arXiv subjects

Ryshard-Pavel Kostecki

Publications and source records attributed to Ryshard-Pavel Kostecki.

3 recordsLinked to original sources

arXiv2026

Four negations and the spectral presheaf

Using Vakarelov's theory of lattice logics with negation, we introduce the (co)quasiintuitionistic logic, and prove its soundness and completeness with respect to the class of (co)quasiintuitionistic algebras. Combining these algebras together, we obtain biquasiintuitionistic algebras and the biquasiintuitionistic logic. Their further extension with the Skolem algebra structure defines Akchurin algebras and the respective logic, which is a product of biquasiintuitionistic and biintuitionistic logics, featuring four distinct negations. Next we generalise the framework of spectral presheaves (which is a main object in the Butterfield--Isham--Döring topos theoretic approach to quantum mechanics) to arbitrary complete orthocomplemented lattices, and show that the orthocomplementation determines two negation operators on the spectral presheaf (one paraconsistent, another paracomplete), equipping the set of all closed-and-open subpresheaves of a spectral presheaf with the structure of a biquasiintuitionistic algebra. Combined with the generic Skolem (i.e. Heyting and Brouwer) algebra structure of this set, this gives a particular instance of an Akchurin algebra. We also show that the underlying orthocomplemented lattice can be reconstructed as an internal object of the spectral presheaf, resulting as the image of a double coquasiintuitionistic (resp., quasiintuitionistic) negation monad (resp., comonad). Finally, we prove a no-go theorem for the claim that the spectral presheaf is a model of a dialectical (or any other) relevance logic.

math.LO↗

arXiv2026

Towards postquantum Vaĭnberg--Brègman relative entropies

We develop a new approach to construction of the Vaĭnberg--Brègman relative entropies over nonreflexive Banach spaces, based on nonlinear embeddings into reflexive Banach spaces. We apply it to derive some new families of Vaĭnberg--Brègman relative entropies over some radially compact base normed spaces in spectral duality, and to establish their basic properties. In particular, we prove (left and right) generalised pythagorean theorem and norm-to-norm continuity of the left entropic projections for a family of Vaĭnberg--Brègman relative entropies induced on preduals of any W*-algebras (resp., semifinite JBW-algebras) using Mazur maps into noncommutative (resp., nonassociative) $L_p$ spaces, on preduals of semifinite W*-algebras using Kaczmarz maps into noncommutative Orlicz spaces, and on (resp., positive parts of) unit spheres of commutative $L_1$ spaces (resp., trace class operators) using Lozanovskiĭ factorisation maps. We also prove left generalised pythagorean theorem for a family of Vaĭnberg--Brègman relative entropies over preduals of generalised spin factors. Additionally, we characterise strict convexity, Gateaux differentiability, Radon--Riesz--Shmul'yan property, and reflexivity of the Morse--Transue--Nakano and Orlicz norms on noncommutative Orlicz spaces, establish Lipschitz--Hölder continuity of the nonassociative Mazur map on positive parts of unit balls, and introduce a new class of $L_p$ spaces over spectrally dual order unit spaces.

math-ph↗

arXiv2026

Vaĭnberg--Brègman relative entropy and quasinonexpansive operators

We review the theory of Vaĭnberg--Brègman relative entropies and quasinonexpansive operators on reflexive Banach spaces, and obtain several new results. We also develop an extension of this theory to nonreflexive Banach spaces, which is a joint generalisation of the reflexive Banach space approach and the finite-dimensional information geometric approach. In the reflexive case, we study generalised pythagorean inequality, as well as norm-to-norm, uniform, and Lipschitz--Hölder continuity, of (left and right) entropic projections, proximal maps, and resolvents. We also provide a detailed study of a special (`gauge') family of Vaĭnberg--Brègman geometries and operators that is tightly related with the geometric properties of the underlying Banach space norm. The extended theory belongs to the intersection of convex theoretic and homeomorphic approaches to nonlinear analysis. Its models are constructed, using integration theory on order unit spaces, via nonlinear embeddings into reflexive rearrangement invariant spaces. E.g., we compute the exponent parameters of Lipschitz--Hölder continuity of the extended entropic projections and resolvents, and establish composability of a suitable class of nonlinear quasinonexpansive operators, over normal state spaces of JBW- and W*-algebras, determined by `gauge' Vaĭnberg--Brègman geometries over, respectively, nonassociative and noncommutative L$_p$ spaces, and extended via Mazur embeddings. Other examples of extended Vaĭnberg--Brègman geometries feature the (commutative and noncommutative) Lozanovskiĭ factorisation map, generalised spin factors, finite dimensional base normed spaces, and convex spectral functions on unitarily invariant ideals of compact operators. We also discuss several categories of entropic projections and quasinonexpansive operators naturally appearing in this framework.

math.FA↗