Class Of Equations Research Articles

Gain total graphs and their spectra via G-phases and group representations

We introduce a definition for the total graph of a gain graph (Γ,ψ) on a group G by using G-phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice of the G-phase associated with ψ. More precisely, we consider the left-right multiplication action of G on the space of all G-phases, proving that its orbits consist of sets of G-phases inducing the same switching equivalence class of gain functions on Γ or, equivalently, on its total graphs. Moreover, we prove that the stabilizers of this action are isomorphic to the centralizers of the sets of gains of closed walks in the associated gain graphs. Our construction is consistent with the existing notions of total graph for signed graphs and, in analogy with the signed case, we are able to explicitly compute the spectrum of the total graph of a regular gain graph over an arbitrary group G, where the spectrum is defined by means of a unitary representation of the gain group G.

Optimal allocation method for MIES-based shared energy storage using cooperative game theory and CSP

To further promote the efficient use of energy storage and the local consumption of renewable energy in a multi-integrated energy system (MIES), a MIES model is developed based on the operational characteristics and profitability mechanism of a shared energy storage station (SESS), considering concentrating solar power (CSP), integrated demand response, and renewable energy output uncertainty. We propose a corresponding MIES model based on co-operative game theory and the CSP and an optimal allocation method for MIES shared energy storage. The model considers the maximum operating benefit of the SESS as the upper objective function and the minimum operating cost of the MIES as the lower objective function. First, the Karush–Kuhn–Tucker conditions of the lower-layer model are transformed into constraints of the upper-layer model, and the Big-M method is used to linearize the nonlinear problem and convert the two-layer nonlinear model into a single-layer linear model. Second, based on the Nash negotiation theory, the benefits of each IES in the MIES are allocated. Finally, the fuzzy chance constraints are used to relax the power balance constraints, and the trapezoidal fuzzy numbers are transformed into a deterministic equivalence class to assess the impact of renewable energy output uncertainty on system operation. The validity and rationality of the proposed two-layer model are verified through simulation, and the results demonstrate that the proposed shared storage capacity leasing model can effectively reduce the total operation cost, increase the profitability of the shared storage operator, and increase the utilization rate of the SESS.

A Novel Spark-Based Attribute Reduction and Neighborhood Classification for Rough Evidence.

Neighborhood classification (NEC) algorithms have been widely used to solve classification problems. Most traditional NEC algorithms employ the majority voting mechanism as the basis for final decision making. However, this mechanism hardly considers the spatial difference and label uncertainty of the neighborhood samples, which may increase the possibility of the misclassification. In addition, the traditional NEC algorithms need to load the entire data into memory at once, which is computationally inefficient when the size of the dataset is large. To address these problems, we propose a novel Spark-based attribute reduction and NEC for rough evidence in this article. Specifically, we first construct a multigranular sample space using the parallel undersampling method. Then, we evaluate the significance of attribute by neighborhood rough evidence decision error rate and remove the redundant attribute on different samples subspaces. Based on this attribute reduction algorithm, we design a parallel attribute reduction algorithm which is able to compute equivalence classes in parallel and parallelize the process of searching for candidate attributes. Finally, we introduce the rough evidence into the classification decision of traditional NEC algorithms and parallelize the classification decision process. Furthermore, the proposed algorithms are conducted in the Spark parallel computing framework. Experimental results on both small and large-scale datasets show that the proposed algorithms outperform the benchmarking algorithms in the classification accuracy and the computational efficiency.

Using a Respondent-Type Matching-to-Sample Exclusion Training Procedure to Establish Equivalence Responding

An exclusion training procedure involves presenting a sample stimulus with an unrelated comparison stimulus that is presented alongside other comparison stimuli that previously have acquired a negative relation to the sample stimulus. Due to the already established negative comparisons, the participant selects the unrelated stimulus, establishing a relation between the two stimuli. Of the large body of research on exclusion, there has been no research conducted that has combined respondent conditioning with an exclusion training procedure. This study used a respondent-type matching-to-sample (ReTMTS) exclusion training procedure with probe trials to train 3, three-member equivalence classes. A–B relations were trained using the ReTMTS procedure, and A–C relations were trained via exclusion using the ReTMTS procedure. Of the 10 participants who reached the test phase, only 2 failed to reach the criterion required to demonstrate equivalence responding. These findings are discussed in the context of previous research on exclusion training.

The virtual spectrum of linkoids and open curves in 3-space

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.

