ARTICLE | doi:10.20944/preprints201708.0036.v1
Subject: Mathematics & Computer Science, Other Keywords: fuzzy set; ordered category; category of fuzzysets
Online: 9 August 2017 (06:39:47 CEST)
In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms -- of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.
Subject: Mathematics & Computer Science, General Mathematics Keywords: Category; Algebra; State; Category Algebra; State on Category; Noncommutative Probability; Quantum Probability; GNS representation
Online: 31 May 2021 (08:31:29 CEST)
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.
ARTICLE | doi:10.20944/preprints202010.0505.v1
Subject: Social Sciences, Organizational Economics & Management Keywords: Sharing Economy; Category Formation; Emergence; Social Movement; Similarity Clustering; Truce; Radial Category; Identity Legitimation; Stakeholders; Business Models.
Online: 26 October 2020 (08:50:38 CET)
The Sharing Economy (SE) has dawn great attention from several stakeholders in society in the last five years. While business actors are interested in financial opportunities to meet consumer needs, new business models, the academia and governmental organizations are concerned with potential unintended effects on the society and environment. In the process of making a clearer comprehension of the SE phenomenon, researchers have identified that, despite its notable global growth, there still persists a lack of a more solid ground in understanding its origins and respective mechanisms under which it has been evolving over time as a category. In this research, we address the problematics of the origins and ascendency of the SE by examining the process by which the SE is arising as a new category, searching for conceptual clarification and pinpointing the legitimacy granted by key stakeholders. Our guiding research questions are: (1) how the SE was formed and evolved as a market category; and (2), as a market category, is the SE legitimate? Additionally, we attempt to identify the nature of the SE as a category. To answer these questions, we conducted an historical analysis of the expression SE and its equivalents. This paper deepens the discussion about the nature of the SE by providing evidence that (i) the SE has predominantly been formed by emergence processes, comprising social movement, similarity clustering and truce components. It is the combination of all these aforementioned processes that renders the SE a special case of market category formation, which, in turn, has been allowing communication, entrepreneurship, regulation and research about what really is the SE, and despite the evident lack of agreements regarding both the label and its content; (ii) there is a generalized legitimacy granted to the SE by a vast number of stakeholders, even though still lacking on the consolidation of socio-political legitimation, and (iii) the nature of the SE seems to fall in a metaphorical approach, particularly, the notion of radial categories.
ARTICLE | doi:10.20944/preprints202011.0412.v1
Subject: Biology, Anatomy & Morphology Keywords: Process; ontological category; life concept; essential feature
Online: 16 November 2020 (10:49:11 CET)
Although increasing knowledge about biological systems has advanced exponentially in recent decades, it is surprising to realize that the very definition of Life keeps presenting theoretical challenges. Even if several lines of reasoning seek to identify the essence of life phenomenon, most of these thoughts contain fundamental problem in their basic conceptual structure. Most concepts fail to identify necessary and sufficient features to define life. Here, we analyzed the main conceptual framework regarding theoretical aspects supporting life concepts, such as (i) the physical, (ii) the cellular and (iii) the molecular approaches. Based on ontological analysis, we propose that Life should not be positioned under the ontological category of Matter. Yet, life should be better understood under the top-level ontology of “Process”. Exercising an epistemological approach, we propose that the essential characteristic pervading each and every living being is the presence of organic codes. Therefore, we explore theories in biosemiotics in order to propose a clear concept of life as a macrocode composed by multiple inter-related coding layers. Therefore, we suggest a clear distinction between the concept of life and living beings, a distinction that is not evident in theoretical terms. From the proposed concept, we suggest that the evolutionary process is a fundamental characteristic for life’s maintenance but not to its definition. The current proposition opens a fertile field of debate in astrobiology, biosemiotics and robotics.
ARTICLE | doi:10.20944/preprints201809.0021.v1
Subject: Physical Sciences, General & Theoretical Physics Keywords: Einstein; manifold; Hilbert space; Abelian; dagger category
Online: 3 September 2018 (10:51:51 CEST)
The unexploited unification of general relativity and quantum physics is a painstaking issue that prevents physicists to properly understanding the whole of Nature. Here we propose a pure mathematical approach that introduces the problem in terms of group theory. Indeed, we build a cyclic groupoid (a nonempty set with a binary operation defined on it) that encompasses both the theories as subsets, making it possible to join together two of their most dissimilar experimental results, i.e., the commutativity detectable in our macroscopic relativistic world and the noncommutativity detectable in the quantum, microscopic world. This approach, combined with the Connes fusion operator, leads to a mathematical framework useful in the investigation of relativity/quantum mechanics relationships.
