Regular Representation Research Articles

Analysing the use of particle verbs in German as a foreign language: Unproductive particle separation and a proposal for a data-driven learning intervention

This study addresses the long-observed phenomenon of foreign language learners avoiding particle verbs from a novel perspective. Departing from the decades-old debate about the uncertain grammatical status of particle verbs, recent studies have proposed to reconceptualize particle verbs not as words or phrases but as constructions. Constructions are holistic form–meaning representations of usage regularities, which emerge from lexical patterns in language use. In accordance with this understanding, we analyse the use of particle verbs in a corpus of native and non-native German and compare lexical distributions across speaker groups, proficiency levels, and linguistic contexts. Of particular importance is the comparison of contexts of joint particle–verb conjunction (e.g. an-lächeln ‘to smile at’) with those of syntactic particle separation (e.g. lächelt an ‘smiles at’), as these contexts appear to be primarily responsible for learning difficulties and avoidance behaviors. Our findings reveal that contexts of particle separation tend to be lexically sparse and formulaic, contrasting with the more lexically diverse joint contexts. We argue that the joint construction emerges as the default for particle verb usage, ‘crowding out’ the less productive separate construction which becomes confined to specific lexical contexts in non-native language use. Building upon these findings, we use corpus material to develop data-driven learning activities that aim to prevent learners from becoming entrenched in fossilized lexical patterns and to facilitate the acquisition of productive constructions.

Generalized solutions of boundary value problems ofdynamics of thermoelastic rods

We consider spatially one-dimensional boundary value problems of uncoupled thermoelasticity, which can be used to study various rod structures under thermal heating conditions. This model well describes thermodynamic processes at low strain rates. Here, a unified methodology has been developed for solving various boundary value problems typical of practical applications. The task is set to determine the thermo-stressed state of a thermoelastic rod under various boundary conditions at its ends, as well as the acting forces and heat sources along the entire length of the rod.Shock elastic waves arising in such structures under the action of shock loads are considered. Based on the method of generalized functions, generalized solutions of non-stationary and stationary direct and semi-inverse boundary value problems in the class of generalized vector functions of slow growth are constructed. Regular integral representations of generalized solutions are also given, which provide analytical solutions to the posed boundary value problems. The peculiarity of the constructed solutions makes them convenient for engineering applications because make it possible to study the influence of each boundary condition, as well as the acting power and heat sources separately, which is very important when assessing the strength properties of rod structures.

Regular and semi-regular representations of groups by posets

Regular and semi-regular representations of groups by posets

Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups

Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups

A renormalization scheme for semi-regular continued fractions

In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T^slow and T^fast: these maps are defined for (x,y)∈[0,1], where x is the number for which a semi- regular continued fraction representation is developed by T^slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map T^fast is a “sped up" version of the map T^slow, and we show that T^fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, T^slow preserves no such measure, but does preserve an infinite, σ-finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y under rotation by x modulo one. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada’s α-continued fractions.

Realizing regular representations of finite groups

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E(X) and its group of homeomorphisms Aut(X) are isomorphic to G, and the action of G on the n-th homology group Hn(X) is the regular representation. We also discuss other representations.

Van der Corput’s difference theorem for amenable groups and the left regular representation

We establish a connection between two variants of van der Corput’s Difference Theorem (vdCDT) for countably infinite amenable groups G and the ergodic hierarchy of mixing properties of a unitary representation U of G. In particular, we show that one variant of vdCDT corresponds to subrepresentations of the left regular representation, and another variant of vdCDT corresponds to the absence of finite dimensional subrepresentations. We then obtain applications for measure preserving actions of countably infinite abelian groups.

Using Schur Rings to Produce GRRs for Dihedral Groups

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following: Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(Dn, {abr, abs, abt}) is a GRR of Dn. We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

Branching rules for the restriction of regular representations of GL2(o/℘ r ) to SL2(o/℘ r )

Let o be a compact discrete valuation ring with maximal ideal ℘ such that the finite residue field o / ℘ has characteristic p. For r ≥ 2 and p = 2 , we obtain the branching rules for the restriction of a regular representation of GL 2 ( o / ℘ r ) to SL 2 ( o / ℘ r ) . These results have different behavior than that of the known case of p ≠ 2.

A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

Let C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document} be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type (1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1,\\, 4)$$\\end{document} over C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document} are naturally parametrized by the closed points of Spec(C((ε))[x])∪˙{1,2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spec}(\\mathbb {C}(\\!(\\varepsilon )\\!)[x]){\\dot{\\cup }}\\{1,\\,2\\}$$\\end{document}. Moreover, we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document}. This suggests a model of those algebras in the sense of the work of Geiss et al. (Invent Math 209(1):61–158, 2017; Math Z 295:1245–1277, 2020).

Generation and deployment of honeytokens in relational databases for cyber deception

Despite considerable investments in database security, global statistics indicate an exponential increase in data breaches. Organizations are often unaware of data breaches for weeks, months, or even years. Sufficient for adversaries to compromise and ex-filtrate business or mission-critical data. Recent research suggests using honeytokens for early detection of data breaches in organizations. Existing honeytoken generation methods rely on regular expressions, rule mining, constraint satisfaction, or representation learning, which are complex and limited to a few attributes. We created a framework for generating and deploying honeytokens in relational databases that actively monitor sensitive records and quickly detect data breaches and their misuse. To generate the honeytoken we have used the hierarchical machine learning algorithm which uses a recursive technique to model the parent–child relationships of multi-table databases. The proposed method enables the organization to take remedial action to reduce the impact of data breaches and complement existing database security solutions.

