Class Of Equations Research Articles

Green groupoids of 2-Calabi–Yau categories, derived Picard actions, and hyperplane arrangements

We present a construction of (faithful) group actions via derived equivalences in the general categorical setting of algebraic 2-Calabi–Yau triangulated categories. To each algebraic 2-Calabi–Yau category C \mathscr {C} satisfying standard mild assumptions, we associate a groupoid G C \mathscr {G}_{ \mathscr {C} } , named the green groupoid of C \mathscr {C} , defined in an intrinsic homological way. Its objects are given by a set of representatives m r i g C mrig\mathscr {C} of the equivalence classes of basic maximal rigid objects of C \mathscr {C} , arrows are given by mutation, and relations are given by equating monotone (green) paths in the silting order. In this generality we construct a homomorphsim from the green groupoid G C \mathscr {G}_{ \mathscr {C} } to the derived Picard groupoid of the collection of endomorphism rings of representatives of m r i g C mrig\mathscr {C} in a Frobenius model of C \mathscr {C} ; the latter canonically acts by triangle equivalences between the derived categories of the rings. We prove that the constructed representation of the green groupoid G C \mathscr {G}_{ \mathscr {C} } is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of C \mathscr {C} come from hyperplane arrangements. If Σ 2 ≅ i d \Sigma ^2 \cong id and C \mathscr {C} has finitely many equivalence classes of basic maximal rigid objects, we prove that G C \mathscr {G}_{ \mathscr {C} } is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.

Loss Function Information Fusion and Decision Rule Deduction of Three-Way Decision by Constructing Interval-Valued $q$-Rung Orthopair Fuzzy Integral

Three-way decision is consistent with human cognitive processes and applied to different information systems. However, since the real world is more imprecision, hesitation, and ambiguity, experts can select interval fuzzy information to represent their evaluations. In the evaluation process, experts sometimes may not guarantee the interval fuzzy information satisfy the constraint between membership and nonmembership, which further impacts make decisions. Interval-valued <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -rung orthopair fuzzy set (IVq-ROFS) are proposed to improve the fault-tolerant ability and ensure the experts’ evaluation make sense. In this case, this article generalizes IVq-ROFS to three-way decision, which consists of the construction of three-way decision and the aggregation of interval continuity of IVq-ROFS. For considering the aggregation of interval continuity of IVq-ROFS, we first investigate the IVq-ROFS integral to aggregate continuous IVq-ROFSs by developing simplified interval-valued <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -rung orthopair fuzzy set. Particularly, in order to obtain integrable region, we design <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -amendment to amend integral region of IVq-ROFS for removing unmeaning point in advance. To simplify the calculation, we further utilize Taylor series of the integral function to replace the complicated integral function. Second, a new equivalence class method is constructed to support three-way decision. Third, with the aid of the integral of IVq-ROFS and equivalence class method, we availably fuse the loss functions of the equivalence class and obtain the classifying rules of three-way decision by comparing the expected losses. Finally, we elaborate three-way decision by using a case of medical diagnosis about nephropathy and prove the validity of the investigated model.

A covariant Stinespring theorem

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.

Measuring the impact of anonymization on real-world consolidated health datasets engineered for secondary research use: Experiments in the context of MODELHealth project.

IntroductionElectronic Health Records (EHRs) are essential data structures, enabling the sharing of valuable medical care information for a diverse patient population and being reused as input to predictive models for clinical research. However, issues such as the heterogeneity of EHR data and the potential compromisation of patient privacy inhibit the secondary use of EHR data in clinical research.ObjectivesThis study aims to present the main elements of the MODELHealth project implementation and the evaluation method that was followed to assess the efficiency of its mechanism.MethodsThe MODELHealth project was implemented as an Extract-Transform-Load system that collects data from the hospital databases, performs harmonization to the HL7 FHIR standard and anonymization using the k-anonymity method, before loading the transformed data to a central repository. The integrity of the anonymization process was validated by developing a database query tool. The information loss occurring due to the anonymization was estimated with the metrics of generalized information loss, discernibility and average equivalence class size for various values of k.ResultsThe average values of generalized information loss, discernibility and average equivalence class size obtained across all tested datasets and k values were 0.008473 ± 0.006216252886, 115,145,464.3 ± 79,724,196.11 and 12.1346 ± 6.76096647, correspondingly. The values of those metrics appear correlated with factors such as the k value and the dataset characteristics, as expected.ConclusionThe experimental results of the study demonstrate that it is feasible to perform effective harmonization and anonymization on EHR data while preserving essential patient information.

