The approach of multi-dimensional SUSY Quantum Mechanics is used in an explicit construction of exactly solvable 3-body ( and quasi-exactly-solvable N -body ) matrix problems on a line. From intertwining relations with time-dependent operators, we build exactly solvable non-stationary scalar and 2 × 2 matrix 3-body models which are time-dependent extensions of the Calogero model. Finally, we in...

Symbolic model checking of various important properties like reachability, containment and equivalence of constraint automata could be unsolvable problems in general. This paper identifies several classes of constraint automata for which these properties can be guaranteed to be solvable by reformulating them as the evaluation problem of solvable or approximately solvable classes of constraint l...

This paper focuses on the fuzzy Volterra integro-differential equation of nth order of the second-kind with nonlinear fuzzy kernel and initial values. The derived integral equations are solvable, the solutions of which are unique under certain conditions. The existence and uniqueness of the solutions are investigated in a theorem and an upper boundary is found for solutions. Comparison of the e...

Let cS(d) denote the minimal composition length of all finite solvable groups with solvable (or derived) length d. We prove that:

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

We prove that the terms of the derived series of a free solvable group are definable by existential formulae. We use this result to prove some ‘model theoretic’ results about free solvable groups. For example, we prove that if Hilbert’s 10th problem has a negative answer for the field of the rationals, then the universal theory of a noncyclic free solvable group of class > 3 is undecidable. @ 1...

Several explicit examples of quasi exactly solvable ‘discrete’ quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/ sin x potential de...

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have...

By applying the higher order Darboux algorithm to an exactly solvable non Hermitian PT symmetric potential, we obtain a hierarchy of new exactly solvable non Hermitian PT symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is nonlinear pseudo supersymmetry. We also show that this formalism can be used to generate a la...

We describe a “condition” number for the linear complementarity problem (LCP), which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used. Consequently, we develop a class of LCPs solvable in polynomial time. The result suggests that the convexity (or positive semidefiniteness) of the LCP may not be the basic issue that separates LCPs solvable and...