Abstract We investigate the problem of balanced embedding of a non-compact complex manifold into an infinite-dimensional projective space. In this paper, we prove the existence of such an embedding in a model case of a complex space. The strategy is by using a gradient flow in a Hilbert space; both the long-time existence and convergence at infinite time are non-trivial. The long time existence is established by choosing a perturbation of the ordinary differential equation; the convergence depends on a priori bounds that use techniques in the proof of the Tauber–Hardy–Littlewood theorem.

Geometric entanglement entropy on projective Hilbert space

We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu (2020) by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space ℂ P n and on complex hyperbolic space ℂ H n , with n ≥ 2 , separating coordinates necessarily have precisely n ignorable coordinates. In view of results of Boyer et al. (1983, 1985) and later results of Winternitz et al. (1994), this completes the description of separation of variables on ℂ P n for all n and on ℂ H n for n = 2 , 3 .

Transverse beam parameters in particle accelerators are commonly described using the Twiss parameters, which are experimentally accessible yet inherently limited because they neglect correlations between different transverse coordinates. Such correlations frequently arise from uncompensated cathode magnetic fields or misaligned focusing quadrupoles, affecting beam quality and accelerator performance. To address this limitation, we propose and validate a novel diagnostic method for the complete four-dimensional (4D) transverse beam matrix. Our method involves passing the beam through a beamline comprising both conventional and skew quadrupole magnets, followed by downstream measurements of the resulting two-dimensional (2D) beam profiles. These measurements represent distinct 2D projections of the underlying 4D transverse phase-space distribution. By systematically varying quadrupole strengths, multiple independent projections of the beam phase space are obtained. We reconstruct the original 4D beam matrix from these measured projections using an optimization-based least-square fit, providing fast and robust reconstruction regardless of the specific beamline configuration. Through extensive numerical simulations and realistic particle-tracking studies, we demonstrate the diagnostic's accuracy, robustness, and capability to achieve reconstruction uncertainties smaller than measurement errors, particularly when employing sufficient numbers of quadrupole scans. This method presents a powerful and flexible approach for comprehensive beam characterization and accelerator tuning.

Some projective distance inequalities for subvarieties of complex projective spaces and their applications

Stability of pullbacks of foliations on weighted projective spaces

We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply that the equality holds only for smooth points.

Let ℓp ⊗ π X (resp. ℓp ⊗Δ p X) denote the projective (resp. natural tensor product) of ℓp and a Banach space X. We study p-(V ∗) sets, Dunford-Pettis sets, and limited sets in ℓp ⊗ π X (resp. ℓp ⊗Δ p X). Some classes of operators T on ℓp ⊗ π X (resp. ℓp ⊗Δ p X) are characterized in terms of the representing operators Tn . We study pseudo weakly compact, limited completely continuous, DP q-convergent, limited q-convergent operators on ℓp ⊗ π X (resp. ℓp ⊗Δ p X). Applications to some properties on these spaces are given.

Abstract A finite CW-complex X is called P $\mathcal{P}$ -trivial if, for any real vector bundle ξ over X , the total Pontrjagin class p ( ξ ) = 1. In this article, we completely determine when each of the following spaces are P $\mathcal{P}$ -trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces. Furthermore, we show that the 3-fold suspension of a compact orientable surface of genus ≥ 1 $\ge \text{ }1$ is not P $\mathcal{P}$ -trivial.

Accurate registration of multi-temporal LiDAR point clouds is essential for reliable monitoring of mining subsidence. Systematic errors in point clouds acquired at different times can arise from GNSS/INS positioning drift, sensor calibration bias, and differences in observation geometry. These errors typically manifest as global reference shifts or gradual distortions. When such errors are superimposed on real terrain changes, they can mask subsidence signals and introduce observational pseudo-differences, thereby increasing the difficulty of separating actual subsidence from artifacts. To address this issue, this study proposes Robust-Registration-Based Systematic Error Correction for Time-Series Point Clouds (RR-SEC), which establishes a consistent reference framework across epochs. The method does not assume that stable areas remain strictly unchanged. Instead, it identifies regions whose local change patterns are more temporally consistent using an information entropy analysis of multi-temporal differences. Under complex terrain, the method selects points with lower difference entropy as stable control points and uses them to constrain the registration process. It then performs Generalized Iterative Closest Point (GICP) rigid registration under these constraints to estimate the overall three-dimensional translation and rotation between point clouds from different periods. The estimated transformation is applied to the entire point cloud to correct inter-epoch reference mismatches and unify the coordinate reference across all epochs. Comprehensive validation using simulated complex terrain data containing rigid reference biases and non-rigid deformations, as well as UAV LiDAR data collected from the MuduChaideng Coal Mine, shows that, compared with the baseline GICP method, RR-SEC reduces alignment errors. It decreases the mean residual in stable areas by approximately 85%. The subsidence values computed from the corrected point clouds are more consistent with measured values, and the spatial deformation patterns are easier to interpret. RR-SEC demonstrates robust performance and can serve as a practical approach to improve the accuracy of deformation monitoring in mining areas and potentially other geoscientific applications.