Efficient Syndrome Decoder for Heavy Hexagonal QECC via Machine Learning

Error syndromes for heavy hexagonal code and other topological codes such as surface code have typically been decoded by using Minimum Weight Perfect Matching– (MWPM) based methods. Recent advances have shown that topological codes can be efficiently decoded by deploying machine learning (ML) techniques, in particular with neural networks. In this work, we first propose an ML-based decoder for heavy hexagonal code and establish its efficiency in terms of the values of threshold and pseudo-threshold for various noise models. We show that the proposed ML-based decoding method achieves ~ 5 × higher values of threshold than that for MWPM. Next, exploiting the property of subsystem codes, we define gauge equivalence for heavy hexagonal code, by which two distinct errors can belong to the same error class. A linear search-based method is proposed for determining the equivalent error classes. This provides a quadratic reduction in the number of error classes to be considered for both bit flip and phase flip errors and thus a further improvement of ~ 14% in the threshold over the basic ML decoder. Last, a novel technique based on rank to determine the equivalent error classes is presented, which is empirically faster than the one based on linear search.

Ideal classes of orders in quaternion algebras

We provide an algorithm that, given any order O in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right O-ideals, including the non-invertible ones. The theory is developed for a more general kind of algebras.

New Classification Method for Equivalence Classes on S1/Z2 and T2/Z3 Orbifolds

Abstract In 5D and 6D U(N) and SU(N) gauge theories compactified on S1/Z2 and T2/Z3 orbifolds, we propose a new method to classify the equivalence classes (ECs) of boundary conditions (BCs) without depending on the structure of gauge transformations. Some of the BCs are connected through gauge transformations and constitute ECs, each of which contains physically equivalent BCs. Previous methods for classifying ECs have been used for specific gauge transformations. In this paper, we show that a geometric property of orbifolds significantly narrows down the possibilities of connecting BCs and completes the classification of ECs.

Double-local conditional probability based fast calculation method for approximation regions of local rough sets

As an important extension of classical rough sets, local rough set model can effectively process data with noise. How to effectively calculate three approximation regions, namely positive region, negative region and boundary region, is a crucial issue of local rough sets. Existing calculation methods for approximation regions are based on conditional probability, the time complexity is O (|X||U||C|). In order to improve the computational efficiency of three approximation regions of local rough sets, we propose a double-local conditional probability based fast calculation method. First, to improve the computational efficiency of equivalence class, we define the double-local equivalence class. Second, based on the double-local equivalence class, we define the double-local conditional probability. Finally, given the probability thresholds and a local equivalence class, the monotonicity of double-local conditional probability is proved, on this basis, a double-local conditional probability based fast calculation method for approximation regions of local rough sets is proposed, and the time complexity is O (MAX (|X|2|C|, |X||XC||C|)). Experimental results based on 9 datasets from UCI demonstrate the effectiveness of the proposed method.

On virtual indicability and property (T) for outer automorphism groups of RAAGs

We give a condition on the defining graph of a right-angled Artin group, which implies that its automorphism group is virtually indicable, that is, it has a finite index subgroup that admits a homomorphism onto \mathbb{Z} . We use this as a part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence, we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.

Exploratory Study of Equivalence under Conjoint Select and Reject Control

Carrigan and Sidman Journal of the Experimental Analysis of Behavior, 58, 183-204, (1992) proposed that select and reject arbitrary conditional relations are equivalence relations, each resulting in the emergence of alternative stimulus equivalence classes. The standard matching to sample (MTS) procedure can potentially teach both select and reject conditional relations, which apparently would prevent the emergence of equivalence relations, although there is extensive evidence for the emergence of equivalence with the standard MTS procedure. One possibility is that participants trained with the standard MTS procedure predominantly learn one type of control over the other. Experiment 1 explores the implementation of a Detached-MTS procedure, which separately trains the select and reject conditional relations involved in the Standard-MTS procedure. The results suggest that the emergence of equivalence relations may be compatible with conjoint select and reject conditional control. In Experiment 2, the Detached-MTS procedure succeeds in replicating the emergence of equivalence under exclusive select control but not under exclusive reject control, which conditions the findings of Experiment 1. The sources of control associated with the emergence of equivalence in the standard MTS procedure and some methodological issues of the Detached-MTS procedure are discussed.

On input and Langlands parameters for epipelagic representations

A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of p p -adic groups. The input for this construction is a pair ( λ , χ ) (\lambda , \chi ) where λ \lambda is a stable vector in a certain representation coming from a Moy–Prasad filtration, and χ \chi is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet.

The effects of a training package to teach note taking on the formation of equivalence classes.