ARTICLE | doi:10.20944/preprints201706.0055.v1
Subject: Arts & Humanities, Theory Of Art Keywords: aesthetics; mathematical structure; category theory; natural intelligence
Online: 12 June 2017 (13:26:59 CEST)
This paper proposes a new approach to investigation into the aesthetics. Specifically, it argues that it is possible to explain the aesthetic and its underlying dynamic relations with axiomatic structure (the octahedral axiom derived category) based on contemporary mathematics – namely, category theory – and through this argument suggests the possibility for discussion about the mathematical structure of the aesthetic. If there was a way to describe the structure of aesthetics with the language of mathematical structures and mathematical axioms – a language completely devoid of arbitrariness – then we would make possible a synthetical argument about the essential human activity of “the aesthetics”, and we would also gain a new method and viewpoint on the philosophy and meaning of the act of creating a work of art and artistic activities. This paper presents one hypothesis as a first step in constructing the science of dynamic generative aesthetics based on axiomatic functionalism, which is in turn based on a new interdisciplinary investigation into the functional structure of aesthetics.
ARTICLE | doi:10.20944/preprints202109.0445.v1
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: Many-sorted partial algebra; free completion; category of completions; weakly initial object; comma category of objects over a completion; Schmidt construction; Schmidt homomorphism; twisted morphism category; Schmidt endofunctor; functoriality of the Schmidt construction
Online: 27 September 2021 (11:45:35 CEST)
After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labeled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this we associate to an ordered pair (α,f), where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism in D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥαG,0(f):Schα(f)B. We then prove that there exists an endofunctor, ΥαG,0, of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards we prove that, for every Σ-completion G=(D,G,ρ), there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of ΥG, sends a morphism of Σ-completion in Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0.
ARTICLE | doi:10.20944/preprints202107.0530.v1
Subject: Mathematics & Computer Science, Artificial Intelligence & Robotics Keywords: robot tactile; convolution neural network; attribute strength identification; category identification; robot operating system
Online: 23 July 2021 (09:27:35 CEST)
Objectives: In order to solve the problem that most of the existing research focuses on the binary tactile attributes of objects,which ignores the tactile attribute strength and category recognition,an attribute strength and category recognition method based on convolutional neural network matrix-label is proposed. Methods:Firstly,in the data preparation stage,we preprocess the raw data and determine the matrix labels to build the haptic dataset.Secondly,in the feature extraction stage,we fuse the haptic data of two fingers and use the convolutional neural network to extract the attribute strength features.Finally,in the attribute strength and category recognition stage,all channel haptic data is fused to predict the attribute strength and category.Results:We compared with the multi-label convolutional neural network method in terms of elastic strength,hardness strength and category,and compared the attribute strength recognition capabilities of the two methods using novel objects outside the haptic dataset.The results show that the accuracy of the last 20 iterations of the matrix-label method has an average elastic strength of 96.73%,hardness strength of 97.34%,and category of 96.67%.The performance is better.When the Euclidean distance between the prediction of the novel object and the real label is less than 1,the accuracy of the elastic strength is best to reach 100%,and the hardness strength is best to reach 100%.The performance is better. Conclusions:The effectiveness of the method has been verified.Comparing with the convolutional neural network method,our method can effectively recognize the attribute strength and category of objects.
REVIEW | doi:10.20944/preprints201912.0179.v1
Subject: Life Sciences, Biophysics Keywords: globular set; category theory; multidimensional; visual recognition; drug-resistant epilepsy; transcranial magnetic stimulation.