Distance–transitive digraphs: Descendant-homogeneity, property [formula omitted] and reachability

A digraph is distance–transitive if for every integer s≥0, the automorphism group of the digraph is transitive on pairs (u,v) of vertices for which there is a directed path of length s from u to v, but no directed path of length t<s. This implies vertex and edge transitivity but is weaker than being highly-arc-transitive. A digraph is said to have property Z if it has a homomorphism onto the two-way infinite directed path. In a digraph D, an edge e′ is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e′. Reachability is an equivalence relation on the edge set of D and if D is transitive on edges, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs.We investigate the class of infinite distance–transitive digraphs of finite out-valency. First, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold for the class of distance–transitive digraphs. Second, we show that if D is a weakly descendant–homogeneous digraph in our class then either (1) D has property Z and the reachability relation is not universal; or (2) D does not have property Z, the reachability relation is universal and D has infinite in-valency. Finally, we describe some distance–transitive, weakly descendant–homogeneous digraphs for which the reachability relation is not universal.

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries

Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.

Independence equivalence classes of paths

The independence equivalence class of a graph G is the set of graphs that have the same independence polynomial as G. A graph whose independence equivalence class contains only itself, up to isomorphism, is independence unique. Beaton, Brown and Cameron [2] showed that paths with an odd number of vertices are independence unique and raised the problem of finding the independence equivalence class of paths with an even number of vertices. The problem is completely solved in this paper.

Staphylococcus epidermidis RP62A's Metabolic Network: Validation and Intervention Strategies.

Increasingly, systems biology is gaining relevance in basic and applied research. The combination of computational biology with wet laboratory methods produces synergy that results in an exponential increase in knowledge of biological systems. The study of microorganisms such as Staphylococcus epidermidis RP62A enables the researcher to understand better their metabolic networks, which allows the design of effective strategies to treat infections caused by this species or others. S. epidermidis is the second most commoncause of infection in patients with joint implants, so treating its proliferation seems vital for public health. There are different approaches to the analysis of metabolic networks. Flux balance analysis (FBA) is one of the most widespread streams of research. It allows the study of large metabolic networks, the study their structural properties, the optimization of metabolic flux, and the search for intervention strategies to modify the state of the metabolic network. This work presents the validation of the Staphylococcus epidermidis RP62A metabolic network model elaborated by Díaz Calvo et al. Then, we elaborate further on the network analysis’s essential reactions. The full set of essential reactions (including a previously unobserved one) was computed, and we classified them into equivalence classes. Some proposals to intervene in the network and design knock-outs by studying minimal cut sets of small length are also introduced. In particular, minimal cut sets related to the medium (including exchange reactions associated with medium metabolites) have been computed. In this sense, the unique external MCS (composed of cysteine and sulfate ion) has been found, and all hybrid MCS (based on knocking out both internal and exchange reactions) of length two have also been computed. The paper also points out the possible importance of these new intervention strategies.

Rigidity of valuative trees under henselization

Let $(K,v)$ be a valued field and let $(K^h,v^h)$ be the henselization determined by the choice of an extension of $v$ to an algebraic closure of $K$. Consider an embedding $v(K^*)\hookrightarrow\Lambda$ of the value group into a divisible ordered abelian group. Let $T(K,\Lambda)$, $T(K^h,\Lambda)$ be the trees formed by all $\Lambda$-valued extensions of $v$, $v^h$ to the polynomial rings $K[x]$, $K^h[x]$, respectively. We show that the natural restriction mapping $T(K^h,\Lambda)\to T(K,\Lambda)$ is an isomorphism of posets. As a consequence, the restriction mapping $T_v\to T_{v^h}$ is an isomorphism of posets too, where $T_v$, $T_{v^h}$ are the trees whose nodes are the equivalence classes of valuations on $K[x]$, $K^h[x]$ whose restriction to $K$, $K^h$ are equivalent to $v$, $v^h$, respectively.

On some symmetries of the base n expansion of 1∕m : the class number connection

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Furthermore, we obtain corollaries connecting the $ 3 $ rank of the class number of the real quadratic field $ \mathbb{Q}(\sqrt{m}) $ to the $ 3 $ divisibility of the number of certain quadratic residues modulo $ m $. Secondly, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain distribution results of $ m $ modulo any prime $ p $ that properly divides $ n+1 $. Our result implies that the primes $ m $ for which $ n $ is a primitive root belong to one of two equivalence classes modulo any prime $ p $ as above.

