arXiv subjects

Pelle Brooke Borgeke

Publications and source records attributed to Pelle Brooke Borgeke.

3 recordsLinked to original sources

arXiv2026

PBNF-transform as a formulation of Propositional Calculus, II

Here we show, in the second paper in a series of articles, methods to calculate propositional statements with algebraic polyno mials as symbols for the connectives, which here are named operators. In the first article, we explained this formulation of the Propositional Calculus. In short, we transform to a dual space, which we here refer to as a polynomial family, which is another shape of DBNF. We name the polynomial families as PBNF, which stands for Polynomial Boolean Normal Form. We just use the one law of inference, the rule of Substi tution. We can use different polynomial families in the House of PBNF, depending on the statement form, making it even simpler. It is also pos sible to find new theorems and generalize older ones, for example, those given by Church and Barkley Rosser (see follow-up article) concerning duality.

math.LO↗

arXiv2026

PBNF-transform as a formulation of Propositional Calculus, I

Here, in a series of articles, we show methods for calculating propositional statements using algebraic polynomials as symbols for the connectives, which are named operators. These polynomials originate from the transformation between the principles of duality and the Disjunctive Boolean Normal Form, DBNF, and they appear if we use a geometrization in the unit square and simple algebraic methods, modulo 2. This we call the PBNF-transform. PBNF stands for Polynomial Boolean Normal Form as these families are based on DBNF involved here. In the first paper in this series, we show that statements can be mapped bijectively into different polynomial families g(p,q) belonging to H(g)$, which we call the The House of PBNF. We can also replace the connectives of logic with PBNF, as the polynomials are, in fact, a geometrization of these connectives; the systems are isomorphic. The benefit of this formulation of the Propositional Calculus(PC) is a near trivialization of the methods. No axioms are needed, no truth tables, just a list of polynomials (which in themselves are self-explanatory), the only law of inference is the rule of Substitution.

math.LO↗

arXiv2026

Subprincipal Controlled Quasimodes and Spectral Instability

Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<h<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal intersections of bicharacteristics, where the principal symbol has double multiplicity, $p=dp=0$, in a small neigborhood $Ω$. Because of this fact, we instead study the subprincipal symbol b, and we can conclude that we get transport equations depending on b where sign changes for the imaginary part of b give approximative solutions with small support. These modes are used to estimate spectral instability, or the pseudospectrum. We also investigate the possibility that we can factorize the model operator as $P(h)=h^2P_1P_2,$ in this way actually annihilating the subprincipal symbol, thus there is no condition for the imaginary part of b. In a follow-up article, we examine different cases for more complex operators with tangential intersections of bicharacteristics, thereby generalizing the findings here.

math.AP↗