Equivalence of u–p and ζ–ψ formulations of the time-dependent Navier–Stokes equations

This paper deals with the non-stationary incompressible Navier--Stokes equations for two-dimensional flows expressed in terms of the velocity and pressure and of the vorticity and streamfunction. The equivalence of the two formulations is demonstrated, both formally and rigorously, by virtue of a condition of compatibility between the boundary and initial values of the normal component of velocity. This condition is shown to be the only compatibility condition necessary to allow for solutions of a minimal regularity, namely H¹ for the velocity, as in most current numerical schemes relying on spatial discretizations of local type.     

The number of ideals in a quadratic field

LetK be a quadratic Geld, and letR(N) be the number of integer ideals inK with norm at most AT. Letx with conductork be the quadratic character associated withK. Then |R(N)−NL(1,x)|⩽Bk ^50/73 N ^23/73(logN)^461/146 forN ≥Ak, whereA andB are constants. ForN ≥Ak ^c,C sufficiently large, the factork ^50/73 may be replaced by (d(k))^4/73 k ^46/73. Dedicated to the memory of Professor K G Ramanathan     

A characterization of an order ideal in Riesz spaces

In this paper we give a characterization of order ideals in Riesz spaces.

     

The core of zero-dimensional monomial ideals

The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it. We relate the core to adjoints and first coefficient ideals, and in dimension two and three we give explicit formulas.     

A note on δ-I-sets

We characterize δ-I-sets, discuss the relation between δ-sets, δ-I-sets and I _δ-sets and generalize some of the results.     

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

粗糙子群和粗糙子环

文[1]研究了半群中的理想,首次提出了粗半群和粗理想的概念.本文继续研究了群中的粗糙子群和粗正规子群.同时,在环中引入了粗子环的概念.最后,研究了粗糙集的同态问题.     

A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.