Lipschitz geometry of pairs of normally embedded Hölder triangles

We consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded Hölder triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called sigma tau -pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.

Complete classification of steerability under local filters and its relation with measurement incompatibility

Quantum steering is a central resource for one-sided device-independent quantum information. It is manipulated via one-way local operations and classical communication, such as local filtering on the trusted party. Here, we provide a necessary and sufficient condition for a steering assemblage to be transformable into another via local filtering. We characterize the equivalence classes with respect to filters in terms of the steering equivalent observables (SEO), first proposed to connect the problem of steerability and measurement incompatibility. We provide an efficient method to compute the extractable steerability that is maximal via local filters and show that it coincides with the incompatibility of the SEO. Moreover, we show that there always exists a bipartite state that provides an assemblage with steerability equal to the incompatibility of the measurements on the untrusted party. Finally, we investigate the optimal success probability and rates for transformation protocols (distillation and dilution) in the single-shot scenario together with examples.

The boundary at infinity of the curve complex and the relative Teichmüller space

In this paper we study the boundary at infinity of the curve complex \mathcal{C}(S) of a surface S of finite type and the relative Teichmüller space \mathcal{T}_{\mathrm{el}}(S) obtained from the Teichmüller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) are quasi-isometric, and Masur–Minsky have shown that \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) are hyperbolic in the sense of Gromov. We show that the boundary at infinity of \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) is the space of topological equivalence classes of minimal foliations on S .

Primitively universal quaternary quadratic forms

A (positive definite and integral) quadratic form f is said to be universal if it represents all positive integers, and is said to be primitively universal if it represents all positive integers primitively. We also say f is almost primitively universal if it represents almost all positive integers primitively. Conway and Schneeberger proved (see [1]) that there are exactly 204 equivalence classes of universal quaternary quadratic forms. Recently, Earnest and Gunawardana proved in [4] that among 204 equivalence classes of universal quaternary quadratic forms, there are exactly 152 equivalence classes of almost primitively universal quaternary quadratic forms. In this article, we prove that there are exactly 107 equivalence classes of primitively universal quaternary quadratic forms. We also determine the set of all positive integers that are not primitively represented by each of the remaining 152−107=45 equivalence classes of almost primitively universal quaternary quadratic forms.

Reexamining nodality in equivalence classes.

Equivalence classes are defined according to the substitutability, or functional similarity, of the stimuli within a stimulus class. Several studies have demonstrated that the degree of functional similarity between stimuli in a class is dependent, in part, upon the number of nodes (intervening stimuli) between the stimuli. Higher nodal number is related to lower functional similarity. This effect is referred to as "nodality." There are three key factors that have not been simultaneously controlled for in the relevant studies: priming effects, reinforcement during training, and multiple stimulus functions of stimuli (sample, comparison, or both). In the present experiment, controlling for these factors, two 6-member, 4-node equivalence classes were established, and a within-class preference assessment was used to evaluate nodality. Of 12 participants, five achieved criterion accuracy (90%) during testing. These participants demonstrated nodality, showing preference for stimuli that were nodally proximal to a sample in the preference test. When distal comparisons were chosen, participants took longer, on average, to make the selection compared to selections of proximal stimuli. These findings are consistent with earlier studies demonstrating nodality, which suggests that nodality is a robust phenomenon and not an artifact of the factors that were controlled for in the present study.

A new algorithm for compression of partially commutative alphabets

In this paper, we will address the problem of how to compress sequences defined in partially commutative alphabets. The use of partial order between symbols may result in far more efficient compression schemes than the most commonly used compression techniques, which use total order. We suggest a new algorithm for compressing sequences defined in alphabets with a partial order. This new algorithm is named the Lempel–Ziv with Partial Order (POLZ) and stands for a Lempel–Ziv (LZ78) generalization. We will show how to compress and decompress sequences by utilizing the POLZ. In addition, we establish the methods for determining the upper bounds for sequence complexity by using the notion of equivalence classes. We will also demonstrate that POLZ is asymptotically optimal, achieving an even greater compression rate when compared to the LZ78 when applied to sequences emanating from concurrent processes. Furthermore, we will show that the complexity obtained by using the LZ78 is also an upper bound for the POLZ complexity. We have developed examples in robotics and real-time systems to show how to use and the benefits of using our algorithm for compression within the context of concurrent systems. When the concurrency of the processes is taken into account, all experimental results show a significant compression gain.

