Representation Of Properties Research Articles

A comparative analysis of strategies for multiplication and division of fractions between elementary preservice teachers in the United States and Korea

The present comparative analysis study conducted an examination of 46 U.S. and 30 South Korean elementary preservice teachers (PSTs), assessing their overall performances and comparing problem-solving strategies across 11 fraction multiplication and division problems. The tasks spanned across three domains: posing story problems, solving word problems, and illustrating visual models. This study’s purpose is to relay common misconceptions held by elementary PSTs regarding fraction multiplication and division between the U.S. and South Korea. The overall performance results indicated that South Korean PSTs demonstrated higher proficiency in all three areas (i.e., posing story problems, solving word problems, and illustrating visual models). However, both groups of U.S. and South Korean PSTs encountered the most difficulty in representing visual models. Strategy analysis highlighted that the application of properties of multiplication and clearer representation of the unit were key factors contributing to the success of South Korean PSTs. Such results contribute to the growing literature supporting the importance of elementary PSTs’ influence on students to develop necessary foundational skills in mathematics. The study also discusses implications for teacher educators based on these findings.

Rewording Theoretical Predictions at Colliders with Vacuum Amplitudes.

We propose multiloop vacuum amplitudes in the loop-tree duality (LTD) as the optimal building blocks for efficiently assembling theoretical predictions at high-energy colliders. This hypothesis is strongly supported by the manifestly causal properties of the LTD representation of a vacuum amplitude. The vacuum amplitude in LTD, acting as a kernel, encodes all the final states contributing to a given scattering or decay process through residues in the on-shell energies of the internal propagators. Gauge invariance and the wave function renormalization of the external legs are naturally incorporated. This methodological approach, dubbed LTD causal unitary, leads to a novel differential representation of cross sections and decay rates that is locally free of ultraviolet and infrared singularities at all orders in perturbation theory. Threshold singularities also match between different phase-space residues. Most notably, it allows us to conjecture for the first time the local functional form of initial-state collinear splitting functions. The fulfillment of all these properties provides a theoretical description of differential observables at colliders that is well defined in the four physical dimensions of the space-time.

A quantified local-to-global principle for Morse quasigeodesics

Kapovich, Leeb and Porti (2014) gave several new characterizations of Anosov representations \Gamma \to G , including one where geodesics in the word hyperbolic group \Gamma map to “Morse quasigeodesics” in the associated symmetric space G/K . In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples.

Coorbit Theory of Warped Time-Frequency Systems in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}

Warped time-frequency systems have recently been introduced as a class of structured continuous frames for functions on the real line. Herein, we generalize this framework to the setting of functions of arbitrary dimensionality. After showing that the basic properties of warped time-frequency representations carry over to higher dimensions, we determine conditions on the warping function which guarantee that the associated Gramian is well-localized, so that associated families of coorbit spaces can be constructed. We then show that discrete Banach frame decompositions for these coorbit spaces can be obtained by sampling the continuous warped time-frequency systems. In particular, this implies that sparsity of a given function f in the discrete warped time-frequency dictionary is equivalent to membership of f in the coorbit space. We put special emphasis on the case of radial warping functions, for which the relevant assumptions simplify considerably.

Category-guided multi-interest collaborative metric learning with representation uniformity constraints

Multi-interest collaborative metric learning has recently emerged as an effective approach to modeling the multifaceted interests of a user in recommender systems. However, two issues remain unexplored. (1) There is no explicit guidance for the matching of an item against multiple interest vectors of a user. (2) The desired property of item representations with respect to their categories is overlooked, resulting in that different categories of items are mixed up in the latent space. To overcome these issues, we devise a Category-guided Multi-interest Collaborative Metric Learning model (CMCML) with representation uniformity constraints. CMCML is designed as a novel category-guided Mixture-of-Experts (MoE) architecture, where the gating network leverages the item category to guide the matching of an item against multiple interest vectors of a user, encouraging items with the same category to approach the same interest vector. In addition, we design a user multi-interest uniformity loss and a category-aware item uniformity loss: The former aims to avoid representation degeneration by enlarging the difference among multiple interest vectors of the same user; the latter is tailored to push different categories of items apart in the latent space. Quantitative experiments on Ciao, Epinions and TaFeng demonstrate that our CMCML improves the value of NDCG@20 by 12.41%, 10.89% and 10.39% respectively, compared to other state-of-the-art collaborative metric learning methods. Further qualitative analyses reveal that our CMCML yields a better representation space where items from distinct categories are arranged in different regions with high density.

