Adaptive partitioning techniques for index 1 IDEs

Many implicit differential equations (IDEs) modelling practical problems can be partitioned into loosely coupled subsystems. In this paper the objective of the partitioning is to permit the numerical integration of one time step to be performed as the solution of a sequence of small subproblems. This reduces the computational complexity compared to solving one large system and permits efficient parallel execution under appropriate conditions. The subsystems are integrated using methods based on low order backward differentiation formulas.     

Slices of Essentially Algebraic Categories

This paper is a contribution to the theory of functor slices of J. Sichler and V. Trnková. For every ordinal α we introduce a basket , prove that every essentially algebraic category of height α is a slice of , characterize small slices of and give a common generalization of known results about slices of the algebraic basket .      

Slices of the unitary spread

We prove that slices of the unitary spread of Q⁺(7,q)\mathcal{Q}^{+}(7,q), q≡2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of PΓO ⁺(8,q) fixing the unitary spread. When q is even, there is a connection between spreads of Q⁺(7,q)\mathcal{Q}^{+}(7,q) and symplectic 2-spreads of PG(5,q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173–194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q⁺(7,q)\mathcal{Q}^{+}(7,q), q=2^2h+1. Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151–165, 1982. When q=3 ^h , we classify, up to the action of the stabilizer in PΓO(7,q) of the unitary spread of Q(6,q), those among its slices producing spreads of the elliptic quadric Q^-(5,q)\mathcal{Q}^{-}(5,q).     

Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1

Summary For the numerical solution of non-stiff semi-explicit differentialalgebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods.     

Three-step alternating iterations for index 1 and non-singular matrices

Numerical Algorithms - Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to...     

Properties of finite-difference schemes for singular integrodifferential equations of index 1

Systems of integrodifferential equations with a singular matrix multiplying the highest derivative of the unknown vector function are considered. An existence theorem is formulated, and a numerical solution method is proposed. The solutions to singular systems of integrodifferential equations are unstable with respect to small perturbations in the initial data. The influence of initial perturbations on the behavior of numerical processes is analyzed. It is shown that the finite-difference schemes proposed for the systems under study are self-regularizing.     

G a Invariants and Slices

Criteria for local triviality of algebraic actions of the additive group of complex numbers on complex affine space are extended to more general varieties. Finite generation of rings of invariants of locally trivial actions on factorial affine varieties is dicussed, giving some sufficient conditions for finite generation and examples where finite generation fails. A missing hypothesis in a theorem of Miyanishi is identified, and an example is given to demonstrate the necessity of the hypothesis. The corrected theorem is shown to hold for a class of triangular G _a actions on C⁴, with the consequence that all these actions are conjugate to translations. A new criterion for an action to be conjugate to a global translation is given.     

Sarkisov links for index 1 Fano 3-folds in codimension 4

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.     

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

The paper deals with the numerical treatment of stochastic differential-algebraic equations of index one with a scalar driving Wiener process. Therefore, a particularly customized stochastic Runge-Kutta method is introduced. Order conditions for convergence with order 1.0 in the mean-square sense are calculated and coefficients for some schemes are presented. The proposed schemes are stiffly accurate and applicable to nonlinear stochastic differential-algebraic equations. As an advantage they do not require the calculation of any pseudo-inverses or projectors. Further, the mean-square stability of the proposed schemes is analyzed and simulation results are presented bringing out their good performance.     

Representation and index theory for C1-algebras generated by commuting isometries

We discuss here representation and Fredholm theory for C¹-algebras generated by commuting isometries. More particularly, for n commuting isometries {V_j: 1 ? j ? n} on separable Hilbert space we give a representation resembling the well-known representation for a single isometry. Our representation permits an analysis of the C¹-algebras $\text{Ol}$=$\text{Ol}$(V_j:1?j?n) generated by the {V_j}. The commutator ideal in $\text{Ol}$ is identified precisely and, under certain additional hypotheses, the Fredholm operators in $\text{Ol}$ are also precisely determined. Finally, we obtain formulas in terms of topological data for the index of Fredholm operators in some interesting algebras of the type $\text{Ol}$(V_j:1?j?n).     

The bigraphical t-wise balanced designs of index 1

The t-wise balanced designs of index 1 whose point set are the edges of K_m,n, the complete bipartite graph, and that have the subgroup of all automorphisms that fix the two independent sets of K_m,n as an automorphism group are studied. All such designs are found. © 1994 John Wiley & Sons, Inc.