Hasse Diagram Research Articles

On Hasse diagrams connected with the 1-oversupercritical poset (1,2,7)

On Hasse diagrams connected with the 1-oversupercritical poset (1,2,7)

Folding orthosymplectic quivers

Folding identical legs of a simply-laced quiver creates a quiver with a non-simply laced edge. So far, this has been explored for quivers containing unitary gauge groups. In this paper, orthosymplectic quivers are folded, giving rise to a new family of quivers. This is realised by intersecting orientifolds in the brane system. The monopole formula for these non-simply laced orthosymplectic quivers is introduced. Some of the folded quivers have Coulomb branches that are closures of minimal nilpotent orbits of exceptional algebras, thus providing a new construction of these fundamental moduli spaces. Moreover, a general family of folded orthosymplectic quivers is shown to be a new magnetic quiver realisation of Higgs branches of 4d mathcal{N} = 2 theories. The Hasse (phase) diagrams of certain families are derived via quiver subtraction as well as Kraft-Procesi transitions in the brane system.

Coulomb branch global symmetry and quiver addition

To date, the best effort made to simply determine the Coulomb branch global symmetry of a theory from a 3d mathcal{N} = 4 quiver is by applying an algorithm based on its balanced gauge nodes. This often gives the full global symmetry, but there have been many cases seen where it instead gives only a subgroup. This paper presents a method for constructing several families of 3d mathcal{N} = 4 unitary quivers where the true global symmetry is enhanced from that predicted by the balance algorithm, motivated by the study of Coulomb branch Hasse diagrams. This provides a rich list of examples on which to test improved algorithms for unfailingly identifying the Coulomb branch global symmetry from a quiver.

Excluding a Ladder

A ladder is a $2 \times k$ grid graph. When does a graph class $\mathcal{C}$ exclude some ladder as a minor? We show that this is the case if and only if all graphs $G$ in $\mathcal{C}$ admit a proper vertex coloring with a bounded number of colors such that for every $2$-connected subgraph $H$ of $G$, there is a color that appears exactly once in $H$. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph $H$ of $G$, there must be a color that appears exactly once in $H$. The minimum number of colors in a centered coloring of $G$ is the treedepth of $G$, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length $k$ has a path of length $k+1$. We show that every $3$-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a $2 \times k$ grid has a $2 \times (k+1)$ grid minor. Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension.

Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems

We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry language whose primary letters are the alphabet. For proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets, etc.); for music, one is dealing with clear-cut repetition units called musical forms and for poems the structure consists of grammatical forms (names, verbs, etc.). We show in this paper that the mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r counts the number of types of such secondary non local segments. The number of conjugacy classes of a given index (also the number of graph coverings over a base graph) of a group fp is found to be close to the number of conjugacy classes of the same index in the free group Fr−1 on r−1 generators. In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to the secondary structure organizing the foldings. Then, we look at the musical forms employed in the classical and contemporary periods. Finally, we investigate in much detail the group structure of a small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.

Non-kissing complexes and tau-tilting for gentle algebras

We interpret the support τ \tau -tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g \mathbf {g} -vector fan and prove that it is the normal fan of a non-kissing associahedron.

Central sleep apnea treatment in patients with heart failure with reduced ejection fraction: a network meta-analysis.

