Categorical Interpretation Research Articles

A categorification of cluster algebras of type B and C through symmetric quivers

We express cluster variables of type Bn and Cn in terms of cluster variables of type An. Then we associate a cluster tilted bound symmetric quiver Q of type A2n−1 to any seed of a cluster algebra of type Bn and Cn. Under this correspondence, cluster variables of type Bn (resp. Cn) correspond to orthogonal (resp. symplectic) indecomposable representations of Q. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.

Laurent family of simple modules over quiver Hecke algebras

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring $\mathcal {A}_q(\mathfrak {n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition vector of a simple module $X$ and categorical interpretation of (co)degree of $[X]$. As applications of such $\mathbb {Z}\mspace {1mu}$-vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $\Lambda$-invariants of $R$-matrices in the quiver Hecke algebra theory.

Examining a sentiment algorithm on session patient records in an eating disorder treatment setting: a preliminary study.

Clinicians collect session therapy notes within patient session records. Session records contain valuable information about patients' treatment progress. Sentiment analysis is a tool to extract emotional tones and states from text input and could be used to evaluate patients' sentiment during treatment over time. This preliminary study aims to investigate the validity of automated sentiment analysis on session patient records within an eating disorder (ED) treatment context against the performance of human raters. A total of 460 patient session records from eight participants diagnosed with an ED were evaluated on their overall sentiment by an automated sentiment analysis and two human raters separately. The inter-rater agreement (IRR) between the automated analysis and human raters and IRR among the human raters was analyzed by calculating the intra-class correlation (ICC) under a continuous interpretation and weighted Cohen's kappa under a categorical interpretation. Furthermore, differences regarding positive and negative matches between the human raters and the automated analysis were examined in closer detail. The ICC showed a moderate automated-human agreement (ICC = 0.55), and the weighted Cohen's kappa showed a fair automated-human (k = 0.29) and substantial human-human agreement (k = 0.68) for the evaluation of overall sentiment. Furthermore, the automated analysis lacked words specific to an ED context. The automated sentiment analysis performed worse in discerning sentiment from session patient records compared to human raters and cannot be used within practice in its current state if the benchmark is considered adequate enough. Nevertheless, the automated sentiment analysis does show potential in extracting sentiment from session records. The automated analysis should be further developed by including context-specific ED words, and a more solid benchmark, such as patients' own mood, should be established to compare the performance of the automated analysis to.

Beyond Protest

Abstract The systematic protests by the nuncio Fabio Chigi against the peace treaties negotiated during the Congress of Westphalia (1643–1649) have long been interpreted as a papal diplomatic failure, at a time when the ideal of Christianity was definitively giving way to a Europe made up of states. This paper seeks to move away from this categorical interpretation of the phenomenon. By studying the correspondence of the nuncios stationed in Madrid, Paris, Brussels and Vienna, the aim is to analyse how the peace of Westphalia was received, and determine whether this reception differs from the protest politics pursued by Fabio Chigi in Münster. It emerges that the nunciatures had varied reactions to the negotiations and the treaties, within fragmented contexts where the peace in Germany was just one of many current events, which were dominated by ongoing conflicts. The primary characteristic of these reactions is that they result from the local activities of each nuncio. By focusing on the variety of reactions, this study offers a polycentric and interconnected understanding of papal diplomacy at the time of the Congress of Westphalia. It aims to provide a better comprehension of the ‚agency‘ of the apostolic nuncios in redefining the diplomatic culture of the Holy See, through their differing receptions, practices and uses of the notion of peace.

Deformed Cartan matrices and generalized preprojective algebras II: general type

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß–Leclerc–Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura–Pestun in their study of quiver mathcal {W}-algebras.

The Ore Extension of Group-Cograded Hopf Coquasigroups

The aim of this paper is the Ore extension of group-cograded Hopf coquasigroups. This paper first shows a categorical interpretation and some examples of group-cograded Hopf coquasigroups, and then provides a necessary and sufficient condition for the Ore extension of group-cograded Hopf coquasigroups to be group-cograded Hopf coquasigroups. Finally, a certain isomorphism between Ore extensions are considered.

