TWO-POINT PROBLEM FOR LINEAR SYSTEMS OF PARTIAL DIffERENTIAL EQUATIONS

The problem with two nodes on the selected variable $t$ and periodicity conditions in other coordinates $x_1,\ldots,x_p$ for linear partial differential equations is investigated. The conditions of solvability problem in the spaces of smooth functions with exponential behavior of Fourier coefficients are established. The estimates for characteristic determinants of the problem are proved.

About the class of approximation by generalized characters

Given a certain class K1 of algebraic structures. We study a problem of finding a class K2 of algebraic structures such that the class K1 is approximable into K2 with respect to various predicates by generalized characters from K1 to K2. The problem of minimization of approximation is also considered. Some theorems related to the problem of constructing an approximation class are obtained. The problem in question is much more complicated and actual than the approximation problem we have been studying before (see [2]-[6]). The results of the description of the approximation class play an important role in studying the solvability problem of the predicate P in the class of semigroups K. In particular, if the approximation class consists of finite semigroups, then this problem is solved positively. Even more difficult is the problem of determining the necessary conditions that class is an approximation class for a given class K.

Solvable problems or problematic solvability? Problem conceptualization in transdisciplinary sustainability research and a possible epistemological contribution

Problems are a major focal point in transdisciplinary sustainability research (TSR). As a text analysis shows, the term “problem” is most commonly used in the context of analyzing research processes that are directed towards societal problem-solving. At the same time, these findings imply that TSR does not follow the idea that problems are solvable. Instead, TSR should transgress the general tension between the solution imperative and the insolvability of complex problems by rather tackling each problem as situated and specific.Problem orientation plays a significant role in emerging transdisciplinary sustainability research (TSR), where the assumption of solvability resonates with the term “problem” yet is not questioned from a sustainability perspective. This paper questions the meaning of “problems” in and for TSR from a discourse studies perspective. The results of a collocation and concordance analysis of the term “problem(s)” in GAIA articles show that sustainability-oriented problem-solving is explicated normatively as a key research goal. In the analyzed articles, emphasis is put on how to proceed towards this goal through research process analysis. This paper begins by analyzing the meaning of “problems” before seeking to orientate TSR in terms of how knowledge could be conceptualized. This is supported by the epistemological concept of the problematic, which originates from 20th century French philosophy. It proves helpful to discuss how TSR can be epistemologically grasped, and thereby strengthened in its transformative potential.

The complexity of the equation solvability and equivalence problems over finite groups

We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a [Formula: see text]-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic presentation of these finite solvable groups.

Sixteen practically solvable systems of difference equations

Closed-form formulas for general solutions to sixteen hyperbolic-cotangent-type systems of difference equations of interest are obtained, showing their practical solvability and completely solving a solvability problem for some concrete values of delays.

RANK LOGIC IS DEAD, LONG LIVE RANK LOGIC!

Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. WhileFPRcan express most of the known queries that separateFPCfromPtime, almost nothing was known about the limitations of its expressive power.In our first main result we show that the extensions ofFPCby rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic${\text{FPR}}^{\text{*}}$with an operator that uniformly expresses the matrix rank over finite fields is more expressive thanFPR.One important step in our proof is to consider solvability logicFPSwhich is the analogous extension ofFPCby quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.

The equation solvability problem over supernilpotent algebras with Mal’cev term

In 2011, Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper, he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal’cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal’cev term that have co-NP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth’s question is negative in general (assuming P[Formula: see text]NP).

On the Complexity of Solving Restricted Word Equations

We investigate the complexity of the solvability problem for restricted classes of word equations with and without regular constraints. The solvability problem for unrestricted word equations remains [Formula: see text]-hard, even if, on both sides, between any two occurrences of the same variable no other different variable occurs; for word equations with regular constraints, the solvability problems remains [Formula: see text]-hard for equations whose two sides share no variables or with two variables, only one of which is repeated. On the other hand, word equations with only one repeated variable (but an arbitrary number of variables) and at least one non-repeated variable on each side, can be solved in polynomial-time.