Learning representations for robust human–robot interaction

AbstractThis article summarizes the author's presentation in the New Faculty Highlight at the Thirty‐Eighth AAAI Conference on Artificial Intelligence. It discusses the desired properties of representations for enabling robust human–robot interaction. Examples from the author's work are presented to show how to build these properties into models for performing tasks with natural language guidance and engaging in social interactions with other agents.

Beyond the root: Geometric characterization for the diagnosis of syndromic heritable thoracic aortic diseases

Syndromic heritable thoracic aortic diseases (sHTAD), such as Marfan (MFS) or Loeys–Dietz (LDS) syndromes, involve high risk of life threatening aortic events. Diagnosis of syndromic features alone is difficult, and negative genetic tests do not necessarily exclude a genetic or hereditary condition. Periodic 3D imaging of the aorta is recommended in patients with aortic disease. Thus, an imaging-based approach aimed at identifying unique features of aortic geometry can be highly effective for diagnosing sHTAD and assessing risk. In this study, we present a method that can help identify the manifestations of sHTAD by focusing on the entire geometry of the thoracic aorta, rather than only using measurements of dilation of the aortic root. We analyze the geometric phenotype of 97 patients with genetically confirmed sHTAD (79 MF and 18 LDS) and of 45 healthy volunteers, using 3D aorta meshes obtained from phase contrast-enhanced magnetic resonance angiograms computed from 4D flow cardiac magnetic resonance. We build a geometric encoding of the aorta, based on a vessel coordinate system, and use several mathematical models to discriminate between controls and patients with sHTAD: a baseline scenario, based on aortic root dimensions only, a descriptor typically used in sHTAD patients; a low dimensional scenario, with a reduce encoding using principal component analysis; and a high-dimensional scenario, which included the full coefficient representation for geometry encoding, aiming to capture finer geometric details. The results indicate that considering the anatomy of the whole thoracic aorta can improve predictive ability. We achieve precision and sensitivity values over 0.8, with a specificity of over 70% in all the models used, while a single value classifiers (based only on aortic root diameter) demonstrated a trade-off between sensitivity and specificity. Using the mathematical properties of the vessel coordinate system representation, feature importance is mapped onto a set of anatomical traits that are used by the models to do the classification, thus providing interpretability of the results. This analysis indicates that in addition to the diameter of the aortic root, aortic elongation and a narrowing of the descending thoracic aorta may be markers of positive sHTAD.

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.

Cross-modality sub-image retrieval using contrastive multimodal image representations

In tissue characterization and cancer diagnostics, multimodal imaging has emerged as a powerful technique. Thanks to computational advances, large datasets can be exploited to discover patterns in pathologies and improve diagnosis. However, this requires efficient and scalable image retrieval methods. Cross-modality image retrieval is particularly challenging, since images of similar (or even the same) content captured by different modalities might share few common structures. We propose a new application-independent content-based image retrieval (CBIR) system for reverse (sub-)image search across modalities, which combines deep learning to generate representations (embedding the different modalities in a common space) with robust feature extraction and bag-of-words models for efficient and reliable retrieval. We illustrate its advantages through a replacement study, exploring a number of feature extractors and learned representations, as well as through comparison to recent (cross-modality) CBIR methods. For the task of (sub-)image retrieval on a (publicly available) dataset of brightfield and second harmonic generation microscopy images, the results show that our approach is superior to all tested alternatives. We discuss the shortcomings of the compared methods and observe the importance of equivariance and invariance properties of the learned representations and feature extractors in the CBIR pipeline. Code is available at: https://github.com/MIDA-group/CrossModal_ImgRetrieval.