On homological mirror symmetry for chain type polynomials

We consider Takahashi’s categorical interpretation of the Berglund–Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed \(A_{\infty }\)-categories, which may be of independent interest. We give a full proof of this claim on the B-side. On the A-side we give a detailed sketch of an argument, which falls short of a full proof because of certain missing foundational results in Fukaya–Seidel categories, most notably a generation statement.

『중용장구』 20장 해석에 대한 비교 고찰

Chapter 20 of Juongyongjanggu has been criticized not only for the problem of sentence division but also for the interpretation of the concepts contained in it. This paper compares and analyzes the interpretation of Ju-hee and Wang-buji related to the ‘three good ways’, ‘three knowledges, three practices’, and ‘three closer’ among the main concepts shown in Chapter 20. Ju-hee used the dual perspective of categorical interpretation and graded interpretation in connecting and interpreting the ‘three good ways’ and the ‘three knowledges, three practices’ according to a perspective on knowledge and practices. This causes the complexity of interpretation, which has become one of the important problems in the history of Juongyong interpretation. There are many people who raised questions about Ju-hee’s interpretation, but this paper examined Wang-buji’s opinion from a relatively perspective.
 In the interpretation of Chapter 20 of Jungyongjanggu, it is believed that the reason why Ju-hee interpreted it from two perspectives is to emphasize the character of knowledge and practice. On the other hand, Wang-buji rejects Ju-hee’s inter-pretation and provides his own interpretation by his ideological diagram that emphasizes the unity of knowledge and practice, and the priority of “practice” Although he accepts most of Ju-hee’s views in the interpretation of the Saseo, he is not following Ju-hee’s view in this matter.

τ-tilting finiteness of non-distributive algebras and their module varieties

We treat the τ-tilting finiteness of minimal representation-infinite algebras and particularly the non-distributive ones. Building upon the new results of Bongartz, we fully determine which algebras in this family are τ-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the minimal representation-infinite biserial algebras. Consequently, we obtain nontrivial explicit conditions for τ-tilting infiniteness of a large family of algebras. This also produces concrete families of “minimal τ-tilting infinite algebras”, recently studied in our work.We further use our results to establish a conjectural connection between the τ-tilting theory and two geometric notions (the dense orbit property and Schur-representation finiteness) introduced by Chindris, Kinser and Weyman while studying module varieties. We verify the conjectures for the algebras studied in this note: For the minimal representation-infinite algebras which are non-distributive or biserial, if Λ has the dense orbit property, then Λ is τ-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is τ-tilting finite. This gives a categorical interpretation of Schur-representation-finiteness over this family of algebras.

Interobserver and Intraobserver Reliability of Classification Systems for Radiographic Complications After Radial Head Arthroplasty

Although several classifications are used to assess radiographs following radial head arthroplasty (RHA), including the Popovic classification for radiolucency, the Chanlalit classification for stress shielding (SS), the Brooker classification for heterotopic ossification (HO), and the Broberg-Morrey classification for radiocapitellar arthritis, little is known about the reliability of these classification systems. The purpose of this study was to determine the interobserver and intraobserver reliability of these classifications. Six orthopedic surgeons at various levels of training reviewed elbow radiographs of 20 patients who underwent RHA and classified them according to the Popovic, Chanlalit, Brooker, and Broberg-Morrey classifications for radiolucency, SS, HO, and RHA, respectively. Four weeks after initial review, radiographic reviews were repeated. Reliability was measured using the Fleiss kappa and the intraclass correlation coefficient. Agreement was interpreted as none (<0), slight (0.01-0.2), fair (0.21-0.4), moderate (0.41-0.6), substantial (0.61-0.8), and almost perfect (0.81-1) based on agreement among attending surgeons. Among fellowship-trained attending surgeons, interobserver reliability was slight for SS (Chanlalit) and the categorical interpretation of radiolucency (Popovic), fair for radiocapitellar arthritis (Broberg-Morrey) and HO (Brooker), and substantial for the ordinal interpretation of radiolucency (Popovic). Residents had a higher interobserver reliability than attending physicians when using the Brooker classification. Mean intraobserver reliability was fair for SS (Chanlalit) and the categorical interpretation of radiolucency (Popovic), moderate for HO (Brooker) and radiocapitellar arthritis (Broberg-Morrey), and almost perfect for the ordinal interpretation of radiolucency (Popovic). Trainees had higher intraobserver reliability than attending surgeons using the SS (Chanlalit) classification. The number of Popovic zones is reliable for communication between physicians, but caution should be taken with the Brooker, Chanlalit, Broberg-Morrey, and categorical interpretation of the Popovic classifications. All the classifications had better intraobserver than interobserver reliability. Reliability of classification systems for radiographic complications after RHA is less than substantial except the number of zones of radiolucency; therefore, caution is required when drawing conclusions based on these classifications.

