相关论文: Objects of Categories as Complex Numbers

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…

One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results…

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the…

We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…

We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \in [-2^{n-1},2^{n-1}] such that \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in…

We present a Bayesian method for the identification and classification of objects from sets of astronomical catalogs, given a predefined classification scheme. Identification refers here to the association of entries in different catalogs…

We can define a module to be an exact functor on a small abelian category. This is explained and shown to be equivalent to the usual definition but it does offer a different perspective, inspired by the notions from model theory of…

We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the…

The concept of {\em complexity} (as a quantity) has been plagued by numerous contradictory and confusing definitions. By explicitly recognising a role for the observer of a system, an observer that attaches meaning to data about the system,…

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…

Complexity of patterns is a key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal…

In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be.…

Current object recognition methods fail on object sets that include both diffuse, reflective and transparent materials, although they are very common in domestic scenarios. We show that a combination of cues from multiple sensor modalities,…

This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into…

The human mind is known to be sensitive to complexity. For instance, the visual system reconstructs hidden parts of objects following a principle of maximum simplicity. We suggest here that higher cognitive processes, such as the selection…

Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…

A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points.…

This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly…

Motivated by some problems proposed by Cuadra and Simson related to flat objects in finitely accessible Grothendieck categories, we study flatness in the more general setting of finitely accessible additive categories. For such category…