Invariant rings of the special orthogonal group have nonunimodal h-vectors

For [Formula: see text] an infinite field of characteristic other than two, consider the action of the special orthogonal group [Formula: see text] on a polynomial ring via copies of the regular representation. When [Formula: see text] has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when [Formula: see text] has positive characteristic, the invariant ring is [Formula: see text]-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for [Formula: see text] as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the [Formula: see text]-invariant and information on the Hilbert series. Indeed, we use this to show that the [Formula: see text]-vector of the invariant ring for [Formula: see text] need not be unimodal.

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

Tensor product and quandle rings of connected quandles of prime order

Let C be field of complex numbers and X be a connected quandle of prime order. We study the regular representation of X by describing the quandle ring C [ X ] as direct sum of right simple ideals. We provide description of tensor product of connected quandles of prime order. We further discuss multiplicity freeness of quandle ring decomposition for connected quandles of order ≤ 47 and prove that C [ X ] decomposes multiplicity free for affine connected quandle X.

Solving deep-learning density functional theory via variational autoencoders

In recent years, machine learning models, chiefly deep neural networks, have revealed suited to learn accurate energy-density functionals from data. However, problematic instabilities have been shown to occur in the search of ground-state density profiles via energy minimization. Indeed, any small noise can lead astray from realistic profiles, causing the failure of the learned functional and, hence, strong violations of the variational property. In this article, we employ variational autoencoders (VAEs) to build a compressed, flexible, and regular representation of the ground-state density profiles of various quantum models. Performing energy minimization in this compressed space allows us to avoid both numerical instabilities and variational biases due to excessive constraints. Our tests are performed on one-dimensional single-particle models from the literature in the field and, notably, on a three-dimensional disordered potential. In all cases, the ground-state energies are estimated with errors below the chemical accuracy and the density profiles are accurately reproduced without numerical artifacts. Furthermore, we show that it is possible to perform transfer learning, applying pre-trained VAEs to different potentials.

On the regular representation of solvable Lie groups with open coadjoint quasi-orbits

On the regular representation of solvable Lie groups with open coadjoint quasi-orbits

The Brain's Sensitivity to Real-world Statistical Regularity Does Not Require Full Attention.

Predictive coding accounts of perception state that the brain generates perceptual predictions in the service of processing incoming sensory data. These predictions are hypothesized to be afforded by the brain's ability to internalize useful patterns, that is, statistical regularities, from the environment. We have previously argued that the N300 ERP component serves as an index of the brain's use of representations of (real-world) statistical regularities. However, we do not yet know whether overt attention is necessary in order for this process to engage. We addressed this question by presenting stimuli of either high or low real-world statistical regularity in terms of their representativeness (good/bad exemplars of natural scene categories) to participants who either fully attended the stimuli or were distracted by another task (attended/distracted conditions). Replicating past work, N300 responses were larger to bad than to good scene exemplars, and furthermore, we demonstrate minimal impacts of distraction on N300 effects. Thus, it seems that overtly focused attention is not required to maintain the brain's sensitivity to real-world statistical regularity. Furthermore, in an exploratory analysis, we showed that providing additional, artificial regularities, formed by altering the proportions of good and bad exemplars within blocks, further enhanced the N300 effect in both attended and distracted conditions, shedding light on the relationship between statistical regularities learned in the real world and those learned within the context of an experiment.

On oriented m $m$‐semiregular representations of finite groups

AbstractA finite group admits an oriented regular representation if there exists a Cayley digraph of such that it has no digons and its automorphism group is isomorphic to . Let be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented ‐semiregular representations using ‐Cayley digraphs. Given a finite group , an ‐Cayley digraph of is a digraph that has a group of automorphisms isomorphic to acting semiregularly on the vertex set with orbits. We say that a finite group admits an oriented ‐semiregular representation (OSR for short) if there exists an ‐Cayley digraph of such that it has no digons and is isomorphic to its automorphism group. Moreover, if is regular, that is, each vertex has the same in‐ and out‐valency, we say is a regular oriented ‐semiregular representation (regular OSR for short) of . In this paper, we classify finite groups admitting a regular OSR or an OSR for each positive integer .

Neural representations of statistical and rule-based predictions in Gilles de la Tourette syndrome.

Gilles de la Tourette syndrome (GTS) is a disorder characterised by motor and vocal tics, which may represent habitual actions as a result of enhanced learning of associations between stimuli and responses (S-R). In this study, we investigated how adults with GTS and healthy controls (HC) learn two types of regularities in a sequence: statistics (non-adjacent probabilities) and rules (predefined order). Participants completed a visuomotor sequence learning task while EEG was recorded. To understand the neurophysiological underpinnings of these regularities in GTS, multivariate pattern analyses on the temporally decomposed EEG signal as well as sLORETA source localisation method were conducted. We found that people with GTS showed superior statistical learning but comparable rule-based learning compared to HC participants. Adults with GTS had different neural representations for both statistics and rules than HC adults; specifically, adults with GTS maintained the regularity representations longer and had more overlap between them than HCs. Moreover, over different time scales, distinct fronto-parietal structures contribute to statistical learning in the GTS and HC groups. We propose that hyper-learning in GTS is a consequence of the altered sensitivity to encode complex statistics, which might lead to habitual actions.

Period Relations Between the Betti–Whittaker Periods for GLn Under Duality

Abstract In this paper, under some regularity conditions, we prove a period relation between the Betti–Whittaker periods associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{n}({{\mathbb{A}}})$ and its contragredient. As a consequence, we prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{2n}({{\mathbb{A}}})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for $\textrm{GSpin}_{2n}^{*} \times \operatorname{GL}_{n^{\prime}}$ by the result of Harder and Raghuram.