In this paper, we generalize our formalism of the elliptic virtual structure constants to hypersurfaces and complete intersections within certain weighted projective spaces possessing a single Kähler class.

This article explores how small mountain tourism destinations (MTDs) in the European Alps are beginning to translate climate change adaptation strategies into concrete, context-specific measures. Drawing on the strategies of 5 Pilot Working Areas (PWAs)—Ala di Stura and Balme (Italy), Werfenweng (Austria), Bohinj (Slovenia), Sattel-Hochstuckli (Switzerland), and Balderschwang (Germany)—of the Interreg Alpine Space project “BeyondSnow,” the study includes one of the first structural analyses of adaptation measures of MTDs. By applying the framework method for applied policy research, a matrix for the structural assessment of these measures was developed, including 3 analytic dimensions: overarching objectives, structural categories, and the distinction between implementation instruments (means) and the destination elements addressed (targets). Regarding overarching objectives, Year-round tourism emerged as the primary objective, followed by Destination cooperation and Sustainable development. Regarding structural categories, Product/concept development predominated as the implementation means, followed by Communication & marketing and Intradestination cooperation, with targets concentrated on Activity infrastructure and Activities & entertainment and complemented by Accommodation & gastronomy, Mobility & accessibility, and Destination community. Reading across the PWAs, means, and targets revealed 3 recurring implementation pathways: offer-led diversification, cooperation-led system alignment and portfolio extension, and distributed activation centered on specific audiences, placing low-barrier actions ahead of selective capital-intensive steps. The article concludes that, due to tourism's multidimensional, cross-sectoral nature, effective adaptation in small MTDs requires systemic approaches spanning products, community participation, and enabling infrastructure. Sequencing feasible, scalable measures, aligned with challenges identified, local governance, and sustainability visions, lays the groundwork for the broader systemic changes that are necessary for the creation of robust adaptation pathways, aimed at safeguarding both the competitiveness and viability of MTDs.

This article examines the representation of orphan caretakers in Francoist Spanish cinema (1956–1963) as a space of social negotiation and ideological projection. Focusing on films such as El pequeño ruiseñor and Rocío de la Mancha, it highlights how narratives of orphanhood reveal shifting attitudes toward class, gender and care. During the 1950s, male characters primarily took on the role of paternal caregivers, but by the 1960s, there was a shift towards more independent female protagonists, symbolizing a change toward self-sufficiency and secular institutions. The article argues that orphanhood becomes a metaphor for the regime’s contested vision of Spanish identity.

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini–Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini–Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.

Abstract. Topicality of methodological aspects of the research of poly-culture development of a leader as a dominant value within the educational spaces of a current higher educational institution is of priority in terms of new challenges and social realia. This leads to developing his pluri-cultural identity as mentor in the novel frameworks of life and activities. The purpose of this article is seen in the conceptualization of the polyculture development of a leader of as a dominant value within the information spaces of both the current polyculture society and the polyphony of the information spaces of a current higher educational institution. Methodology and methods of this research focuses on the system-activity approach as a priority within the frameworks of axiology of education and current scientific Scholarship of polyculture development of a leader in the polyphony of culture-oriented and culture-created activities. Research outcomes present the poly-culture and lingua polyphony of the information and education spaces project of modeling culture – oriented and organization as Self of a personality of a leader within the process of the communication challenges of a current higher educational institution continuum under consideration. In conclusion the prognosis aspect of the phenomena of the multi personal relations and poly-culture & poly-lingua regional context mixture is pronounced to underline the essence of the motives for authentic communication within a current higher educational institution continuum of the Region and overseas.

