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## Invariant vector fields and linear equations on the sphere

### Ukrainian Mathematical Journal (1989-03-01) 41: 264-268 , March 01, 1989

## On symmetric radix representation of Gaussian integers

### BIT Numerical Mathematics (1989-09-01) 29: 563-571 , September 01, 1989

Symmetric radix representation and symmetric mixed-radix representation of Gaussian integers play a significant role in the residue arithmetic of*Z*[*i*]. In the following, known results concerning corresponding representations of integers are generalized. It is shown that for any modulus*m*ε*Z*[*i*] with
$$m\bar m > 1$$
, except for*m*=1±*i*, 2, there exists a unique symmetric*m*-radix representation of Gaussian integers.

## Parameter Restrictions for Distance-Regular Graphs

### Distance-Regular Graphs (1989-01-01) 18: 167-192 , January 01, 1989

In this chapter we collect most of the restrictions on intersection arrays of distance-regular graphs known to us. (A few very basic facts have already been mentioned in §4.1D.) Some of these restrictions are important tools in the theoretical investigation of the properties of distance-regular graphs, like the unimodality of the sequence (*k*_{i})_{i} discussed in §5.1. (We already used this on several occasions.) Various bounds on the diameter in terms of the valency are theoretically important. First we have Terwilliger’s diameter bound for the case where the graph contains a quadrangle; next Ivanov’s theory, which yields abound on the diameter for arbitrary distance-regular graphs with fixed numerical girth, and finally the work by Bannai & Ito, who strive to remove the dependency on the girth from these bounds. Also Godsil proved diameter bounds, but this time in terms of a multiplicity.

## Boundary values of the solutions of degenerate elliptic equations

### Journal of Soviet Mathematics (1989-05-01) 45: 1205-1218 , May 01, 1989

One determines the natural boundary values of the solutions of degenerate elliptic equations. It is shown that the natural boundary values can be expressed in terms of the Dirichlet data with the aid of the classical pseudodifferential operator.

## Codes and curves

### Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (1989-01-01) 357: 22-30 , January 01, 1989

## Temperature states of spin-boson models

### Quantum Probability and Applications IV (1989-01-01) 1396: 149-157 , January 01, 1989

## Interacting Population Reaction-Diffusion Systems, Dirichlet Conditions

### Systems of Nonlinear Partial Differential Equations (1989-01-01) 49: 47-109 , January 01, 1989

We will use the techniques described in the last chapter to study reaction-diffusion systems related to ecology. We consider steady states and stabilities for prey-predator and competing-species systems. In this chapter, we are primarily concerned with the case when values for the species are prescribed on the boundary (i.e., Dirichlet boundary conditions). In the next chapter, more elaborate problems and other boundary conditions are treated, together with certain asymptotic approximations. The special case of zero-flux boundary condition (i.e. homogeneous Neumann condition) is studied in Chapter 7. Numerical approximations and calculations by finite difference is presented in Chapter 6.

## Problemi di estremalita', secondo Dubreil, per ideali perfetti di altezza 2

### Rendiconti del Seminario Matematico e Fisico di Milano (1989-12-01) 59: 81-89 , December 01, 1989

### Sunto

Si studia una famiglia di ideali perfetti, di altezza 2, estremali rispetto ad una diseguaglianza di Dubreil, caratterizzadone gli elementi attraverso proprietà della loro matrice dei gradi.

## The number of lattice rules

### BIT Numerical Mathematics (1989-09-01) 29: 527-534 , September 01, 1989

A lattice rule is a quadrature rule for integration over an*s*-dimensional hypercube that employs*N* abscissas located on a lattice, chosen to conform to certain specifications. In this paper we determine the number*v*_{s}(*N*) of distinct*N*-point*s*-dimensional lattice rules. We show that, in general,
$$v_s (MN) \leqslant v_s (M)v_s (N),$$
equality being attained if and only if*M* and*N* are mutually prime. This is used to establish that, when*L* has prime factor expansion
$$\prod\limits_j {p_j^{t_J } ,} $$
, then
$$v_s (L) = \prod\limits_j {v_s (p_j^{t_J } )'} $$
where
$$v_s (p^t ) = \prod\limits_{i = 1}^{s - 1} {[(p^{i + t} - 1)/(p^i - 1)]; s \geqslant 1,t \geqslant 0.} $$