Association between digital smart device use and myopia: a systematic review and meta-analysis

Excessive use of digital smart devices, including smartphones and tablet computers, could be a risk factor for myopia. We aimed to review the literature on the association between digital smart device use and myopia. In this systematic review and meta-analysis we searched MEDLINE and Embase, and manually searched reference lists for primary research articles investigating smart device (ie, smartphones and tablets) exposure and myopia in children and young adults (aged 3 months to 33 years) from database inception to June 2 (MEDLINE) and June 3 (Embase), 2020. We included studies that investigated myopia-related outcomes of prevalent or incident myopia, myopia progression rate, axial length, or spherical equivalent. Studies were excluded if they were reviews or case reports, did not investigate myopia-related outcomes, or did not investigate risk factors for myopia. Bias was assessed with the Joanna Briggs Institute Critical Appraisal Checklists for analytical cross-sectional and cohort studies. We categorised studies as follows: category one studies investigated smart device use independently; category two studies investigated smart device use in combination with computer use; and category three studies investigated smart device use with other near-vision tasks that were not screen-based. We extracted unadjusted and adjusted odds ratios (ORs), β coefficients, prevalence ratios, Spearman's correlation coefficients, and p values for associations between screen time and incident or prevalent myopia. We did a meta-analysis of the association between screen time and prevalent or incident myopia for category one articles alone and for category one and two articles combined. Random-effects models were used when study heterogeneity was high (I2>50%) and fixed-effects models were used when heterogeneity was low (I2≤50%). 3325 articles were identified, of which 33 were included in the systematic review and 11 were included in the meta-analysis. Four (40%) of ten category one articles, eight (80%) of ten category two articles, and all 13 category three articles used objective measures to identify myopia (refraction), whereas the remaining studies used questionnaires to identify myopia. Screen exposure was measured by use of questionnaires in all studies, with one also measuring device-recorded network data consumption. Associations between screen exposure and prevalent or incident myopia, an increased myopic spherical equivalent, and longer axial length were reported in five (50%) category one and six (60%) category two articles. Smart device screen time alone (OR 1·26 [95% CI 1·00-1·60]; I2=77%) or in combination with computer use (1·77 [1·28-2·45]; I2=87%) was significantly associated with myopia. The most common sources of risk of bias were that all 33 studies did not include reliable measures of screen time, seven (21%) did not objectively measure myopia, and nine (27%) did not identify or adjust for confounders in the analysis. The high heterogeneity between studies included in the meta-analysis resulted from variability in sample size (range 155-19 934 participants), the mean age of participants (3-16 years), the standard error of the estimated odds of prevalent or incident myopia (0·02-2·21), and the use of continuous (six [55%] of 11) versus categorical (five [46%]) screen time variables INTERPRETATION: Smart device exposure might be associated with an increased risk of myopia. Research with objective measures of screen time and myopia-related outcomes that investigates smart device exposure as an independent risk factor is required. None.