The long root geometry A n , { 1 , n } ( 𝕂 ) for the special linear group SL ( n + 1 , 𝕂 ) admits an embedding in the (projective space of) the vector space of the traceless square matrices of order n + 1 with entries in the field 𝕂 , usually regarded as the natural embedding of A n , { 1 , n } ( 𝕂 ) . S. Smith and H. Völklein in [10] have proved that the natural embedding of A 2 , { 1 , 2 } ( 𝕂 ) is relatively universal if and only if 𝕂 is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of A 2 , { 1 , 2 } ( 𝕂 ) which covers the natural one, but that information is not sufficient to exhaustively describe it. The “if” part of Smith-Völklein’s result also holds true for any n , as proved by Völklein in [13] in his investigation of the adjoint modules of Chevalley groups. In this paper we give an explicit description of the relatively universal embedding of A n , { 1 , n } ( 𝕂 ) which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to 𝔡 + n 2 + 2 n where 𝔡 is the transcendence degree of 𝕂 over its minimal subfield (if char ( 𝕂 ) = 0 ) or the generating rank of 𝕂 over 𝕂 p (if char ( 𝕂 ) = p > 0 ). Accordingly, both the “if” and the “only if” part of Smith-Völklein’s result hold true for every n ≥ 2 .

Although care and gender mainstreaming are increasingly recognized as key dimensions of sustainable urban planning, an analysis of their implementation in Mexico reveals the conceptual and material limitations of anthropocentric approaches to care within public space projects. In this article, we argue that ecofeminist and posthumanist perspectives on care help foreground the spatial and environmental dimensions of Everyday Care Environments (ECEs), highlighting ecosystemic interdependencies that remain largely overlooked in research focused on domestic, feminized, and family-based aspects of care work. Through qualitative research based on documentary analysis of local urban planning instruments and gender initiatives in Mexico City (CDMX) in the last 25 years, this article identifies persistent gaps in the integration of care work, safety, mobility, and intersectional perspectives into sustainable urban policy and practice. The findings offer insights for developing planning strategies capable of creating ECE that foster More-than-Human socio-environmental understandings of care, while advancing nature-based and ecosystem-oriented approaches to spatial justice.

Abstract We investigate experimentally the single-particle motion in water of silica colloidal beads half-coated with carbon under the action of a converging laser beam. The beads are self-propelled in this medium by means of self-thermophoresis resulting from local heating as a result of light absorption by their carbon cap. Within a certain laser power range, we find that these particles exhibit a quasi-two-dimensional active motion near a solid surface with stochastic rotational reversals when propelling themselves away from the region of maximum intensity, which leads to a stable trapping with oscillatory-like behavior inside the illuminated region. The orientation autocorrelation function of this type of confined active motion displays damped oscillations whose characteristic frequency increases with increasing propulsion speed, thus resulting in four regimes of translational motion depending on the observation timescale: thermal diffusion, ballistic motion, oscillatory behavior, and confinement. Our experimental findings are well described by a minimal phenomenological model that includes the nonlinear effect of a torque that reorients the particle toward the center of the optical confinement, which in combination with rotational diffusion gives rise to the observed orientational changes that allow their oscillatory trapping inside the light field. We also show that a similar active trapping mechanism emerges in the case of Janus colloidal rods, even though the periodicity is hindered by their three-dimensional rotation in the laser beam.

Hilbert polynomials have positivity properties under favorable conditions.We establish a similar "K-theoretic positivity" for matroids.As an application, for a multiplicityfree subvariety of a product of projective spaces such that the projection onto one of the factors is generically finite onto its image, we show that a transformation of its K-polynomial is Lorentzian.This partially answers a conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montao.As another application, we show that the h * -vector of a simplicially positive divisor on a matroid is a Macaulay vector, affirmatively answering a question of Speyer for a new infinite family of matroids.

We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of $\mathbb P^n \times \mathbb P^{n+1}$ play a central role in the study of the birational geometry of $X^{n,n+1}_s$, its blowup in $s$ points in general position. We show that $X^{n,n+1}_s$ is log Fano, and we compute its effective and movable cones, for $s\le n+2$ and $n\ge 1$ and for $s\le n+3$ and $n\le 2$, and we compute the effective and movable cones of $X^{3,4}_6$.