Search Results (640)

We give a proof of Cartan’s Theorem on totally geodesic submanifolds for real analytic manifolds endowed with a real analytic, torsion-free, affine connection. We apply the theorem to real analytic Hadamard manifolds and, more generally, to real analytic manifolds with a torsion-free, analytic, affine connection, such that at a manifold point $p\in M$, the exponential map is a real analytic diffeomorphism from the tangent space $T_p\left(M\right)$ to M. Examples of manifolds with this property are statistical manifolds with a cubic form divisible by the metric, as was recently proven. We also give examples of totally geodesic submanifolds obtained as fixed points of affine transformations of M and, moreover, as certain submanifolds of connected Lie groups with the 0-connection of Cartan–Schouten. Finally, we also determine all connected complete totally geodesic surfaces of the Riemannian manifold $({\mathcal{P}}_2,g)$ of symmetric positive definite $2\times 2$ real matrices, endowed with the trace metric g. Full article

Background and Objectives: The optimal oocyte yield in gonadotropin-releasing hormone (GnRH) antagonist in vitro fertilization (IVF) cycles remains debated, and data specific to antagonist protocols are limited. This study evaluated the discriminative and independent predictive performance of oocyte count for clinical pregnancy in GnRH antagonist IVF cycles. Materials and Methods: This retrospective cohort included 1171 women undergoing their first GnRH antagonist IVF cycle with fresh embryo transfer at a single tertiary center (September 2007–December 2021). The primary outcome was an institutional composite pregnancy outcome (sustained β-hCG positivity with subsequent ongoing intrauterine pregnancy or live birth; biochemical and ectopic pregnancies were negative). Patients were grouped by oocytes retrieved (1–5, 6–10, 11–15, ≥16). Performance was assessed with logistic regression, ROC with 2000-iteration bootstrap, integrated discrimination improvement (IDI), continuous net reclassification improvement (NRI), and restricted cubic spline. Predefined subgroup analyses by age, regimen, and antral follicle count tertile were performed. Results: A positive outcome occurred in 430 patients (36.7%). After adjustment, oocyte count was not an independent predictor (adjusted odds ratio 0.999, 95% CI 0.979–1.020; p = 0.96). The full model (AUC 0.564, 95% CI 0.529–0.598) did not outperform oocyte count alone (AUC 0.532; bootstrap p = 0.11). IDI (0.011) and NRI (0.135) were statistically detectable but clinically trivial. Spline regression showed no non-linearity (p = 0.47). Findings were consistent across subgroups, and the narrow confidence interval excluded per-oocyte effects ≥1.10. Conclusions: In GnRH antagonist IVF cycles, oocyte count showed weak discriminatory performance and was not independently associated with fresh-cycle pregnancy. Oocyte yield should be interpreted alongside—rather than as a substitute for—established parameters such as age and ovarian reserve. The principal clinical value of a higher oocyte response may lie in cumulative rather than fresh-cycle success. Live-birth outcomes were not available, and the source institution was permanently closed in 2025; these limitations define the boundary of inference. Full article

Useful quantum neural networks should not merely explore large Hilbert spaces but should organise their expressive capacity according to the symmetries of the learning problem. We introduce symmetry-organised complexity as an ansatz-level, representation-theoretic trajectory diagnostic for quantum neural networks. The diagnostic combines symmetry-sector organisation, cross-irreducible representation organised complexity, and symmetry metastability into a composite index, which is then multiplied by a compliance factor that penalises apparent complexity arising from symmetry violation. This compliance factor is defined at the level of the implemented trainable generators rather than as a representation-independent channel metric. The representation-theoretic basis of the construction is that, for an exactly equivariant network, the effective trainable operators lie in the commutant of the group action and are controlled by multiplicity dimensions rather than by the full Hilbert-space dimension. We show that joint sector collapse and state freezing force the index to vanish under an explicit multiplicity–purity condition and that networks with identical qubit and parameter counts can have different values of the index. Two analytically tractable four-qubit examples with excitation number and total spin symmetry illustrate how the diagnostic separates sector-collapsed, symmetry-organised, and symmetry-breaking behaviour. A controlled $U\left(1\right)$-compatible teacher–student classification task further shows that, in this validation setting, the ordering of the composite index across equivariant, hybrid, and non-equivariant ansatze agrees with the ordering of generalisation accuracy. The framework is most informative when the relevant symmetry of the learning problem is known. Full article

