A Study on the Space and Energy Dependent Reactor Kinetics, with Direct Physical Interpretation of the Effective Neutron Lifetime and Criticality Factor

First, the concept of neutron importance is introduced. It is assumed that each of the neutrons produced by fission in a chain reactor possesses an importance proportional to the number of its descendants. Secondly, on the basis of the law of conservation, a transport equation of the neutron importance is derived. Then, the effective neutron lifetime is defined as the mean interval of successive fission events in the course of the importance transport. The consistent definition of the criticality factor is the neutron multiplicity during the effective neutron lifetime so defined. After defining the basic reactor kinetics parameters, such as the effective neutron lifetime and criticality factor, the persistent time behavior of nuclear chain reactors has been investigated. The kernel form reactor equation is used because of its physical intelligibility. The formulas obtained are applicable to any reactor, provided that the neutron flux and its adjoint function is known either analytically or numerically

Bounded Solutions of the Boltzmann Equation in the Whole Space

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category

Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff

In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every $L^1$ weak solution to the Cauchy problem with finite moments of all order acquires the $C^\infty$ regularity in the velocity variable for the positive time

Noise analysis of single-qumode Gaussian operations using continuous-variable cluster states

We consider measurement-based quantum computation that uses scalable continuous-variable cluster states with a one-dimensional topology. The physical resource, known here as the dual-rail quantum wire, can be generated using temporally multiplexed offline squeezing and linear optics or by using a single optical parametric oscillator. We focus on an important class of quantum gates, specifically Gaussian unitaries that act on single modes, which gives universal quantum computation when supplemented with multi-mode operations and photon-counting measurements. The dual-rail wire supports two routes for applying single-qumode Gaussian unitaries: the first is to use traditional one-dimensional quantum-wire cluster-state measurement protocols. The second takes advantage of the dual-rail quantum wire in order to apply unitaries by measuring pairs of qumodes called macronodes. We analyze and compare these methods in terms of the suitability for implementing single-qumode Gaussian measurement-based quantum computation.Comment: 25 pages, 9 figures, more accessible to general audienc

Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces

Optimal Lp–Lq convergence rates for the compressible Navier–Stokes equations with potential force

AbstractIn this paper, we are concerned with the optimal Lp–Lq convergence rates for the compressible Navier–Stokes equations with a potential external force in the whole space. Under the smallness assumption on both the initial perturbation and the external force in some Sobolev spaces, the optimal convergence rates of the solution in Lq-norm with 2⩽q⩽6 and its first order derivative in L2-norm are obtained when the initial perturbation is bounded in Lp with 1⩽p<6/5. The proof is based on the energy estimates on the solution to the nonlinear problem and some Lp–Lq estimates on the semigroup generated by the corresponding linearized operator