Class Of Equations Research Articles

Group theoretical classification of SIC-POVMs

The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated {1.7×106,1.1×105,169,50} Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension.

On the category of Harish-Chandra block modules

If Γ is a subalgebra of A, then an A-module is called a Harish-Chandra module if it is the direct sum of its generalized weight spaces with respect to Γ. Drozd, Futorny, and Ovsienko [4] defined a generalization of a central subalgebra called a Harish-Chandra subalgebra and showed that when Γ is a Harish-Chandra subalgebra of A the structure of Harish-Chandra A-modules can be described using information about the relationship between A and the cofinite maximal ideals of Γ.We extend the results of [4] by dropping the assumption that Γ is quasicommutative. We facilitate this by introducing an equivalence relation ∼ on the set cfs(Γ) of cofinite maximal ideals of Γ. We define Harish-Chandra block modules with respect to ∼ to be A-modules that are the direct sum of so called block spaces corresponding to the equivalence classes cfs(Γ)/∼. If Γ is a Harish-Chandra block subalgebra of A with respect to ∼, then the structure of Harish-Chandra block modules can be described based on the relationship between A and cfs(Γ)/∼. In particular, we give a decomposition of the category of Harish-Chandra block modules and the collection of isomorphism classes of irreducible Harish-Chandra block modules. Furthermore, we define a category A on cfs(Γ)/∼, and show the category of profinite A-modules is equivalent to the category of Harish-Chandra block modules. Taking Γ to be noetherian and quasicommutative, and ∼ to be the equality relation, we recover (in fact, a slight refinement of) results from [4]. Lastly, we provide a sufficient condition for when there are a finite number of isomorphism classes of simple Harish-Chandra block modules with a given support.

Polynomial Equivalence of Complexity Geometries

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds. In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.

Transformations establishing equivalence across neural networks: When have two networks learned the same task?

Transformations are a key tool in the qualitative study of dynamical systems: transformations to a normal form, for example, underpin the study of instabilities and bifurcations. In this work, we test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using diffusion maps with a Mahalanobis-like metric. If the construction succeeds, the two networks can be thought of as belonging to the same equivalence class. We first discuss transformation functions between only the outputs of the two networks; we then also consider transformations that take into account outputs (activations) of a number of internal neurons from each network. Whitney's theorem dictates the number of (generic) measurements from one of the networks required to reconstruct each and every feature of the second network. The construction of the transformation function relies on a consistent, intrinsic representation of the network input space. We illustrate our algorithm by matching neural network pairs trained to learn (a) observations of scalar functions, (b) observations of two-dimensional vector fields, and (c) representations of images of a moving three-dimensional object (a rotating horse). We also demonstrate reconstruction of a network's input (and output) from minimal partial observations of intermediate neuron activations. The construction of equivalences across different network instantiations clearly relates to transfer learning and will also be valuable in establishing equivalence between different machine learning-based tools.

Insertion Algorithms for Gelfand $$S_n$$-Graphs

Abstract The two tableaux assigned by the Robinson–Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard tableaux. Beissinger found a concise way of formulating this restricted map, which involves adding an extra cell at the end of a row after a Schensted insertion process. We show that by changing this algorithm slightly to add cells at the end of columns rather than rows, one obtains a different bijection from involutions to standard tableaux. Both maps have an interesting connection to representation theory. Specifically, our insertion algorithms classify the molecules (and conjecturally the cells) in the pair of W-graphs associated with the unique equivalence class of perfect models for a generic symmetric group.

Sums of squares in function fields over henselian discretely valued fields

Let n∈N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2n(g+1). We show that this bound is optimal. Moreover, in the case where n=1, we show that if F/K is a hyperelliptic function field such that the order of this quotient group is 2g+1, then F is nonreal.

Primitively 2-universal senary integral quadratic forms

For a positive integer m, a (positive definite integral) quadratic form is called primitively m-universal if it primitively represents all quadratic forms of rank m. It was proved in [9] that there are exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms.

LEXICAL AND SEMANTIC CORRESPONDENCE IN POLITICAL DISCOURSE ON INTERNATIONAL ARMED CONFLICTS

Discourse affects the semantics of lexical units. Discursively dependent semantics presents a challenge for intercultural communication. This study proposes a method for identifying the discursively dependent semantics of lexical units through an analysis of the discourse on the conflict in Syria, presented in the Russian and French press. The proposed methodology is based on the analysis of the structure of lexical-semantic fields (LSF) of lexical units included in the discourse, and allows us to achieve technological efficiency in the analysis and formalize the results. As a result of the comparative analysis of the cooccurrence of lexical units within the French- and Russian-language discourses, it was found that the lexical units forming the classes of conditional equivalence have significant differences at the level of the structure of lexical-semantic fields due to the peculiarities of media coverage of the Syrian conflict in Russia and France. The difference at the level of LSF of the same-named units significantly affects the semantics of lexical units and, thus, modifies its pragmatic effect. These data should be taken into account when translating.

Transforming Ontology Web Language Elements into Common Terminology Service 2 Terminology Resources.

Communication and cooperation are fundamental for the correct deployment of P5 medicine, and this can be achieved only by correct comprehension of semantics so that it can aspire to medical knowledge sharing. There is a hierarchy in the operations that need to be performed to achieve this goal that brings to the forefront the complete understanding of the real-world business system by domain experts using Domain Ontologies, and only in the last instance acknowledges the specific transformation at the pure information and communication technology level. A specific feature that should be maintained during such types of transformations is versioning that aims to record the evolution of meanings in time as well as the management of their historical evolution. The main tool used to represent ontology in computing environments is the Ontology Web Language (OWL), but it was not created for managing the evolution of meanings in time. Therefore, we tried, in this paper, to find a way to use the specific features of Common Terminology Service-Release 2 (CTS2) to perform consistent and validated transformations of ontologies written in OWL. The specific use case managed in the paper is the Alzheimer's Disease Ontology (ADO). We were able to consider all of the elements of ADO and map them with CTS2 terminological resources, except for a subset of elements such as the equivalent class derived from restrictions on other classes.

Алгоритмы перечисления симметричных хордовых диаграмм

Numerous applications require exhaustive lists of strings, which may be subject to various requirements, for example, such as their non-equivalence under the action of symmetry groups on them. The necklace is the equivalence class of r-ary strings under rotation. An unlabeled r-ary necklace is an equivalence class of r-ary strings under rotation and permutation of alphabetic characters. Let G be a symmetry group acting on a given set of necklaces T. A necklace x  T is called symmetric if there exists an element g  G, g  e such that gx = x. Other definitions of the symmetry of necklaces are possible, equivalent to the above. All of them, in one way or another, are involved in determining the symmetry of the figure compared to the unlabelled necklace. A flat figure is called symmetrical if it is self-aligned during movements of space R2 i. e. during its isometric transformations. A special case of figures and, accordingly, necklaces are chord diagrams. Chord diagrams represent an object of research that is interesting from different aspects (knot theory, Feynman diagrams, representations of Lie algebras), and has been studied by many authors. The article presents algorithms for enumerating symmetric chord diagrams and shows their connection with the knot theory using simple examples. The list of symmetrical chord diagrams may, in particular, be of interest in the field of mathematical poetry, which studies the dynamics of poetic stanzas horizontally (rhythm, syllabic volume of verses) and vertically (rhyme schemes, refrains, and other stylistic devices).

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.