Effective note taking may enhance learning outcomes for students and serve as a directly observable form of mediation within a test context. Frampton et al. (2023) used stimulus fading to teach note taking in the form of a graphic organizer (GO) during matching-to-sample baseline relations training (MTS-BRT). Moderately high yields were observed with young adults despite the use of linear series training, abstract stimuli, and five-member classes. The present study taught the same note taking strategy using an intervention package including video illustration, voice-over instructions, and feedback to eight college students. Participants were taught to construct the GO during MTS-BRT with three three-member classes of familiar stimuli. Then the effects of MTS-BRT alone with three five-member classes of abstract stimuli was evaluated. Participants efficiently completed training with familiar stimuli and passed the posttest on the first attempt. With the abstract stimuli, participants engaged in GO construction during MTS-BRT and the six participants that demonstrated high levels of fidelity to the trained note taking strategy passed the posttest on the first attempt. These results replicate findings from Frampton et al. while using a more efficient intervention package. Benefits of teaching overt mediation responses are discussed as well as future directions for translation to applied contexts.

The geometry of inflationary observables: Lifts, flows, equivalence classes

The Eisenhart lift allows formulating the dynamics of a scalar field in a potential as pure geodesic motion in a curved field-space manifold involving an additional fictitious vector field. Making use of the formalism in the context of inflation, we show that the main inflationary observables can be expressed in terms of the geometrical properties of a two-dimensional uplifted field-space manifold spanned by the time derivatives of the scalar and the temporal component of the vector. This allows to abstract from specific potentials and models and describe inflation solely in terms of the flow of geometric quantities. Our findings are illustrated through several inflationary examples previously considered in the literature.

Twisted conjugacy in $\mathrm{SL}_{n}$ and $\mathrm{GL}_{n}$ over subrings of $\overline{\mathbb{F_p}}(t)$

Let \varphi\colon G\to G be an automorphism of an infinite group G . One has an equivalence relation \sim_{\varphi} on G defined as x\sim_{\varphi} y if there exists a z\in G such that y=zx\varphi(z^{-1}) . The equivalence classes are called \varphi -twisted conjugacy classes, and the set G/{\sim}_{\varphi} of equivalence classes is denoted by \mathcal{R}(\varphi) . The cardinality R(\varphi) of \mathcal{R}(\varphi) is called the Reidemeister number of \varphi . We write R(\varphi)=\infty when \mathcal{R}(\varphi) is infinite. We say that G has the R_{\infty} -property if R(\varphi)=\infty for every automorphism \varphi of G . We show that the groups G=\mathrm{GL}_{n}(R), \mathrm{SL}_{n}(R) have the R_{\infty} -property for all n\ge 3 when F[t]\subset R\subsetneq F(t) , where F is a subfield of \overline{\mathbb{F_p}} . When n\ge 4 , we show that any subgroup H\subset \mathrm{GL}_{n}(R) that contains \mathrm{SL}_{n}(R) also has the R_{\infty} -property.

Birational classification of toric orbifolds

We give a complete classification of the torus-equivariant birational equivalence classes of smooth proper toric Deligne–Mumford stacks with trivial generic stabilizer in terms of their associated stacky fans.

Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

Analysis of Fuzzy Vector Spaces as an Algebraic Framework for Flag Codes

Flag codes are a recent network coding strategy based on linear algebra. Fuzzy vector subspaces extend the notions of classical linear algebra. They can be seen as abstractions of flags to the point that several fuzzy vector subspaces can be identified to the same flag, which naturally induces an equivalence relation on the set of fuzzy vector subspaces. The main contributions of this work are the methodological abstraction of flags and flag codes in terms of fuzzy vector subspaces, as well as the generalisation of three distinct equivalence relations that originated from the fuzzy subgroup theory and study of their connection with flag codes, computing the number of equivalence classes in the discrete case, which represent the number of essentially distinct flags, and a comprehensive analysis of such relations and the properties of the corresponding quotient sets.

Path Integral over Equivalence Classes for Quantum Dynamics with Static Disorder.

An efficient, fully quantum mechanical, real-time path integral method for including the effects of static disorder in the dynamics of systems coupled to common or local harmonic baths is presented. Rather than performing a large number of demanding calculations for different realizations of the system Hamiltonian, the influence of the bath is captured through a single evaluation of the path sum by grouping the system paths into equivalence classes of fixed system amplitudes. The method is illustrated with several analytical and numerical examples that show a variety of nontrivial effects arising from the interplay among coherence, dissipation, thermal fluctuations, and geometric phases.

Q-Painlevé equations on cluster Poisson varieties via toric geometry

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type.