Reflection Equation Research Articles

Integrating Transformer Architecture and Householder Transformations for Enhanced Temporal Knowledge Graph Embedding in DuaTHP

The rapid advancement of knowledge graph (KG) technology has led to the emergence of temporal knowledge graphs (TKGs), which represent dynamic relationships over time. Temporal knowledge graph embedding (TKGE) techniques are commonly employed for link prediction and knowledge graph completion, among other tasks. However, existing TKGE models mainly rely on basic arithmetic operations, such as addition, subtraction, and multiplication, which limits their capacity to capture complex, non-linear relationships between entities. Moreover, many neural network-based TKGE models focus on static entities and relationships, overlooking the temporal dynamics of entity neighborhoods and their potential for encoding relational patterns, which can result in significant semantic loss. To address these limitations, we propose DuaTHP, a novel model that integrates Transformer blocks with Householder projections in the dual quaternion space. DuaTHP utilizes Householder projections to map head-to-tail entity relations, effectively capturing key relational patterns. The model incorporates two Transformer blocks: the entity Transformer, which models entity–relationship interactions, and the context Transformer, which aggregates relational and temporal information. Additionally, we introduce a time-restricted neighbor selector, which focuses on neighbors interacting within a specific time frame to enhance domain-specific analysis. Experimental results demonstrate that DuaTHP significantly outperforms existing methods in link prediction and knowledge graph completion, effectively addressing both semantic loss and time-related issues in TKGs.

The Power of Randomly Concrete Examples and Numerical Demonstrations

Householder reflector is a must-taught special unitary matrix in numerical linear algebra. It has plentiful theoretical as well as numerical properties, and it plays a significant role in practical algorithms. We use Householder reflectors as the linking thread to teach techniques and concepts on three topics: Deriving needed formulas, avoiding numerical cancelation errors, and implementing efficient code. Each topic is based on simple yet “randomly” concrete numerical examples, which are easy to construct and demonstrate via common MATLAB commands. We use the examples to bring forth salient as well as less-known features of a Householder reflector and related concepts. We also construct an extended version of a Householder reflector to address the perplexing question on the number of complex signs allowed in the reflector. Our numerical demonstrations, although simple in nature, can help students see the big picture of certain methods and concepts. The demonstrations are also designed to arouse students' interest so that they become more attentive in learning the more subtle technical details.

Approximation for reflection and transmission at a thin isotropic poroelastic bed between two isotropic elastic half‐spaces

AbstractSeismic responses from a horizontal poroelastic layer provide chances to detect fluids and characterize reservoirs. The poroelastic layer can be considered a thin poroelastic bed if the layer's thickness is less than about one‐eighth of the P‐wave wavelength. Most previous theoretical studies on the reflection and transmission of waves in a model containing a thin poroelastic bed employ fluid or poroelastic medium as the overlying media. Existing approximate formulas of PP‐wave reflection coefficients are given for P‐wave normal‐incidence. Thus, this paper derived the wave reflection and transmission approximate formulas of a thin poroelastic bed between two elastic half‐spaces with P‐wave oblique incident. First, we illustrated the exact reflection and transmission matrix equations for P‐wave incidents based on poroelasticity theory and the boundary conditions. Assuming the poroelastic bed's thickness is far less than wavelengths of S‐ and P‐waves, approximate reflection and transmission formulas are expanded in Taylor series centred at value of the parameter defined as the product of angular frequency, thickness and slowness. Numerical results show that the thinner the poroelastic layer, the closer the approximate reflection and transmission coefficients are to the exact ones. The approximate formulas are valid for small and medium angles. Approximated PP‐wave reflection and transmission coefficients are closer to the exact values than those of the converted waves, which is caused by the fact that P‐wave has a lower slowness than S‐wave.

