The subclass of directed series-parallel graphs plays an important role in computer science. To determine whether a graph is series-parallel is a well studied problem in algorithmic graph theory. Fast sequential and parallel algorithms for this problem have been developed in a sequence of papers. For series-parallel graphs methods are also known to solve the reachability and the decomposition p...

We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

Let Γ be a finitely generated linear group over a field of characteristic 0. Suppose that every solvable subgroup of Γ is polycyclic. Then any solvable subgroup of Γ is separable. This conclusion is false without the hypothesis that every solvable subgroup of Γ is polycyclic.

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in Ref. [20], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce mos...

Lattice systems with certain Lie algebraic or quantum Lie algebraic symmetries are constructed. These symmetric models give rise to series of integrable systems. As examples the An-symmetric chain models and the SU(2)-invariant ladder models are investigated. It is shown that corresponding to these An-symmetric chain models and SU(2)invariant ladder models there are exactly solvable stationary ...

From the partial differential Calogero’s (three-body) and Smorodinsky-Winternitz (superintegrable) Hamiltonians in two variables we separate the respective angular Schrödinger equations and study the possibilities of their “minimal” PT symmetric complexification. The simultaneous loss of the Hermiticity and solvability of the respective angular potentials V (φ) is compensated by their replaceme...

Virasoro-type symmetries and their roles in solvable models are reviewed. These symmetries are described by the two-parameter Virasoro-type algebra Virp,q by choosing the parameters p and q suitably. Based on talks presented by H.A. and J.S. at the Nankai-CRM joint meeting on the quantum deformed Virasoro algebra, Tianjin, China, August 19-24, 1996. To appear in the CRM series in mathematical p...

One main goal in Systems Biology is the understanding and modelling of regulatory processes on molecular level. An important step to achieve this goal is the study of gene regulatory networks. Models for such networks based on piecewise linear differential equations have the advantage of being analytically solvable. A parameter estimation approach is proposed which is based on hinging hyperplan...

Training a Support Vector Machine (SVM) requires the solution of a very large quadratic programming (QP) problem. This paper proposes an algorithm for training SVMs: Sequential Minimal Optimization, or SMO. SMO breaks the large QP problem into a series of smallest possible QP problems which are analytically solvable. Thus, SMO does not require a numerical QP library. SMO’s computation time is d...

A problem is defined to be exactly solvable if its solution can be obtained by solving a sequence of polynomials using radicals. Therefore, if a problem is not exactly solvable, then we have to use approximation schemes for solving the problem. It has been proved that the shortest network problem in space is not exactly solvable even if the network spans only four points and even if the topolog...