An inverse semigroup from uniform structures on doubles

It was shown by the second named author that the equivalence classes of metrics on the two copies of a metric space X form an inverse semigroup M(X). Here we generalize this result to the uniform spaces. We define the composition of uniform structures on the two copies of a uniform space X and show that the equivalence classes of uniform structures are a semigroup U(X). We also give a characterization of idempotents in U(X). Our main result is that this semigroup is an inverse semigroup. We also compare this semigroup with Md(X) when (X,d) is a metric space.

AF-algebras with lattice-ordered K0: Logic and computation

Following Bratteli's original definition, an AF-algebra A is the norm closure of the union of an ascending sequence of finite-dimensional C⁎-algebras, all with the same unit. Elliott proved that the unital dimension (Grothendieck) group K0(A) uniquely determines A up to isomorphism. AF-algebras with lattice-ordered K0, (AFℓ-algebras) have a preeminent role in the AF-algebraic literature. The Elliott classifier E(A) of any AFℓ-algebra A—i.e., the unit interval of K0(A), or equivalently, the countable set of Murray-von Neumann equivalence classes of projections of A—has the structure of a countable MV-algebra. Every countable MV-algebra arises as E(A) for some AFℓ-algebra A. Since E(A) is the Lindenbaum algebra of some countable set of formulas in Łukasiewicz logic Ł∞, every Ł∞-formula ϕ naturally codes an equivalence class ϕA of projections in A. The deductive-algorithmic machinery of Ł∞ can be thus applied to decide, among many others, the following basic problems, with ⪯ the Murray-von Neumann order:Is ϕA⪯ψA?IsϕA={0}? Is ϕA central ? For any AFℓ-algebra A we construct mutual polytime reductions of the first two problems. The third problem is polytime reducible to both problems. If E(A) is finitely generated and A either has no quotient isomorphic to C or is simple, then the first two problems are polytime reducible to the third one. The complexity of these problems is polytime for many AFℓ-algebras in the literature, including the Behncke-Leptin algebras Am,n, the Farey AFℓ-algebra M1 and all its principal quotients, the CAR algebra and Glimm's universal UHF algebra, and every Effros-Shen algebra Fθ for θ∈[0,1]∖Q a real algebraic integer, or for θ=1/e. For any fixed n=1,2,… we consider the class Kn of AFℓ-algebras presented by generators X1,…,Xn and relations θ1=0,…,θm=0, for an arbitrary finite set Θ={θ1,…,θm} of Ł∞-formulas in the variables X1,…,Xn. We prove that all decision problems studied in this paper are polytime decidable for Kn—uniformly over the presentation Θ.

The enumeration of extreme rigid honeycombs

Rigid honeycombs were introduced by Knutson et al. (J Am Math Soc 17:19–48, 2004), and they were shown in Bercovici et al. (J Funct Anal 258:1579–1627, 2010) to be sums of extreme rigid honeycombs, with uniquely determined summands up to permutations. Two extreme rigid honeycombs are essentially the same if they have proportional exit multiplicities and, up to this identification, there are countably many equivalence classes of such honeycombs. We describe two ways to approach the enumeration of these equivalence classes. The first method produces a (finite) list of all rigid tree honeycombs of fixed weight by looking at the locking patterns that can be obtained from a certain quadratic Diophantine equation. The second method constructs arbitrary rigid tree honeycombs from rigid overlays of two rigid tree honeycombs with strictly smaller weights. This allows, in principle, for an inductive construction of all rigid tree honeycombs starting with those of unit weight. We also show that some rigid overlays of two rigid tree honeycombs give rise to an infinite sequence of rigid tree honeycombs of increasing complexity but with a fixed number of nonzero exit multiplicities. This last result involves a new inflation/deflation construction that also produces other infinite sequences of rigid tree honeycombs.

P-adic polylogarithms and p-adic Hecke L-functions for totally real fields

Abstract The purpose of this article is to newly define the p-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of p-adic L-functions interpolating nonpositive values of Hecke L-functions of the totally real field in terms of special values of these p-adic polylogarithms.