Class Of Equations Research Articles

Graded contractions of the [formula omitted]-grading on [formula omitted]

Graded contractions of the Z23-grading on the complex exceptional Lie algebra g2 are classified up to equivalence and up to strong equivalence. The non-toral fine Z23-grading is highly symmetric, with all the homogeneous components Cartan subalgebras. This makes possible a combinatorial treatment based on certain nice subsets of the set of 21 edges of the Fano plane. There are 24 such nice sets up to collineation. Each of these is the support of an admissible graded contraction, one of which is present in every equivalence class of graded contractions. Each nice set gives rise to a single Lie algebra, except for three of the cases in which families depending on one or two parameters are found. In particular, a large family of 14-dimensional Lie algebras arise, most of which are solvable. The properties of each of these Lie algebras are studied.

Frege on Identity and Identity Statements: 1884/1903

ABSTRACT In this essay, I first solve solve a conundrum and then deal with criteria of identity, Leibniz's definition of identity and Frege's adoption of it in his (failed) attempt to define the cardinality operator contextually in terms of Hume's Principle in Die Grundlagen der Arithmetik. I argue that Frege could have omitted the intermediate step of tentatively defining the cardinality operator in the context of an equation of the form ‘NxF(x) = NxG(x)'. Frege considers Leibniz's definition of identity to be purely logical, although without saying why it is in line with his logicist project. I argue that the universal criterion of identity that Frege takes from Leibniz's definition and the specific criterion of identity for cardinal numbers embodied in Hume’s Principle (namely equinumerosity) work hand in hand in the tentative contextual definition of the cardinality operator. Yet the interplay between the two criteria is powerless to prevent the emergence of the Julius Caesar problem, let alone suggests how it could be solved. The final explicit definition of the cardinality operator that Frege sets up still rests on the identity criterion of equinumerosity since cardinal numbers are defined as equivalence classes of that relation.

Embedding calculus and grope cobordism of knots

We show that embedding calculus invariants evn are surjective for long knots in an arbitrary 3-manifold. This solves some remaining open cases of Goodwillie–Klein–Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots evn are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the p-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral.Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariantevnof a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension 3, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.

Formation of equivalence classes with the presentation of stimuli for fractions of a second

Studies have shown that it is possible to establish conditional relations among stimuli presented for very brief durations in a matching-to-sample task. ObjectivesIn the present investigation, we asked whether it is possible to establish equivalence relations following the training of conditional relations among stimuli presented for fractions of a second. MethodsTwo experiments were conducted. Adults performed a successive Go/No-go discrimination task in which one element of a compound stimuli was presented for durations as short as 33 ms. Stimuli were combinations of geometric figures and colored backgrounds. In the conditional discrimination training, touching only the correct stimuli was followed by reinforcement. Emergent stimulus relations were tested under extinction. ResultsPresenting either figures or colors with durations as short as 33 ms promoted the acquisition of baseline conditional relations that generated the emergence of corresponding equivalence classes. ConclusionsStimuli, with durations as short as 33 ms, exerted discriminative control over human participants and had the capability to establish equivalence classes, which holds significant importance for symbolic learning.

Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases

A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G×H symmetry gives rise to anomalous LRE upon measuring G-charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way. Published by the American Physical Society 2024

Trace Conservation Laws in T2/Zm Orbifold Gauge Theories

Abstract Gauge theory compactified on an orbifold is defined by gauge symmetry, matter contents, and boundary conditions (BCs). There are equivalence classes (ECs), each of which consists of physically equivalent BCs. We propose the powerful necessary conditions, trace conservation laws (TCLs), which achieve a sufficient classification of ECs in U(N) and SU(N) gauge theories on T2/Zm orbifolds (m = 2, 3, 4, 6). The TCLs yield the equivalent relations between the diagonal BCs without relying on an explicit form of gauge transformations. The TCLs also show the existence of off-diagonal ECs, which consist only of off-diagonal matrices, on T2/Z4 and T2/Z6. After the sufficient classification, the exact numbers of ECs are obtained.

Bayesian network structure learning with a new ensemble weights and edge constraints setting mechanism

Bayesian networks (BNs) are highly effective in handling uncertain problems, which can assist in decision-making by reasoning with limited and incomplete information. Learning a faithful directed acyclic graph (DAG) from a large number of complex samples of a joint distribution is currently a challenging combinatorial problem. Due to the growing volume and complexity of data, some Bayesian structure learning algorithms are ineffective and lack the necessary precision to meet the required needs. In this paper, we propose a new PCCL-CC algorithm. To ensure the accuracy of the network structure, we introduce the new ensemble weights and edge constraints setting mechanism. In this mechanism, we employ a method that estimates the interaction between network nodes from multiple perspectives and divides the learning process into multiple stages. We utilize an asymmetric weighted ensemble method and adaptively adjust the network structure. Additionally, we propose a causal discovery method that effectively utilizes the causal relationships among data samples to correct the network structure and mitigate the influence of Markov equivalence classes (MEC). Experimental results on real datasets demonstrate that our approach outperforms state-of-the-art methods.