Background/Objectives: Abdominal lymphatic malformations (ALM) in children are rare vascular anomalies with heterogeneous presentation and challenging operative anatomy. Existing classification systems are largely descriptive and provide limited guidance for predicting resectability. We developed the LIE scoring system, an imaging-based classification incorporating Location, Intestinal involvement, and Extent, to enable structured preoperative risk stratification. Materials and Methods: We performed a retrospective, multicenter cohort study of pediatric patients with ALM treated at eight tertiary referral centers between 2010 and 2024. Lesions were assigned LIE scores based on preoperative imaging. Scores ranged from 0 to 5, with greater weight assigned to diffuse disease. Patients were categorized as resectable (0–2), limited resectable (3), or high risk (≥4). We performed multivariable logistic regression and receiver operating characteristic (ROC) analyses. Results: Fifty-nine patients were included. Complete resection rates decreased with increasing score (85.7%, 66.7%, and 25%; p for trend = 0.003). Higher scores were associated with increased risk of incomplete excision (OR 5.75, 95% CI 1.04–33.13). Multivariable analysis revealed consistent associations of extent and intestinal involvement with incomplete excision. ROC analysis demonstrated modest discriminative ability (AUC 0.62). Adjunctive therapies were more frequently used in higher-score groups. Conclusions: The LIE scoring system provides a clinically applicable framework for preoperative risk stratification in pediatric ALM. Despite modest predictive performance, it reflects operative complexity and may support surgical planning and patient counseling. Full article

Nonlinear optimization is a central component of visual odometry and simultaneous localization and mapping (SLAM), but its repeated small- and medium-scale linear algebra operations are difficult to deploy efficiently on embedded hardware. This paper presents a synthesizable C++ library for AMD/Xilinx Vitis high-level synthesis (HLS) that provides field-programmable gate array (FPGA)-oriented dense linear algebra kernels and Lie-group primitives on SO(3) and SE(3). The library supports configurable scalar types, including IEEE floating point, posit arithmetic, and reduced-precision floating-point formats, enabling design-space exploration between numerical accuracy and hardware cost. The proposed kernels are integrated into the back-end of a monocular direct mesh-based visual SLAM system and evaluated on an AMD/Xilinx Kria KV260 platform. Compared with the software reference running on the embedded processor, the integrated FPGA implementation reduces the end-to-end optimization iteration time from 32.0 ms to 8.9 ms, corresponding to a speed-up of 3.6×, and reduces the dominant kernel latency from 25.0 ms to 4.9 ms. The most resource-efficient reduced-precision configuration reduces lookup table (LUT) usage by 29.6%, flip-flop (FF) usage by 25.7%, block random-access memory (BRAM) usage by 25.9%, and digital signal processor (DSP) usage by 38.6% relative to the floating-point hardware baseline, while keeping the relative trajectory error within 1.42%. The results show that Lie-group-aware optimization back-ends can be mapped to embedded FPGAs efficiently when fixed-size algebraic kernels, synthesis-aware memory structures, and configurable arithmetic are considered together. Full article

Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which is obtained from a general time-dependent quadratic Hamiltonian. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three conserved quantities, which are time-dependent quadratic forms in position and momentum, and trivial, 0. All obtained invariant quantities are represented by auxiliary conditions, which are two linearly independent solutions of a homogeneous differential equation of the equations of motion. Invariant variables associated with an invariant consisting of the linearity of x and p are defined. It shows that, if the motion of the system is oscillatory, the Poisson bracket of the two invariant variables is obtained as i, and in the case of monotonic motion, it is obtained as 1. Full article

