nodes. On the basis of the theory of three-wise balanced designs and partitionable candelabra systems, several infinite classes of -LELDs are constructed. As a consequence, the existence problem for optimal routings with levelled minimum optical indices is solved for nearly a third of the cases.     

First order logic without equality on relativized semantics

Let $\alpha \ge 2$ be any ordinal. We consider the class ${\mathsf{Drs}}_{\alpha }$ of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of ${\mathsf{Drs}}_{\alpha }$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class ${\mathsf{Drs}}_{\alpha }$ corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.     

High room-temperature pyroelectric response of MgO-modified Pb0.99(Zr0.95Ti0.05)0.98Nb0.02O3 ceramics

The dielectric and pyroelectric responses of MgO-modified Pb_0.99(Zr_0.95Ti_0.05)_0.98Nb_0.02O₃ ceramics were investigated near F_R(LT)–F_R(HT) phase transition. It was found that MgO additive reduced the F_R(LT)–F_R(HT) phase transition temperature from 41 °C to room temperature (24 °C). Superior room-temperature pyroelectric properties were obtained in the composition of 0.10 wt% MgO addition without DC bias. The largest pyroelectric coefficient, 65 × 10⁻⁸ C cm⁻² K⁻¹, was detected. Accordingly, the detectivity figures of merit F_d had maximum values of 20 × 10⁻⁵ Pa^−1/2, and especially the voltage responsivity F_v = 0.91 m²C⁻¹ is the highest value reported so far among all pyroelectric materials. It shows promising potential for application in uncooled pyroelectric infrared detector.