Morita equivalence classes of 2‐blocks with abelian defect groups of rank 4

AbstractWe classify all 2‐blocks with abelian defect groups of rank 4 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Broué's abelian defect group conjecture holds for all blocks under consideration here.

On a Theorem of Lafforgue

Abstract We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.

Protecting privacy and enhancing utility: A novel approach for personalized trajectory data publishing using noisy prefix tree

In recent years, the widespread adoption of location-based software has significantly improved people’s daily lives. However, this convenience has brought about an increasingly severe concern: the risk of data privacy breaches. To tackle this issue, a novel scheme for personalized trajectory data publishing is proposed, leveraging a noisy prefix tree structure. The scheme begins by constructing multiple trajectory equivalence classes based on the spatiotemporal characteristics of trajectories, followed by the calculation of distinct Hilbert curve orders for each equivalence class. Subsequently, the Hilbert curve is employed to partition the location points within each equivalence class, with the aid of a scoring function that selects optimal center points to replace other location points effectively. Additionally, appropriate encoding levels are determined for each partitioned region, facilitating the conversion of the center points into binary encoded characters using Hilbert-Geohash. This conversion process safeguards against data privacy leakage by obfuscating the original latitude and longitude coordinates. Ultimately, a noisy prefix tree is constructed to store the binary encoding in its nodes. To ensure data privacy preservation, a novel privacy budget allocation approach is introduced, applying suitable Laplace noise to each node. The effectiveness of the proposed algorithm is confirmed through experimental comparisons with existing schemes, employing real datasets and demonstrating its notable performance in terms of both data privacy and utility.

Research on Optimization of International Multimodal Transportation Routes Based on Uncertainty Theory

Delays in international multimodal marine transport can result in subsequent connection delays, leading to prolonged transportation time and increased uncertainty. To identify more reliable and cost-effective robust transportation routes, this study establishes a constrained planning model for multimodal transportation routes. Employing uncertainty theory, the model is transformed into a deterministic equivalent class model and ultimately incorporates an adaptive differential evolution (ADE) algorithm. Results of a case study indicate that: (i) the proposed model, compared with deterministic models, exhibits greater robustness and better aligns with practical transportation scenarios, resulting in substantial reductions in actual transportation time and cost; (ii) the solution efficiency of the ADE algorithm surpasses that of the genetic algorithm and Dijkstra algorithm; (iii) the start time of transportation and the confidence level of uncertainty also play crucial roles in influencing route selection. Therefore, decision-makers should consider a multifaceted approach when formulating transportation routes.

RoCK blocks for affine categorical representations

Given a categorical action of a Lie algebra, a celebrated theorem of Chuang and Rouquier proves that the blocks corresponding to weight spaces in the same orbit of the Weyl group are derived equivalent, proving an even more celebrated conjecture of Broué for the case of the symmetric group. In many cases, these derived equivalences are t -exact and thus induce equivalences of abelian categories between different blocks. We call two such blocks “Scopes equivalent.” In this paper, we describe how Scopes equivalence classes for any affine categorification can be classified by the chambers of a finite hyperplane arrangement, which can be found through simple Lie theoretic calculations. We pay special attention to the largest equivalence classes, which we call RoCK, and show how this matches with recent work of Lyle on Rouquier blocks for Ariki–Koike algebras. We also provide Sage code that tests whether blocks are RoCK and finds RoCK blocks for Ariki–Koike algebras.

Closed FRW holography: a time-dependent ER=EPR realization

We extend a recent de Sitter holographic proposal and entanglement entropy prescription to generic closed FRW cosmologies in arbitrary dimensions, and propose that for large classes of bouncing and Big Bang/Big Crunch cosmologies, the full spacetime can be encoded holographically on two holographic screens, associated to two antipodal observers. In the expanding phase, the two screens lie at the apparent horizons. In the contracting phase, there is an infinite number of possible trajectories of the holographic screens, which can be grouped in equivalence classes. In each class the effective holographic theory can be derived from a pair of “parent” screens on the apparent horizons. A number of cases including moduli dominated cosmologies escape our discussion, and it is expected that two antipodal observers and their associated screens do not suffice to reconstruct these cosmologies. The leading contributions to the entanglement entropy between the screens arise from a minimal extremal trapped or anti-trapped surface lying in the region between them. This picture entails a time-dependent realization of the ER=EPR conjecture, where an effective geometrical bridge connecting the screens via the minimal extremal surface emerges from entanglement. For the Big Crunch contracting cases, the screens disentangle and the geometrical bridge closes off when the minimal extremal trapped sphere hits the Big Crunch singularity at a finite time before the collapse of the Universe. Semiclassical, thermal corrections are incorporated in the cases of radiation dominated cosmologies.

Almost isotropy-maximal manifolds of non-negative curvature

We extend the equivariant classification results of Escher and Searle for closed, simply connected, Riemannian n n -manifolds with non-negative sectional curvature admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove that any such manifold is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to three.