Adaptive servo-ventilation (ASV) is contraindicated for the treatment of central sleep apnea (CSA) in patients with heart failure with reduced ejection fraction (HFrEF), limiting treatment options. Though continuous positive airway pressure (CPAP), bi-level PAP with back-up rate (BPAP-BUR), and transvenous phrenic nerve stimulation (TPNS) are alternatives, not much is known about their comparative efficacies, which formed the basis of conducting this network meta-analysis. We sought to analyze their comparative effectiveness in reducing apnea hypopnea index (AHI). Additionally, we also studied their comparative effectiveness on subjective daytime sleepiness as assessed by Epworth sleepiness score (ESS). Randomized controlled trials (RCTs) from PubMed were analyzed in a network meta-analysis and relative superiority was computed based on P-score ranking and Hasse diagrams. Network meta-analysis based on 8 RCTs showed that when compared to guideline-directed medical therapy (GDMT-used as a common comparator across trials), reduction in AHI by ASV (- 26.05 [- 38.80; - 13.31]), TPNS (- 24.90 [- 42.88; - 6.92]), BPAP-BUR (- 20.36 [- 36.47; - 4.25]), and CPAP (- 16.01 [- 25.42; - 6.60]) were statistically significant but not between the interventions. Based on 6 RCTs of all the interventions, only TPNS showed a statistically significant decrease in ESS (- 3.70 (- 5.58; - 1.82)) when compared to GDMT, while also showing significant differences when compared with ASV (- 3.20 (- 5.86; - 0.54)), BPAP-BUR (- 4.00 (- 7.33; - 0.68)), and CPAP (- 4.45 (- 7.75; - 1.14)). Ranking of treatments based on Hasse diagram, accounting for both AHI and ESS as outcomes for relative hierarchy showed relative superiority of both ASV and TPNS over BPAP-BUR and CPAP. Results indicated relative superiority of TPNS and ASV to BPAP-BUR and CPAP in their effects on AHI and ESS.

Effect of Pharmacological and Neurostimulation Interventions for Cognitive Domains in Patients with Bipolar Disorder: A Systematic Review and Network Meta-Analysis of Randomized Controlled Trials.

IntroductionThe priority of interventions to alleviate cognitive deficits in patients with bipolar disorder (BD) is inconclusive. We systematically evaluate the efficacy of pharmacological or neurostimulation interventions for cognitive function in BD through a network meta-analysis.MethodsThe PubMed, PsycINFO, Embase, and Cochrane Library databases were searched from database inception to September 30, 2021. Following PRISMA guidelines, all eligible studies were randomized controlled trials of adult bipolar patients that provided detailed cognitive outcomes. Studies were excluded if participants limited to comorbid substance use disorder or the intervention was a psychotherapy. Network meta-analysis comparing different interventions was conducted for 8 cognitive domains. Partially ordered set with Hasse diagram was used to resolve conflicting rankings between outcomes. The study was preregistered on PROSPERO database (CRD42020152044).ResultsTotal 21 RCTs including 42 tests for assessing intervention effects on cognition were retrieved. Adjunctive erythropoietin (SMD = 0.61, 95% CI = 0.00–1.23), Withania somnifera (SMD = 0.58, 95% CI = 0.03–1.13), and galantamine (SMD = 1.22, 95% CI = 0.10–2.35) was more beneficial for attention, working memory, and verbal learning in euthymic BD patients than treatment as usual, respectively. Hasse diagram suggested ranking of choice when multiple domains were combined.ConclusionConsiderable variability in measurements of cognitive domains in BD was observed, and no intervention resulted in superior benefits across all domains. We suggested interventions priority can be tailored according to individual patients’ cognitive deficits. As current findings from relatively small and heterogeneous dataset, future trials with consensus should be applied for building further evidence.

EHreact: Extended Hasse Diagrams for the Extraction and Scoring of Enzymatic Reaction Templates.

Data-driven computer-aided synthesis planning utilizing organic or biocatalyzed reactions from large databases has gained increasing interest in the last decade, sparking the development of numerous tools to extract, apply, and score general reaction templates. The generation of reaction rules for enzymatic reactions is especially challenging since substrate promiscuity varies between enzymes, causing the optimal levels of rule specificity and optimal number of included atoms to differ between enzymes. This complicates an automated extraction from databases and has promoted the creation of manually curated reaction rule sets. Here, we present EHreact, a purely data-driven open-source software tool, to extract and score reaction rules from sets of reactions known to be catalyzed by an enzyme at appropriate levels of specificity without expert knowledge. EHreact extracts and groups reaction rules into tree-like structures, Hasse diagrams, based on common substructures in the imaginary transition structures. Each diagram can be utilized to output a single or a set of reaction rules, as well as calculate the probability of a new substrate to be processed by the given enzyme by inferring information about the reactive site of the enzyme from the known reactions and their grouping in the template tree. EHreact heuristically predicts the activity of a given enzyme on a new substrate, outperforming current approaches in accuracy and functionality.