Fourier matrices for G(d,1,n) from quantum general linear groups

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group G(d,1,n). The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table S of the cyclic group of order d. The representations of the quantum universal enveloping algebra of the general linear Lie algebra glm, with quantum parameter an even root of unity of order 2d, provide a categorical interpretation of the matrix ⋀mS. We also prove some positivity conjectures of Cuntz at the decategorified level.

CONCEPTUAL AND TERMINOLOGICAL APPARATUS OF PROBLEM RESEARCH OF INNOVATIVE CULTURE DEVELOPMENT OF THE FUTURE MANAGER OF THE GENERAL SECONDARY EDUCATIONAL INSTITUTION IN THE MASTER’S TRAINING PROCESS

Formulation of the problem. The need for a categorical interpretation of the thesaurus is due to methodological priorities for scientific, objectivity, systemic, integrity, unity, complexity and logic of the study. The correctness of the use of relevant concepts and categories is determined by the accuracy of their definitions, fully depends on the verification of the presented innovative culture concept of the leader. Consequently, there is a need for a comprehensive research analysis of conceptual and terminological apparatus of problem research of innovative culture development of the future manager of the general secondary educational institution in the master’s training process. Materials and Methods: Solving the highlighted aim, a set of methods of scientific research adequate to them were used, theoretical: a comparative analysis of conceptual and terminological apparatus of problem research of innovative culture development of the future manager of the general secondary educational institution in the master’s training process; a systematic analysis of conceptual and terminological apparatus of problem research of innovative culture development of the future manager of the general secondary educational institution in the master’s training process. Results. The article presents the innovation culture development of the future manager of general secondary educational institution as a systematically organized, respectively providing process in master's training, determined by the needs of society, state and customers of educational services and institutionalized, which substantiates the relationship and interdependence of innovation culture, which correlate with the content and pedagogically appropriate methodological systems of professional training at higher education. It is determined the IC development of the future manager of general secondary educational institution. Conclusions. Thus, modern theories of social development make it possible to identify the specifics of the innovative culture development and give grounds to conclude that the innovation and innovation culture development is a natural, necessary, historically determined and self-organized process of objectification and institutionalization of innovative content at the social and individual levels. It is important to understand that the emergence of innovation and innovation culture is possible only if the dialectical and synergetic nature of life is taken into account. For the innovation culture development is exceptional importance the openness of educational systems, the presence of the environment and mechanisms for the natural struggle of contradictory and opposing forces, the absence of authoritarian and artificial interference in the institutionalization of innovation culture.

СУТНІСТЬ ІННОВАЦІЙ ТА ІННОВАЦІЙНОГО ПІДПРИЄМНИЦТВА ЗА УМОВ ЗМІНИ СПОЖИВЧИХ ПЕРЕВАГ ТА МАРКЕТИНГОВОЇ ПОВЕДІНКИ ПІДПРИЄМСТВ

The article examines the essence of innovation and innovative entrepreneurship in terms of changing consumer preferences and marketing behavior of enterprises. It has been proven that innovation is the best way of scientific and technological re-equipment, enrichment of the population and gradual diversification of production. An “explosion” of innovative activity is predicted, which can be carried out by effectively functioning entrepreneurship, capable of providing an innovative economy of the future, based on a dramatic increase in productivity, an increase in employment and the strengthening of social standards. To determine the priorities of further effective development of entrepreneurship in the innovation economy systematized terminological apparatus that forms this phenomenon. In particular, the theoretical reflection of the object of a higher level of organization and aggregation of entrepreneurship in the innovation economy in relation to its components is presented as an opportunity to characterize the studied phenomenon as externally determined ontological structuring. This logically reflects the segmentation of entrepreneurship in the innovation economy as a complex whole into relatively independent subject areas, each of which reflects a very specific aspect of its organizational and economic functioning. It is proved that the phenomenon of entrepreneurship, which operates in an innovative economy, has a wide range of basic concepts that characterize it: innovation, entrepreneurship, innovation environment, etc. Representative groups for the definition of innovation in the system of economy “rational use of innovation”, “innovation as a dynamic”, “innovation as a goal-setting process”, “innovation in the context of the foresight of the national economy”. It is concluded that the definition of the category “innovation” is constantly changing, updated and expanded in accordance with modern conditions. This phenomenon can be explained by the fact that changes in the parameters of the innovation system of the national economy are ahead of their scientific understanding and, accordingly, complicate the categorical interpretation of the concept of “innovation”.