Generalized Harish-Chandra morphism on Reflection Equation algebras

Generalized Harish-Chandra morphism on Reflection Equation algebras

Correction to: Householder Transform based Estimation of Signal and Sparsifying Basis from Blind Compressive Measurements

Correction to: Householder Transform based Estimation of Signal and Sparsifying Basis from Blind Compressive Measurements

IFAL-SLAM: an approach to inertial-centered multi-sensor fusion, factor graph optimization, and adaptive Lagrangian method

Abstract In response to the issues of low accuracy, perception degradation, and poor reliability of single-sensor SLAM technologies in complex environments, this study presents a novel IMU-centered multi-sensor fusion SLAM algorithm (IFAL-SLAM) integrating LiDAR, vision, and IMU, based on factor graph elimination optimization (\textbf{I}MU-centered multi-sensor \textbf{F}usion, \textbf{A}daptive \textbf{L}agrangian methods). The proposed system leverages a multi-factor graph model, centering on the IMU, and applies a covariance matrix to fuse visual-inertial and LiDAR-inertial odometries for bias correction, using loop closure factors for global adjustments. To minimize the optimization costs post-fusion, a sliding window mechanism is incorporated, coupled with a QR decomposition elimination method based on Householder transformation to convert the factor graph into a Bayesian network. Finally, an adaptive Lagrangian relaxation method is proposed, employing matrix-form penalty parameters and adaptive strategies to enhance convergence speed and robustness under high rotational dynamics.Experimental results indicate that the proposed algorithm achieves absolute trajectory errors of approximately 0.58 m and 0.24 m in large and small complex scenes, respectively, surpassing classic algorithms in terms of accuracy and reliability.

Rising Female‐Headed Households: Shifts in Living Arrangements or Heightened Gender Symmetry?

AbstractCensuses and surveys predominantly report men as heads of households or reference persons despite women carrying out most domestic and care work. Recent evidence, however, suggests that an increasing number of households are headed by women. Using data from the newly released CORESIDENCE database, which includes over 770 data points from 156 countries worldwide spanning from 1960 to 2021, this study presents the first global map of female headship, traces its recent evolution, and compares female‐headed households with male‐headed ones based on selected household characteristics. The results confirm the widespread increase in female headship in virtually all world regions. Nevertheless, significant cross‐national differences persist, and changes are not uniform across all regions. Spatial and temporal variations in female headship can be attributed, in part, to structural shifts in living arrangements, specifically the decreased presence of adult men in households. Female headship, however, is rising beyond the structural transformation of households. Women are increasingly likely to head households even in the presence of adult men, particularly their male partners. This might be indicative of normative changes towards gender symmetry. We discuss the potential factors behind these transformations and consider their implications for further research and gender equality.

Migration as a Catalyst of Change: The Impact of Indian Labour Migration to the GCC

The idea of emigration as a method of subsistence has long gained traction in India. Emigration and remittances have been praised as financial sources for the transformation of communities and households. Due to its historical links to international migration and its position as one of India's top labor exporting states, Kerala is an especially intriguing state that has been one of the most significant source regions for Indian temporary workers, primarily to the GCC (Gulf Cooperative Council) countries. Kerala is home to almost 2.1 million migrants, and 36% of the state's net state product comes from their contributions. This study looks at empirical data from the socioeconomic discourses surrounding Gulf migration with respect to the emigrants, their households, and the place from where they migrated in order to comprehend how migrant houses differ from non-migrant households. The study covers empirical research on the reasons behind migration, mobility trends, and the issue of using migration as a strategy to lower state unemployment. The paper goes on to discuss the significance of emigration in the socioeconomic transformations of the state.