This paper studies a realization-theoretic problem inside the standard Lorentz-covariant Fourier-dual framework on $L^2\left({\mathbb{R}}^{3,1}\right)$: whether position-space and momentum-space geometric translations can be placed on equal structural footing without leaving the ordinary X- and K-polarized realizations. Working on the common Schwartz core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, we first isolate a Fourier-compatibility obstruction: Fourier transform exchanges geometric translations with character actions, while the Poincaré algebra contains at most one Lorentz-covariant abelian translation ideal. The main result is that, within the resulting Fourier-compatible realization class, the minimal operator-generated Lie algebra is the Lorentz–Heisenberg algebra. We then determine the full center of its universal enveloping algebra, derive the normalized Lorentz-bivector invariants, orbit data, and connected stabilizers in nondegenerate sectors, and show that the orbit variable is a normalized Lorentz bivector rather than a momentum vector. Finally, for fixed spectral elements in the dual translation sectors, we derive the associated scalar, Dirac, and vector equations in position and momentum space and show that, in the regular polarized realizations, the represented Heisenberg sector induces dual local abelian phase groups, compatible covariant derivatives, curvatures, and primary Dirac–Maxwell systems. Full article

Based on the Lie group method (symmetry transformation groups), an analysis of an unsteady (start-up) flow over a flat surface was performed. This approach enabled reducing the number of independent arguments, which significantly simplifies the process of numerical modeling. An unsteady solution was obtained for the velocity profile in the boundary layer. This enabled estimating the dynamics of the velocity profile transformation and its transition to a steady-state mode. It was shown that in the limit of infinite time of the process, the velocity profile tends to the classical steady-state Blasius profile in the boundary layer. The dynamics of the friction coefficient variation over time were elucidated too. Full article

The purpose of our paper is to provide a family of bilinear orthogonal expansions all based upon the same general pattern that is valid for a wide class of special functions. Our first family involves Jacobi, Laguerre, and Hermite polynomials. We give a discrete analogue of these bilinear expansions, the three families of classical orthogonal polynomials being replaced by zonal spherical functions associated with regular distance graphs. Such expansions playing a key role in the field of mathematical statistics, we show how our results apply to this field. We provide generalizations of the well-known Cramér–von Mises and Watson’s statistics, based upon an interpretation of their kernel in terms of the circular Laplacian. The product formula, well-known for zonal functions on Lie groups, is stated for distance-regular graphs, providing an elegant tool for proofs. Examples involving Hahn, q-Hahn, and Krawtchouk polynomials are given. Full article

Object detection from unmanned aerial vehicle (UAV) imagery is essential for applications such as traffic monitoring, disaster response, and urban surveillance, yet most existing methods are developed and evaluated under clear-sky conditions. In real-world UAV operations, adverse weather including fog, rain, and snow introduces severe image degradation that simultaneously disrupts both the geometric and photometric properties of targets. This paper identifies two fundamental bottlenecks underlying this performance collapse: the lack of geometric invariance in standard convolutional operators and the inability of fixed receptive fields to reconstruct features corrupted by atmospheric interference. To address these bottlenecks, we propose SELPNet (Scene Exploration and Large Format Perception Network), a unified framework that integrates geometric alignment and multi-scale contextual perception into the YOLOv13 head. SELPNet consists of two key modules: (1) The Scene Exploration Convolution (SEC) leverages affine Lie group theory to construct a discrete manifold of rotation and scale transformations, actively probing multiple geometric views and selecting the most coherent response via a Maxout mechanism. (2) The Large Format Perception Module (LPM) introduces a dynamic dilation strategy with depthwise separable convolutions, progressively enlarging the receptive field from fine-grained edge preservation to scene-level contextual perception for semantic completion of degraded regions. We further construct and release AWU-OBB, a large-scale benchmark containing over 18,000 oriented bounding box-annotated UAV images across four representative scene categories. Ablation experiments demonstrate that SEC and LPM yield complementary gains, achieving a combined improvement of +4.26% mAP50 over the YOLOv13-n baseline with only 0.11 M additional parameters and 0.2 extra GFLOPs. The source code will be publicly released upon acceptance of this paper. Full article

Indoor mobile robots often experience degraded localization accuracy and robustness when relying on a single positioning modality. In addition, conventional pose computation based on Euler-parameterized transformations can be computationally involved and susceptible to singularities, while practical fusion schemes may not adequately suppress measurement errors. This paper proposes an indoor robot localization method, termed IMU_UWB_ESKF, which tightly fuses inertial and UWB measurements using a Lie-group state representation. IMU- and UWB-derived quantities are formulated on the associated Lie algebra, enabling numerically stable pose propagation and measurement updates. To mitigate sensor noise and reduce drift, a dual-stage Kalman filtering strategy is adopted: an EKF-based measurement correction is first performed, followed by a multi-dimensional error-state Kalman filter for refined fusion. The proposed pipeline is implemented on a wheeled-robot platform under ROS, integrating real-time IMU/UWB parameter extraction, pose transformation, and online state estimation. Experimental results demonstrate stable real-time localization with improved robustness and accuracy under dynamic motion, indicating the method’s applicability to indoor navigation tasks. Full article