Categorical and algebraic aspects of the intuitionistic modal logic IEL― and its predicate extensions

Abstract The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic (S. Artemov and T. Protopopescu. Intuitionistic epistemic logic. The Review of Symbolic Logic, 9, 266–298, 2016). We construct the modal lambda calculus, which is Curry–Howard isomorphic to $\textbf{IEL}^{-}$ as the type-theoretical representation of applicative computation widely known in functional programming.We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a Cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study complete Kripke–Joyal-style semantics for predicate extensions of $\textbf{IEL}^{-}$ and related logics using Dedekind–MacNeille completions and modal cover systems introduced by Goldblatt (R. Goldblatt. Cover semantics for quantified lax logic. Journal of Logic and Computation, 21, 1035–1063, 2011). The paper extends the conference paper published in the LFCS’20 volume (D. Rogozin. Modal type theory based on the intuitionistic modal logic IEL. In International Symposium on Logical Foundations of Computer Science, pp. 236–248. Springer, 2020).

On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules

We discuss finitely graded Iwanaga-Gorenstein (IG) algebras A and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where A is a trivial extension algebra A=Λ⊕C with the grading deg⁡Λ=0,deg⁡C=1. In [17] we gave a necessary and sufficient condition that A is IG in terms of Λ and C by using derived tensor products and derived Homs. For simplicity, we assume that Λ is of finite global dimension in the sequel. In this paper, we show that the condition that A is IG, has a triangulated categorical interpretation. We prove that if A is IG, then the graded stable category CM_ZA of CM-modules is realized as an admissible subcategory of the derived category Db(modΛ). As a corollary, we deduce that the Grothendieck group K0(CM_ZA) is free of finite rank.We give several applications. Among other things, for a path algebra Λ=kQ of an A2 or A3 quiver Q, we give a complete list of Λ-Λ-bimodule C such that Λ⊕C is IG (resp. of finite global dimension) by using the triangulated categorical interpretation mentioned above.

Weyl-Kac character formula for affine Lie algebras in Deligne's categories

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L(X,k) of the parabolic Verma module M(X,k) of level k, where X is an indecomposable object of Deligne's category Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions that the highest weight of X plus the level gives a fundamental weight, t is transcendental, and the base field k has characteristic 0. We compare our result to the partial result in [7, Problem 6.2], and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula, [12, Formula (6.12)].

O pograniczach słowotwórstwa i leksykologii – na wybranych przykładach form gwarowych

The author describes linguistic phenomena on the borderline of word formation and lexicology, mostly semantics and etymology, based on the examples of derivatives functioning in Polish dialects. The article presents examples of homonymy, polysemy and synonymy of derivatives whose both derivational and categorical interpretation must take their lexical meaning into consideration. The subject matter of this paper also includes the results of semantic categorial movements (e.g. names of actions which became names of manners of action: palenie, regulacja) as well as names in case of which the establishment of their derivation type, whether it is semantic or morphological, requires arbitrary decisions (e.g. names of manners of action such as lapka, koziolek). Moreover, derivatives whose analysis requires etymological search are also located on the borderline of word formation and lexicology. They include words with currently vague structure whose motivating words can be indicated through etymological inquiry based on lexicographical collections. While analysing dialect derivatives we are forced to use semantic-lexical and etymological tools and at times even go beyond the language system in search of name-forming factors.

Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.

Categorical Interpretation of Modal Structures under Bisimulation

Abstract In this work we summarise the concept of bisimulation, widely used both in computational sciences and in modal logic, that characterises modal structures with the same behaviour in terms of accessibility relations. Then, we offer a sketch of categorical interpretation of bisimulation between modal structures, which comprise both the structure and the valuation from a propositional language.