Approximate P3 equation analysis in multi-layer slab media: Steady-state and time-domain based on the diffusion model

This study aimed to establish a steady-state and time-domain solutions for the approximate P3 equation in arbitrary multi-layer slab media for light propagation, based on a diffusion model with an isotropic point source in the first layer and extrapolated boundary conditions. Spatially resolved diffuse reflectance and transmittance were calculated using the steady-state approximation of the P3 equation, whereas temporally resolved diffuse reflectance and transmittance were derived from the time-domain approximation. The validity of the approximate P3 solutions was confirmed by comparing their results with Monte Carlo simulations. In steady-state analysis, the approximate P3 equation demonstrated superior accuracy to the diffusion equation for reflectance, particularly at smaller thicknesses. Accuracy further improved as the absorption coefficient and detection distance increased. For transmittance, the approximate P3 equation closely matched the diffusion equation at low thickness, but divergence occurred with higher absorption. In time-domain analysis, the approximate P3 equation aligned closely with Monte Carlo simulations at peak values, while its numerical values were close approximations of the diffusion equation away from the peak. The potential application of the approximate P3 equation in multi-layer media offers significant advancements for optical non-invasive detection and treatment techniques, enabling the extraction of optical parameters from such media.

Quantifying the Carbon Reduction Difference Based on Households' Willingness to Transform Their Lifestyles from the Micro Perspective

Cities are gathering places for residents, industries, and energy consumption, and the carbon emissions generated by urban residents' consumption are becoming increasingly prominent. Under the "dual carbon" goal, how to effectively guide the transformation of urban households' lifestyles and achieve consumption side carbon reduction has become an important issue. Based on questionnaire surveys and face-to-face interviews in the main districts of Hangzhou, the total amount and component of household consumption carbon reduction potential in Hangzhou were calculated by adopting a carbon emission coefficient model, and the relevant factors influencing various types of household consumption behaviors and carbon reduction potential were analyzed through virtual regression models and binary logistic regression. Our results showed that the consumption carbon reduction potential of different groups of residents in the four dimensions of food, clothing, housing, and transportation was significantly different, and electricity consumption had significant direct and indirect impacts on household consumption behaviors and carbon reduction potentials. Age and income were the key factors leading to significant differences in the carbon reduction potential of household consumption. Economic interests and environmental emotional factors were the main factors that affected the transformation of residents' family lifestyle, and their strengths and weaknesses were different. Personal habits, demonstration effects of surrounding people, time costs, etc., were the main reasons affecting residents' willingness to transform their lifestyles. The findings therefore had important implications for climate change mitigation and policy measures associated with lifestyle. On the premise of ensuring stable economic development and improving the well-being of residents, differentiation guides the lifestyle choices of community residents providing reference for better realization of the "double carbon" goal.

Representation theory of the reflection equation algebra II: Theory of shapes

We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the R-matrix associated to the standard q-deformation of GL(N,C) for 0<q<1. We consider the Poisson structure appearing as the classical limit of the R-matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape.

Reflection coefficients for non-welded interface between orthorhombic half-spaces under initial stress

SUMMARY Complex oil and gas reservoirs represented by orthorhombic shale usually have the characteristics of strong anisotropy, wide distribution of non-welded interfaces (i.e. particular boundaries caused by horizontal bedding or fracture development), and ubiquitous initial stress effects. The knowledge of their seismic response characteristics plays an essential role in hydrocarbon exploration in stressed shale reservoirs. However, seismic wave reflection in the orthorhombic media under initial stress remains unclear. To address this issue, we used the theories of acoustoelasticity and elastic anisotropy to derive a stress-dependent effective elastic stiffness tensor. At the same time, we obtain the wave velocity and polarization characteristics directly by solving the Christoffel equation. According to the linear slip theory, we further constructed the exact reflection and transmission coefficient equations for the non-welded interface between isotropic and orthorhombic half-spaces under the initial stress. The effects of the non-welded interface on seismic wave velocity and reflection and transmission coefficients were systematically analysed. Meanwhile, we characterize the law of P-wave reflection coefficient effected by elastic coefficients. Our equations and results potentially lay an equation foundation for orthorhombic reservoirs in high-stress fields and are essential in orthorhombic reservoirs in high-stress geophysical exploration.