Online: 13 December 2019 (10:37:03 CET)
Once a wheat sheaf has been sealed and tied up, its packed down straws display the same orientation and zero-divergence. This observation brings us to the mathematical notion of presheaf, i.e., a topological structure in which diverging functions are locally superimposed. We show how the concepts of presheaves and the correlated globular sets, borrowed from category theory and algebraic topology, allow a well-founded mathematical approach to otherwise elusive activities of the brain. The mathematical assessment of brain functions in terms of presheaves: a) explains why spontaneous random spikes synchronize; b) leads to the counterintuitive intuition of antidromic effects in neuronal spikes: when an entrained oscillation propagates from A to B, changes in B lead to changes in A. We provide testable previsions: a) we suggest the proper locations of transcranial magnetic stimulation’s coils to improve the clinical outcomes of drug-resistant epilepsy; b) we advocate that axonal stimulation by external sources backpropagates and alters the neuronal electric oscillatory frequency. Further, we describe how the hierarchical information transmission inside globular sets provides fresh insights concerning different issues at various coarse-grained scales, such as object persistence, memory reinforcement in spite of random noise, Bayesian inferential circuits.
ARTICLE | doi:10.20944/preprints202003.0236.v1
Subject: Engineering, Civil Engineering Keywords: seismic vulnerability; fuzzy logic system; Interval Type-2 Fuzzy logic; retrofit prioritization; damage category classification
Online: 15 March 2020 (01:54:35 CET)
Rapid Visual Screening (RVS) is a procedure that estimates structural scores for buildings and prioritize their retrofit and upgrade requirements. Despite the speed and simplicity of RVS, many of the collected parameters are non-commensurable and include subjectivity due to visual observations. It might cause uncertainties in the evaluation, which emphasizes the use of a fuzzy-based method. This study aims to propose a novel RVS methodology based on the interval type-2 fuzzy logic system (IT2FLS) to set the priority of vulnerable building to undergo detailed assessment while covering uncertainties and minimizing their effects during evaluation. The proposed method estimates the vulnerability of a building, in terms of Visual Damage Index, considering the number of stories, age of building, plan irregularity, vertical irregularity, building quality, and peak ground velocity, as inputs with a single output variable. Applicability of the proposed method has been investigated using a post-earthquake damage database of 28 reinforced concrete buildings from the Bingöl earthquake in Turkey.
Subject: Behavioral Sciences, Clinical Psychology Keywords: involuntary memories; causal logic and semiotical logic; unconscious; mathematical model of the mind-matter relation; idiotope; category; discrete cofibration
Online: 14 February 2020 (11:44:16 CET)
Using classical clinical observations, we first outline an elementary conceptual model for the Mind Representation System, then move to a more elaborate mathematical model that refers to discrete cofibration with enriched fibers.
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Model-Based Systems Engineering; Category Theory; Object-Process Methodology; Model Analytics; Concept-Model-Graph-View-Concept; Graph Data Structures; Graph Query; Decision Support Matrix; Matrix-Based Analysis
Online: 18 February 2021 (12:27:50 CET)
We introduce the Concept-Model-Graph-View Cycle (CMGVC). The CMGVC facilitates coherent architecture analysis, reasoning, insight, and decision-making based on conceptual models that are transformed into a generic, robust graph data structure (GDS). The GDS is then transformed into multiple views of the model, which inform stakeholders in various ways. This GDS-based approach decouples the view from the model and constitutes a powerful enhancement of model-based systems engineering (MBSE). The CMGVC applies the rigorous foundations of Category Theory, a mathematical framework of representations and transformations. We show that modeling languages are categories, drawing an analogy to programming languages. The CMGVC architecture is superior to direct transformations and language-coupled common representations. We demonstrate the CMGVC to transform a conceptual system architecture model built with the Object Process Modeling Language (OPM) into dual graphs and a stakeholder-informing matrix that stimulates system architecture insight.
ARTICLE | doi:10.20944/preprints202112.0067.v1
Subject: Physical Sciences, Mathematical Physics Keywords: category; topos; presheaf; probability; validity; truth; conditional expectation; measurement; quantum mechanics; information; entropy; reduction; collapse; projection; logic; algebra; Wiener; Bayes; Boole; Heyting; Brownian motion; filter; crible; capacity; reservation; context
Online: 6 December 2021 (12:13:38 CET)
Research for a theory of quantum gravity has recently led to the use of presheaf topos. Quantum uncertainty is linked to the truth values of intuitionistic logic. This paper proposes transposing this model into a classic probability context, that of conditional mathematical expectations. A simulation of Brownian motion is offered for illustrative purposes.