One-sided topological conjugacy of normal subshifts and gauge actions on the associated C*-algebras

The class of normal subshifts includes irreducible nontrivial topological Markov shifts, irreducible nontrivial sofic shifts, synchronized systems, Dyck shifts, β-shifts, substitution minimal shifts, and so on. We will characterize one-sided topological conjugacy classes of normal subshifts in terms of the associated -algebras and its gauge actions with potentials. In particular, the one-sided topological conjugacy class of irreducible nontrivial sofic shifts are characterized in terms of the simple purely infinite Cuntz-Krieger algebras associated with the matrices of its left Fischer cover graphs and gauge actions with potentials. This generalizes the author's previous result for irreducible nontrivial topological Markov shifts and Cuntz-Krieger algebras with gauge actions.

Investigation of Cyber-Security and Cyber-Crimes in Oil and Gas Sectors Using the Innovative Structures of Complex Intuitionistic Fuzzy Relations.

Recently, there has been enormous development due to advancements in technology. Industries and enterprises are moving towards a digital system, and the oil and gas industries are no exception. There are several threats and risks in digital systems, which are controlled through cyber-security. For the first time in the theory of fuzzy sets, this research analyzes the relationships between cyber-security and cyber-crimes in the oil and gas sectors. The novel concepts of complex intuitionistic fuzzy relations (CIFRs) are introduced. Moreover, the types of CIFRs are defined and their properties are discussed. In addition, an application is presented that uses the Hasse diagram to make a decision regarding the most suitable cyber-security techniques to implement in an industry. Furthermore, the omnipotence of the proposed methods is explained by a comparative study.

Combinatorial generation via permutation languages. II. Lattice congruences

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.

Cybersecurity against the Loopholes in Industrial Control Systems Using Interval-Valued Complex Intuitionistic Fuzzy Relations

Technology is rapidly advancing and every aspect of life is being digitalized. Since technology has made life much better and easier, so organizations, such as businesses, industries, companies and educational institutes, etc., are using it. Despite the many benefits of technology, several risks and serious threats, called cyberattacks, are associated with it. The method of neutralizing these cyberattacks is known as cybersecurity. Sometimes, there are uncertainties in recognizing a cyberattack and nullifying its effects using righteous cybersecurity. For that reason, this article introduces interval-valued complex intuitionistic fuzzy relations (IVCIFRs). For the first time in the theory of fuzzy sets, we investigated the relationships among different types of cybersecurity and the sources of cyberattacks. Moreover, the Hasse diagram for the interval-valued complex intuitionistic partial order set and relation is defined. The concepts of the Hasse diagram are used to inspect different cybersecurity techniques and practices. Then, using the properties of Hasse diagrams, the most beneficial technique is identified. Furthermore, the best possible selection of types of cybersecurity is made after putting some restrictions on the selection. Lastly, the advantages of the proposed methods are illuminated through comparison tests.

Transformations and information granularity of knowledge structures in set-based granular computing

Granular computing is a relatively new platform for constructing, describing and processing information or knowledge. For crisp information granulation, the universe is decomposed into granules by binary relations on the universe, say, preorder, tolerance and equivalence relations. A knowledge structure is composed of all information granules induced by a relation that corresponds to the granulation. This paper establishes a novel theoretical framework for the measurement of information granularity of knowledge structures. First, two new relations between knowledge structures are introduced through the use of their respective Boolean relation matrices, where the granular equality relation is defined based on an orthogonal transformation with the transformation matrix being a permutation matrix, and the granularly finer relation is presented by combining the classical finer relation and the orthogonal transformation. Then, it is demonstrated that the simplified knowledge structure base with the granularly finer relation is a partially ordered set, which can be represented by a Hasse diagram. Subsequently, an axiomatic definition of information granularity is proposed to satisfy the constraints regarding these two relations. Moreover, a general form of the information granularity is given, and some existing measures are proved to be its special cases. Finally, as an application of the proposed measure, the attribute significance measure is developed based on the information granularity.