Fragment quantum embedding using the Householder transformation: A multi-state extension based on ensembles

In recent studies by Yalouz et al. [J. Chem. Phys. 157, 214112 (2022)] and Sekaran et al. [Phys. Rev. B 104, 035121 (2021) and Computation 10, 45 (2022)], density matrix embedding theory (DMET) has been reformulated through the use of the Householder transformation as a novel tool to embed a fragment within extended systems. The transformation was applied to a reference non-interacting one-electron reduced density matrix to construct fragments’ bath orbitals, which are crucial for subsequent ground state calculations. In the present work, we expand upon these previous developments and extend the utilization of the Householder transformation to the description of multiple electronic states, including ground and excited states. Based on an ensemble noninteracting density matrix, we demonstrate the feasibility of achieving exact fragment embedding through successive Householder transformations, resulting in a larger set of bath orbitals. We analytically prove that the number of additional bath orbitals scales directly with the number of fractionally occupied natural orbitals in the reference ensemble density matrix. A connection with the regular DMET bath construction is also made. Then, we illustrate the use of this ensemble embedding tool in single-shot DMET calculations to describe both ground and first excited states in a Hubbard lattice model and an ab initio hydrogen system. Finally, we discuss avenues for enhancing ensemble embedding through self-consistency and explore potential future directions.

Complex Palais matrix and a new unitary transform with bounded component norms

Abstract The Palais matrix represents an n n -dimensional rotation between two vectors that is functionally equivalent to the Householder reflection. This study introduces a one-parameter family of unitary transforms, termed the θ \theta transform, which encompasses the transform by the Palais matrix, the Householder reflection, and their unitary extensions. Furthermore, we define the θ ∠ {\theta }_{\angle } transform, a variant of the θ \theta transform featuring bounded component norms. It is demonstrated that the θ ∠ {\theta }_{\angle } transform is computationally efficient and backward stable when one of the vectors has the “one-hot” structure, making it highly valuable for matrix decompositions such as the QR decomposition. In addition, the θ \theta transform exhibits additional characteristics, including its convergence to the identity and the rowwise structure of its backward error.

Unleashing the Power of Decoders: Temporal Knowledge Graph Extrapolation with Householder Transformation

In the realm of artificial intelligence, knowledge graphs (KGs) serve as an essential structured framework, capturing intricate relationships between diverse entities and supporting a broad spectrum of AI applications. Despite their utility, the static characteristic of KGs poses challenges in dynamically evolving information landscapes. This has catalyzed the development of temporal knowledge graphs (TKGs), which introduce a temporal layer to KGs, facilitating the representation of knowledge progression through time. This study zeroes in on the critical task of TKG extrapolation, which is vital for forecasting future occurrences and offering foresight into emerging situations across a variety of fields. Most contemporary approaches to TKG extrapolation are predicated on the symmetrical encoder–decoder paradigm, wherein the processes of representation learning and reasoning are harmoniously intertwined. While the encoder often garners the most attention due to its role in capturing and encoding information, the pivotal role of the decoder, which is often overlooked, is essential for direct inference and the accurate projection of temporal dynamics. To this end, we present the Householder-transformation-based temporal knowledge graph extrapolation (HTKGE) method: a groundbreaking encoder–decoder framework that reimagines the decoder’s contribution to TKG extrapolation. Our approach spotlights an adaptive decoder propelled by Householder transformations, which engage dynamically with the temporal encoding from the encoder. This interaction fosters a nuanced comprehension of the TKG’s temporal trajectory. Our empirical evaluations across four benchmark TKG datasets substantiate HTKGE’s consistent efficacy in TKG extrapolation tasks.

