For solving 3D high order hierarchical FE systems the block SSOR preconditioned CG algorithms based on new stripwise block two-color orderings of degrees of freedom and providing for efficient concurrent/vector implementation are suggested. As demonstrated by numerical results for the 3D Navier equations approximated using hierarchical orderp, 2 ≤p ≤ 5, FE's the convergence rate of such BSSOR-CG algorithms is only slightly dependent onp and mesh nonunformity.

The last five decades have witnessed many developments in the theory of harmonic maps. To become acquainted to some of these, the reader is referred to two reports and a survey paper by Eells and Lemaire [119, 122, 124] about the developments of harmonic maps up to 1988 for details. Several books on harmonic maps [203, 205, 206, 389, 425] are also available. In this chapter, we follow the notions and notations of harmonic maps between Riemannian manifolds by Eells- Sampson [129] in the introduction.We discuss the crucial topics in harmonic maps including fundamentals, regularity, maps of surfaces, maps of KRahler manifolds, maps into groups and Grassmannians, harmonic maps, loop groups, and integrable systems, harmonicmorphisms, maps of singular spaces, and transversally harmonic maps. Since the theory of harmonic maps has been developed over half a century, it is impossible to provide full details. However, we try to present the most important components of the topics.

We consider singular integral operators with rough kernels on the product space of homogeneous groups. We prove L^p boundedness of them for $${p \in (1,\infty)}$$ under a sharp integrability condition of the kernels.

In this chapter we consider a model introduced in Kantor and Kardar [203], where each monomer carries a random charge, and each self-intersection of the polymer is rewarded when the two charges of the associated monomers have opposite sign and is penalized when they have the same sign. This model is a variation on the weakly self-avoiding walk described in Chapters 3 and 4, with a random self-interaction driven by the charges. We will focus on the annealed path measure, of the type defined in (1.5). We will show that the annealed charged polymer is in a collapsed phase, irrespective of its overall charge distribution, and is subdiffusive with a scaling limit that can be computed explicitly, namely, Brownian motion conditioned to stay inside a finite ball. The free energy will be different for neutral and for non-neutral charged polymers, even though the scaling limit is the same. Once more local times will prove to be useful. In particular, the large deviation behavior of the local times of SRW will play a crucial role in the identification of the scaling limit.

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.

A method of solving equations of the form $ {g^{{y_1}}} \cdot h \cdot {g^{{y_2}}} \cdot h \cdot \ldots \cdot {g^{{y_1}}} \cdot h \cdot {g^{{y_{l + 1}}}} = \sigma $ in the symmetric group S_n is proposed, where h is a transposition, g is a full cycle, and σ × S_n. The method is based on building all sets of generalized inversions of the bottom line of the substitution σ by means of a system of Boolean equations associated with σ. An example of solving an equation in a group S₆ is given.

We study oscillations in the discontinuous dynamic system with time delay $$\dot x(t) = - sign x(t - 1) + F(x(t),t), t \geqslant 0$$ . This is a typical model of relay feedback with delay. It is known that stable modes in this system have a bounded oscillation frequency. Here we consider transient processes and obtain the following result: under some restrictions ofF, the average oscillation frequency of any solution becomes finite after a period of time, i.e. super-high-frequency oscillations (with infinite frequency) exist only in a finite time interval. Moreover, we give an effective upper bound on the length of this interval.