Functional equivalence and a cover relation for reaction systems

Reaction systems are a computational model originally introduced to formalize the interactions between biochemical reactions that are the basis of the functioning of the living cell. Subsets of reactions of a reaction system define result functions which leads to a concept of functional equivalence. This equivalence in turn induces a functional cover relation on the reactions of a reaction system which captures redundancies in the system. In this paper, the functional cover relation is transferred to functional equivalence classes of sets of reactions. We introduce so-called atoms as building blocks of reactions. Atoms provide a characterization of functional equivalence classes of sets of reactions and are used to prove that the functional cover relation on functional equivalence classes of sets of reactions is a lattice.

Merging meaningful classes and abstract equivalence classes by exclusion.

The current experiment assessed whether relating abstract stimuli with familiar pictures by exclusion would produce the formation of a meaningful equivalence class. Ten participants learned conditional discrimination relations with abstract stimuli and established equivalence classes (ABC classes). They then learned DA (D1A1, D2A2, and D3A3) conditional discriminations with written words as D stimuli; two words (D1 and D2) were meaningful stimuli in the participants verbal community ("Dentist" and "Baker"), whereas the third (D3) was a pseudoword ("Tabilu"). In testing trials, participants evidenced derived relations between pictures related preexperimentally to D1 and D2 with the experimental equivalence classes related to D1 and D2. For some participants, the decontextualized stimuli were a set of boat pictures (Condition 1), whereas for others they were a set of miscellaneous pictures (Condition 2). Participants in both conditions successfully matched decontextualized pictures (unrelated to dentist and baker contexts) to all abstract stimuli in the class related to D3 (exclusion responding). In Condition 1 the meaning reported to the word Tabilu was similar across participants, but in Condition 2 participants showed more variations to answer to the meaning of Tabilu. These results suggest that exclusion learning can occur under different stimulus control topographies.

Shift-invariant homogeneous classes of random fields

Given an Rd-valued random field (rf) Z(t),t∈T and an α-homogeneous mapping κ we define the corresponding equivalent class of rf's (denoted by Cκ[Z]) which include representers of the same tail measure νZ. When T is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of Z. This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant Cκ[Z]'s. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps U acting on elements of Cκ[Z] turns out to be crucial for properties of functionals of Z. Applications of our findings concern max-stable and symmetric α-stable rf's, their maximal indices as well as their random shift-representations.

$$\mathcal {S}_0$$-equivalence classes, a new direction to find better weightwise perfectly balanced functions, and more

$$\mathcal {S}_0$$-equivalence classes, a new direction to find better weightwise perfectly balanced functions, and more

Rough set Theory-Based group incremental approach to feature selection

Feature selection in a dynamic learning system often encounters challenges from sample variation. Incremental selection techniques using rough set theory (RST) address this challenge, to a certain degree, but two issues remain: (1) high redundancy and (2) low efficiency in processing high-dimensional datasets. To address these issues and further improve feature selection, we develop triple nested equivalence class (TNEC) RST for a group incremental approach to feature selection. In particular, we construct a group TNEC to enable universe-reduction learning to help consistently filter out inconsequential samples and incrementally update dependencies. Furthermore, with TNEC-based incremental partitioning, we develop a novel dependency computing technique to reduce redundant features and avoid repeated learning. We conducted experimental verification using 18 dynamic datasets, including ultra-high-dimensional datasets. Tests on five incremental datasets from each of the 18 original datasets validated that the group TNEC approach significantly outperformed the state-of-the-art methods in terms of classification efficiency, selection accuracy, feature significance, and suitability for ultra-high-dimensional data. In both incremental and static feature selection cases, the TNEC approach extended the ability of evolutionary and other types of heuristic learning algorithms in handling sample variations.

Equivalence class of complete correlation determination of several gray incidence degrees

PurposeThis paper sets out to solve a common and crucial fundamental theoretical problem of gray incidence cluster analysis: to [X]={X|ρ(X,Y)≥1−ε0} constitute an approximate classification, it must first be proven that [X]={X|ρ(X,Y)=1} constitutes a rigorous classification.Design/methodology/approachThis paper does not study the concrete expressions of various incidence degrees but rather the perfect correlation essence of such incidence degrees, that is, sufficient and necessary conditions.FindingsFor any order difference incidence degree, the similarity incidence degree, the direct proportion incidence degree, the parallel incidence degree and the nearness incidence degree, it is proven that the perfect correlation relation is an equivalence relation. The set composed of all sequences Y that are equivalent to sequences X is studied, that is, the equivalence class of X. The structure and mutual relations of these equivalence classes are discussed, and the topological homeomorphism concept of incidence degree is introduced. The conclusion is obtained that the equivalence classes of the two incidence degrees must be the same when the topological homeomorphism is obtained.Research limitations/implicationsIn this paper, only the perfect correlation relation of any order difference incidence degree, the similarity incidence degree, the direct proportion incidence degree, the parallel incidence degree and the nearness incidence degree are studied as equivalent relations.Originality/valueNot only are the research results of several incidence degrees involved in this paper original but also many other effective incidence degrees have not done this basic research, so this paper opens up a research direction with theoretical significance.