The state and error-feedback regulator problems for a boundary control catalytic cracking process

This research work is devoted to boundary control of a catalytic cracking reactor using state and error feedback regulators. The process mathematical model is governed by a set of partial differential equations (PDEs), for which infinite-dimensional representation and spectral properties of the system generator are employed to solve the regulation problems. The primary objective is to track a desired output reference trajectory in the presence of disturbances that are generated by a distributed parameter exosystem. Initially, a state feedback stabilizing regulator is designed to drive the process output towards the reference trajectory. The second objective is to develop a dynamic controller that employs the tracking error as input. The closed-loop plant is shown to be exponentially stable and the tracking error asymptotically approaches zero. The performances of the designed regulators are shown through numerical simulations.

Measuring Disentanglement: A Review of Metrics.

Learning to disentangle and represent factors of variation in data is an important problem in artificial intelligence. While many advances have been made to learn these representations, it is still unclear how to quantify disentanglement. While several metrics exist, little is known on their implicit assumptions, what they truly measure, and their limits. In consequence, it is difficult to interpret results when comparing different representations. In this work, we survey supervised disentanglement metrics and thoroughly analyze them. We propose a new taxonomy in which all metrics fall into one of the three families: intervention-based, predictor-based, and information-based. We conduct extensive experiments in which we isolate properties of disentangled representations, allowing stratified comparison along several axes. From our experiment results and analysis, we provide insights on relations between disentangled representation properties. Finally, we share guidelines on how to measure disentanglement.

Discrete superior dynamics of a generalized chaotic system

In the past few decades, the discrete dynamics of difference maps have attained the remarkable attention of researchers owing to their incredible applications in different domains, like cryptography, secure communications, weather forecasting, traffic flow models, neural network models, and population biology. In this article, a generalized chaotic system is proposed, and superior dynamics is disclosed through fixed point analysis, time-series evolution, cobweb representation, period-doubling, period-3 window, and Lyapunov exponent properties. The comparative bifurcation and Lyapunov plots report the superior stability and chaos performance of the generalized system. It is interesting to notice that the generalized system exhibits superior dynamics due to an additional control parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. Analytical and numerical simulations are used to explore the superior dynamical characteristics of the generalized system for some specific values of parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. Further, it is inferred that the superiority in dynamics of the generalized system may be efficiently used for better future applications.

A non-hierarchical multi-layer multi-configurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces.

The correlation discrete variable representation (CDVR) enables multi-layer multi-configurational time-dependent Hartree (MCTDH) quantum dynamics simulations on general potential energy surfaces. In a recent study [R. Ellerbrock and U. Manthe, J. Chem. Phys. 156, 134107 (2022)], an improved CDVR that can account for the symmetry properties of a tree-shaped wavefunction representation has been introduced. This non-hierarchical CDVR drastically reduces the number of grid points required in the time-dependent quadrature used to evaluate all potential energy matrix elements. While the first studies on the non-hierarchical CDVR approach have been restricted to single-layer calculations, here the complete theory required for the implementation of the non-hierarchical CDVR approach in the multi-layer MCTDH context will be presented. Detailed equations facilitating the efficient recursive computation of all matrix elements are derived, and a new notation adapted to the symmetry properties of the tree-shaped representation is introduced. Calculations studying the non-adiabatic quantum dynamics of photoexcited pyrazine in 24 dimensions illustrate the properties of the non-hierarchical multi-layer CDVR.

