scholarly journals Properties of bigravity solutions in a solvable class

2014 ◽  
Vol 89 (12) ◽  
Author(s):  
Taishi Katsuragawa
Keyword(s):  
Solvable Class

Linear sampling and the ∀∃∀ case of the decision problem

1974 ◽  
Vol 39 (3) ◽  
pp. 519-548 ◽  
Author(s):  
Stål O. Aanderaa ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Combinatorial Problem ◽  
Finite Model ◽  
Unsolvable Problem ◽  
Sampling Problem ◽  
Solvable Class ◽  
Full Class ◽  
Infinite Model ◽  
Intricate Argument ◽  
Linear Sampling

Let Q be the class of closed quantificational formulas ∀x∃u∀yM without identity such that M is a quantifier-free matrix containing only monadic and dyadic predicate letters and containing no atomic subformula of the form Pyx or Puy for any predicate letter P. In [DKW] Dreben, Kahr, and Wang conjectured that Q is a solvable class for satisfiability and indeed contains no formula having only infinite models. As evidence for this conjecture they noted the solvability of the subclass of Q consisting of those formulas whose atomic subformulas are of only the two forms Pxy, Pyu and the fact that each such formula that has a model has a finite model. Furthermore, it seemed likely that the techniques used to show this subclass solvable could be extended to show the solvability of the full class Q, while the syntax of Q is so restricted that it seemed impossible to express in formulas of Q any unsolvable problem known at that time.In 1966 Aanderaa refuted this conjecture. He first constructed a very complex formula in Q having an infinite model but no finite model, and then, by an extremely intricate argument, showed that Q (in fact, the subclass Q2 defined below) is unsolvable ([Aa1], [Aa2]). In this paper we develop stronger tools in order to simplify and extend the results of [Aa2]. Specifically, we show the unsolvability of an apparently new combinatorial problem, which we shall call the linear sampling problem (defined in §1.2 and §2.3). From the unsolvability of this problem there follows the unsolvability of two proper subclasses of Q, which we now define. For each i ≥ 0, let Pi be a dyadic predicate letter and let Ri be a monadic predicate letter.

scholarly journals On a solvable class of product-type systems of difference equations

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6113-6129 ◽  
Author(s):  
Stevo Stevic ◽  
Bratislav Iricanin ◽  
Zdenk Smarda
Keyword(s):  
Closed Form ◽  
Difference Equations ◽  
Type Systems ◽  
Product Type ◽  
Dependent Variables ◽  
Solvable Class ◽  
Solvable Product

It is shown that the following class of systems of difference equations zn+1 = ?zanwbn, wn+1 = ?wcnzdn-2, n ? N0, where a,b,c,d ? Z, ?, ?, z-2, z-1, z0,w0 ? C \ {0}, is solvable, continuing our investigation of classification of solvable product-type systems with two dependent variables. We present closed form formulas for solutions to the systems in all the cases. In the main case, when bd ? 0, a detailed investigation of the form of the solutions is presented in terms of the zeros of an associated polynomial whose coefficients depend on some of the parameters of the system.

A Finitely Solvable Class of Approximating Problems

1977 ◽  
Vol 15 (3) ◽  
pp. 400-406 ◽  
Author(s):  
Gerard G. L. Meyer
Keyword(s):  
Solvable Class

ON CONDITIONALLY QUASI-EXACTLY SOLVABLE CLASS OF QUANTUM POTENTIALS

1995 ◽  
Vol 10 (07) ◽  
pp. 597-604 ◽  
Author(s):  
R. DUTT ◽  
Y. P. VARSHNI ◽  
R. ADHIKARI
Keyword(s):  
Exact Solutions ◽  
New Class ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Constraint Relation ◽  
Solvable Class ◽  
Potential Parameters

A new class of potentials is generated for which exact solutions for one or few states are possible provided one of the potential parameters is assigned a fixed value and there is a constraint relation amongst the others. The method of construction of such conditionally quasi-exactly solvable potentials is illustrated by a few examples.

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