Let X be a compact Riemann surface of genus $g\ge 2$. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal $G_2\left(\mathbb{C}\right)$-bundle, where $G_2\left(\mathbb{C}\right)$ is the exceptional Lie group of automorphisms of the octonions. We prove that the natural inclusion $G_2\left(\mathbb{C}\right)\hookrightarrow \mathrm{SO}(7,\mathbb{C})$ induces a closed embedding of the moduli space ${\mathcal{M}}_{\mathrm{Oct}}\left(X\right)$ into the moduli space ${\mathcal{M}}_{\mathrm{SO}(7,\mathbb{C})}\left(X\right)$ of $\mathrm{SO}(7,\mathbb{C})$-bundles. We further analyze the normal bundle to this embedding, computing its rank as $7(g-1)$ and providing an explicit cohomological description of its fibers, which enables explicit computations of tangent spaces and provides a foundation for deformation theory. As applications of the embedding, we prove that the image is a closed irreducible subvariety not contained in the singular locus of the ambient space, and we derive the Whitney formula $c\left(T_{\mathrm{amb}}\right)=c\left(T\right)·c\left(N\right)$ relating the Chern classes of the tangent bundle of ${\mathcal{M}}_{\mathrm{Oct}}\left(X\right)$, the pullback of the ambient tangent bundle, and the normal bundle over the smooth locus. Full article

The Wigner quasiprobability, along with some of its essentialproperties, is introduced and discussed in two versions, first covering real canonical variables such as $W(q,p)$ and second a pair of complex conjugate coordinates such as $W(\alpha ,{\alpha }^{*})$. The reconstruction of the density operator $\varrho$ of states is also given. Building upon the Susskind–Glogower concept of quantum phase operators, further aspects of phase operator algebras in the quantum optics of a harmonic oscillator are discussed in relation to the realization of the $su(1,1)$ Lie algebra. Coherent phase states $|\epsilon ⟩$ are introduced in analogy to the common coherent states $|\alpha ⟩$ in two ways, as both eigenstates of certain operators and as states generated from a ground state $|0⟩$ by operators of the Lie group SU(1,1). The limiting transition to the non-normalizable Fritz London phase states $|{\mathrm{e}}^{\mathrm{i}\phi }⟩$ on the unit circle and an (over)-completeness relation for the coherent phase states are derived. The Wigner quasiprobability $W(q,p)$ for the coherent phase states is calculated and graphically represented. From the Wigner quasiprobability, a phase distribution $W\left(\phi \right)$ is calculated by integrating over the radius, and its uncertainty is defined and presented. The Hilbert–Schmidt distance is discussed as a measure of the non-classicality of states, where most of our with Viktor Dodonov work was carried out. Full article

Deep learning systems deployed in regulated settings require explanations that are accurate and stable under nuisance transformations, yet classical post hoc transition matrices rely on fidelity-only fitting that fails to guarantee consistent explanations under spatial rotations or other group actions. In this work, we propose Equivariant Transition Matrices, a post hoc approach that augments transition matrices with Lie-group-aware structural constraints to bridge this research gap. Our method estimates infinitesimal generators in the formal and mental feature spaces, enforces an approximate intertwining relation at the Lie algebra level, and solves the resulting convex Least-Squares problem via singular value decomposition for small networks or implicit operators for large systems. We introduce diagnostics for symmetry validation and an unsupervised strategy for regularization weight selection. On a controlled synthetic benchmark, our approach reduces the symmetry defect from 13,100 to 0.0425 while increasing the mean squared error marginally from 0.00367 to 0.00524. On the MNIST dataset, the symmetry defect decreases by 72.6 percent (141.19 to 38.65) with changes in structural similarity and peak signal-to-noise ratio below 0.03 percent and 0.06 percent, respectively. These results demonstrate that explanation-level equivariance can be reliably imposed post-training, providing geometrically consistent interpretations for fixed deep models. Full article