A unified framework for perceived magnitude and discriminability of sensory stimuli

The perception of sensory attributes is often quantified through measurements of sensitivity (the ability to detect small stimulus changes), as well as through direct judgments of appearance or intensity. Despite their ubiquity, the relationship between these two measurements remains controversial and unresolved. Here, we propose a framework in which they arise from different aspects of a common representation. Specifically, we assume that judgments of stimulus intensity (e.g., as measured through rating scales) reflect the mean value of an internal representation, and sensitivity reflects a combination of mean value and noise properties, as quantified by the statistical measure of Fisher information. Unique identification of these internal representation properties can be achieved by combining measurements of sensitivity and judgments of intensity. As a central example, we show that Weber’s law of perceptual sensitivity can coexist with Stevens’ power-law scaling of intensity ratings (for all exponents), when the noise amplitude increases in proportion to the representational mean. We then extend this result beyond the Weber’s law range by incorporating a more general and physiology-inspired form of noise and show that the combination of noise properties and sensitivity measurements accurately predicts intensity ratings across a variety of sensory modalities and attributes. Our framework unifies two primary perceptual measurements—thresholds for sensitivity and rating scales for intensity—and provides a neural interpretation for the underlying representation.

Generalized Balancing and Balancing-Lucas numbers

In this paper, we introduce a generalization of Balancing and Balancing-Lucas numbers. We describe some of their properties also we give the related matrix representation and divisibility properties.

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

PointGAT: A Quantum Chemical Property Prediction Model Integrating Graph Attention and 3D Geometry.

Predicting quantum chemical properties is a fundamental challenge for computational chemistry. While the development of graph neural networks has advanced molecular representation learning and property prediction, their performance could be further enhanced by incorporating three-dimensional (3D) structural geometry into two-dimensional (2D) molecular graph representation. In this study, we introduce the PointGAT model for quantum molecular property prediction, which integrates 3D molecular coordinates with graph-attention modeling. Comparison with other current models in molecular prediction tasks showed that PointGAT could provide higher predictive accuracy in various benchmark data sets from MoleculeNet, including ESOL, FreeSolv, Lipop, HIV, and 6 out of 12 tasks of the QM9 data set. To further examine PointGAT prediction of quantum mechanical (QM) energies, we constructed a C10 data set comprising 11,841 charged and chiral carbocation intermediates with QM energies calculated at the DM21/6-31G*//B3LYP/6-31G* levels. Notably, PointGAT achieved an R2 value of 0.950 and an MAE of 1.616 kcal/mol, outperforming even the best-performing graph neural network model with a reduction of 0.216 kcal/mol in MAE and an improvement of 0.050 in R2. Additional ablation studies indicated that incorporating molecular geometry into the model resulted in markedly higher predictive accuracy, reducing the MAE value from 1.802 to 1.616 kcal/mol. Moreover, visualization of PointGAT atomic attention weights suggested its predictions were interpretable. Findings in this study support the application of PointGAT as a powerful and versatile tool for quantum chemical property prediction that can facilitate high-accuracy modeling for fundamental exploration of chemical space as well as drug design and molecular engineering.

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.

Interpretable representations in explainable AI: from theory to practice

Interpretable representations are the backbone of many explainers that target black-box predictive systems based on artificial intelligence and machine learning algorithms. They translate the low-level data representation necessary for good predictive performance into high-level human-intelligible concepts used to convey the explanatory insights. Notably, the explanation type and its cognitive complexity are directly controlled by the interpretable representation, tweaking which allows to target a particular audience and use case. However, many explainers built upon interpretable representations overlook their merit and fall back on default solutions that often carry implicit assumptions, thereby degrading the explanatory power and reliability of such techniques. To address this problem, we study properties of interpretable representations that encode presence and absence of human-comprehensible concepts. We demonstrate how they are operationalised for tabular, image and text data; discuss their assumptions, strengths and weaknesses; identify their core building blocks; and scrutinise their configuration and parameterisation. In particular, this in-depth analysis allows us to pinpoint their explanatory properties, desiderata and scope for (malicious) manipulation in the context of tabular data where a linear model is used to quantify the influence of interpretable concepts on a black-box prediction. Our findings lead to a range of recommendations for designing trustworthy interpretable representations; specifically, the benefits of class-aware (supervised) discretisation of tabular data, e.g., with decision trees, and sensitivity of image interpretable representations to segmentation granularity and occlusion colour.

From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n-1)⊂S(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(n-1)\\subset S(n)$$\\end{document}. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.