Energy consumption, structural transformation and related carbon dioxide emissions of rural households on the Tibetan plateau

The use of solid fuels presents a dual challenge for rural household energy systems: transitioning to cleaner energy sources while reducing carbon dioxide (CO2) emissions. Particularly in the Tibetan Plateau (TP), which is highly sensitive to global climate change, addressing rural household energy consumption, structural changes, and related CO2 emissions is crucial for implementing targeted measures. Based on household-level survey data, this study analyzes the quantitative and structural changes in energy consumption patterns among rural households on the TP from 2001 to 2021, as well as their related CO2 emissions. The results indicate that: (1) Yak dung is the primary fuel for rural households on the TP and a major contributor to direct household emissions. (2) While cooking has shifted considerably towards a modern clean energy structure, heating continues to rely on solid fuels such as firewood, yak dung, and coal. (3) Commercial energy used by high-income groups for agriculture, transportation, and other purposes has resulted in an overall increase in total household energy consumption and related CO2 emissions. Policymakers must implement more effective measures to accelerate the transition to cleaner, modern energy sources for heating and cooking purposes.

FET-LM: Flow-Enhanced Variational Autoencoder for Topic-Guided Language Modeling.

Variational autoencoder (VAE) is widely used in tasks of unsupervised text generation due to its potential of deriving meaningful latent spaces, which, however, often assumes that the distribution of texts follows a common yet poor-expressed isotropic Gaussian. In real-life scenarios, sentences with different semantics may not follow simple isotropic Gaussian. Instead, they are very likely to follow a more intricate and diverse distribution due to the inconformity of different topics in texts. Considering this, we propose a flow-enhanced VAE for topic-guided language modeling (FET-LM). The proposed FET-LM models topic and sequence latent separately, and it adopts a normalized flow composed of householder transformations for sequence posterior modeling, which can better approximate complex text distributions. FET-LM further leverages a neural latent topic component by considering learned sequence knowledge, which not only eases the burden of learning topic without supervision but also guides the sequence component to coalesce topic information during training. To make the generated texts more correlative to topics, we additionally assign the topic encoder to play the role of a discriminator. Encouraging results on abundant automatic metrics and three generation tasks demonstrate that the FET-LM not only learns interpretable sequence and topic representations but also is fully capable of generating high-quality paragraphs that are semantically consistent.

Exact solution of the [formula omitted] quantum spin chain with open boundary condition

In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.

Quantum Cluster Algebras and 3D Integrability: Tetrahedron and 3D Reflection Equations

Abstract We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock–Goncharov quivers associated with the longest elements of the Weyl groups of type $A$ and $C$, and investigate the cluster transformations corresponding to changing a reduced expression into a “most distant” one. By devising a new realization of the quantum $y$-variables in terms of $q$-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum $y$-variables. Explicit formulas of their matrix elements are also derived for some typical representations.

Fraught triads – treating older women in crisis living with a migrant live-in caregiver and frail husband

ABSTRACT Hiring migrant live-in caregivers to take care of older adults with dementia, chronic disease or frailty has become more common in Europe, North America, the Middle East, and Asia. When older couples hire a non-relative to provide live-in caregiver services for a family member, the attendant household transformation – from couple to triad – gives rise to social and psychological challenges. Although live-in caregivers can be indispensable, living together can be very stressful for older adults and live-in caregiver. This article describes the emotional distress of two older adult wives in their eighties, and the therapeutic methods used to reduce it, including empathic understanding, legitimation of emotions, promotion of open communication, and formulation of a household ‘management strategy,’ establishing specific expectations concerning the parties’ roles and a daily activity schedule. Involvement of older couples’ children was found helpful, and was aided through psychoeducation. Gerontologists, therapists, educators, and social workers who serve the needs of older adults can effectively assist this population in the evolution of their perceptions of their new situation, the regaining of a sense of equilibrium, mastery and control over their lives, and the development of greater tolerance and understanding of migrant live-in caregivers’ work burden, native culture, and individual practices. Professionals should actively screen for risks of burden, stress, depression, and anxiety in households in which live-in caregiver services may be considered. Psychoeducation and preparation of both the older adults and their live-in caregivers entering the new relationship may lessen the incidence of potential stressors.