An Analysis of the 4th Movement of György Ligeti’s Musica Ricercata based on Information Theory and Number Partitioning

In this paper we present a mathematically oriented analysis of the 4th Movement of György Ligeti’s Musica Ricercata (MR). The pitch analysis is based on Theory of Information and rhythm is analyzed through the Theory of Partitions of integer numbers. After a brief historical review of Musica Ricercata and its structure we make an analysis of the pitch distribution along the whole MR4 score, as well the Left and Right hand separately, through Theory of Information. We calculate two Information measures, namely, the Shannon’s Entropy and the Kullback-Leibler Divergence for the three cases. In the second part, about rhythm, we introduce a simple notation for coding rhythm patterns in terms of partitions of an integer number. We show that, with few exceptions, Ligeti used the partitions of number 6 to get rhythm variations on the Right hand against the ostinato on the Left one. In addition, we show the usefulness of the so called Hasse Diagram as a pre-compositional device to generate rhythm patterns.

Graph Coverings and (Im)primitive Homology: Examples of Low Degree

Given a finite covering of graphs , it is not always the case that is spanned by lifts of primitive elements of . In this article, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph covers where the deck group is a finite p-group. For such covers, there is a representation-theoretic criterion for identifying deck groups for which there exist covers with the property. We present an algorithm for determining if a finite p-group satisfies this criterion that uses only the character table of the group. Finally, we provide a complete census of all finite p-groups of rank and order < 1000 satisfying this criterion, all of which are new examples.

Edgeworth Expansions for Centered Random Walks on Covering Graphs of Polynomial Volume Growth

Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends on not only geometric features of the underlying graphs but also the modified harmonic embedding of the graph into a certain nilpotent Lie group. Moreover, we apply the rate of convergence in Trotter's approximation theorem to establish the Berry-Esseen type bound for the random walks.

Optimization of the Design of Diagnostic Systems through Requirement-Based Feature Selection

The classification of faults based on available diagnostic information in large complex mechatronic systems is a well-studied research subject. One challenge is the classification of faults with missing or ambiguous diagnostic information present. In real-world applications, this has the effect of misclassified faults, resulting in costly replacements of functional components. As even the acquisition of diagnostic information is afflicted with costs and part of the diagnostic functions are mandatory, the design of diagnostic content at an early system development stage is essential. The paper presents a procedure to automatically allocate a set of diagnostic functions for a complex mechatronic system that is informative and low in costs. It proposes a granular graph structure called the Diagnostic Cover Graph (DCG) to represent the system functionalities and the respective diagnostic functions. The DCG can represent restrictions on the availability of functionalities, restrictions on the executability of diagnostic functions and potential interdependence between them. On its basis, a decision table is generated, and a minimal test cost feature selection method such as the Test-Cost-Sensitive Quick Reduct (TCSQR) is conducted. The approach guarantees the proposal of a close-to-optimal subset of diagnostic functions. The capability of the presented procedure is demonstrated for the vehicle brake system for several diagnostic scenarios.

Equationally extreme trees

We study extreme semilattices whose Hasse diagrams are trees, and prove that the linearly ordered tree of order n has the minimal average number of solutions of 1-variable equations.

Triadic concept approximation

Formal Concept Analysis is a mathematical theory for knowledge representation as well as data analysis and visualization. It provides a mechanism for understanding and mining data through concept lattice construction and exploration. This theory assists in data processing by providing a framework for applying different analysis techniques. One of its potentials lies in its mathematical foundation that enables the generation, ordering, and visualization of knowledge in the form of formal concepts described in a hierarchy known as a concept lattice. However, as the input dataset grows, the search and especially the visualization and exploration of concepts becomes prohibitive. With the extension of the classical FCA approach to Triadic Concept Analysis, this problem becomes even more evident due to the complexity of the inherent structures in triadic concepts and relationships. In this work, we propose an approach to find triadic concepts when a query in the form of a triple is given, allowing the visualization and exploration of a Hasse diagram of triadic concepts.