Related papers: Scaled-Free Objects
In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation…
We introduce the notion of pre-weight structure on a triangulated category and study the corresponding pseudo-identities. We propose the notion of canonical derived equivalence between algebras that are not necessarily flat, which is…
For an arbitrary rational polyhedron we consider its decompositions into Minkowski summands and, dual to this, the free extensions of the associated pair of semigroups. Being free for the pair of semigroups is equivalent to flatness for the…
Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…
Given commutative, unital rings $A$ and $B$ with a ring homomorphism $A\to B$ making $B$ free of finite rank as an $A$-module, we can ask for a "trace" or "norm" homomorphism taking algebraic data over $B$ to algebraic data over $A$. In…
Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…
This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…
The algebra of linear and quadratic functions of basic observables on the phase space of either the free particle or the harmonic oscillator possesses a finite-dimensional anomaly. The quantization of these systems outside the critical…
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient…
The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous…
A new approach is suggested to characterize algebraically automorphisms of the category of free algebras of a given variety. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category…
We construct and analyze the free Banach $f\!$-algebra $\operatorname{FB{\it f}A}[E]$ generated by a Banach space $E$, extending recent developments on free Banach lattices to the setting of Banach $f\!$-algebras, where multiplication…
Motivated by the application problem of sensor fusion the author introduced the concept of graded set. It is reasoned that in classification problem arising in an information system (represented by information table), a novel set called…
A more realistic object detection paradigm, Open-World Object Detection, has arisen increasing research interests in the community recently. A qualified open-world object detector can not only identify objects of known categories, but also…
The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms…
Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, we explore a pure algebraic…
It is known that the construction of the frame of ideals from a distributive lattice induces a monad whose algebras are precisely the frames and frame homomorphisms. Using the Fakir construction of an idempotent approximation of a monad, we…
We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…