Projective Space Research Articles

Intrinsic-dimension analysis for guiding dimensionality reduction and data fusion in multi-omics data processing.

Multi-omics data have revolutionized biomedical research by providing a comprehensive understanding of biological systems and the molecular mechanisms of disease development. However, analyzing multi-omics data is challenging due to high dimensionality and limited sample sizes, necessitating proper data-reduction pipelines to ensure reliable analyses. Additionally, its multimodal nature requires effective data-integration pipelines. While several dimensionality reduction and data fusion algorithms have been proposed, crucial aspects are often overlooked. Specifically, the choice of projection space dimension is typically heuristic and uniformly applied across all omics, neglecting the unique high dimension small sample size challenges faced by individual omics. This paper introduces a novel multi-modal dimensionality reduction pipeline tailored to individual views. By leveraging intrinsic dimensionality estimators, we assess the curse-of-dimensionality impact on each view and propose a two-step reduction strategy for significantly affected views, combining feature selection with feature extraction. Compared to traditional uniform reduction pipelines in a crucial and supervised multi-omics analysis setting, our approach shows significant improvement. Additionally, we explore three effective unsupervised multi-omics data fusion methods rooted in the main data fusion strategies to gain insights into their performance under crucial, yet overlooked, settings.

The Stability of TiO2 Phases Studied Using r2SCAN in the Hubbard-Corrected Density Functional Theory

Titanium dioxide is a quintessential transition metal oxide with many technologically important applications. With its richness in phases, it has also been a testing ground for numerous theoretical studies including density functional theory. We investigated several phases of TiO2 using the all-electron density functional theory with a regularized–restored strongly constrained appropriately normed (r2SCAN) exchange–correlation functional, a popular choice of meta-generalized gradient approximation (meta-GGA). Specifically, the equilibrium lattice parameters were more accurate than those predicted by GGA and agreed well overall with the experimental data. With increasing pressure, the order of stability was determined as anatase < columbite < rutile < baddeleyite < orthorhombic I < cotunnite, as in the calculations using GGA. Including the Hubbard correction term, the correct ordering between rutile, anatase, and columbite can be achieved, consistent with experimental observations. The necessary U value using r2SCAN is much smaller than that using GGA+U. In addition, the Hubbard correction method using r2SCAN is substantially less sensitive to the size of the local projection space compared to the GGA+U study reported recently. We attribute these significantly improved results to the reduced self-interaction error in the r2SCAN functional.

Evaluation of charge summing correction in CdTe-based photon-counting detectors for breast CT: performance metrics and image quality.

We evaluate the impact of charge summing correction on a cadmium telluride (CdTe)-based photon-counting detector in breast computed tomography (CT). We employ a custom-built laboratory benchtop system using the X-THOR FX30 0.75-mm CdTe detector (Varex Imaging, Salt Lake City, Utah, United States) with a pixel pitch of 0.1mm, operated in both standard mode [single pixel (SP)] and charge summing correction mode [anticoincidence (AC)]. A tungsten anode source operated at 55kVp with 2-mm aluminum external filtration and tube currents of 25, 100, and 200mA with corresponding exposure times of 20, 5, and 2.5ms were employed to study the effects of X-ray fluence and pulse pileup. Performance comparisons between AC and SP modes are performed in both projection and image reconstructed spaces. In the projection space, performance metrics include count rate, energy resolution, uniformity, modulation transfer function (MTF), and noise power spectrum (NPS). In the image space, performance metrics consist of contrast-to-noise ratio (CNR), uniformity, NPS, and iodine quantification accuracy. For both acquisition modes, signal-to-thickness calibration, for gain and beam hardening corrections, is used before image reconstruction. Images are reconstructed via TIGRE CT software using the standard Feldkamp, Davis, and Kress (FDK) filtered back projection algorithm with a Hann filter and reconstructed with a voxel size of 0.081mm. Material decomposition is performed using a standard image-based method. In the detector space, the application of hardware-based charge summing correction enhances spectral resolution and improves the spatial resolution of MTF at lower energy thresholds but introduces anomalous edge enhancement effects and artifacts in the MTF at high fluence. A negative noise correlation was observed in AC mode-acquired images. As expected, the AC acquisition mode results in a decreased detector count rate. In the image space, NPS results displayed elevated noise in low-energy AC images. However, at high energy, noise was comparable between both modes. Greater uniformity was observed in SP mode-acquired images. The largest disparity was observed in the iodine quantification test, where the AC mode demonstrates a much stronger linear relationship between estimated and true iodine concentrations than the SP mode. The results are specific to the studied system, reconstruction parameters, and irradiation conditions limited to 200mA and 0.5 mAs. The AC mode generally provides better energy and MTF resolution at low energy thresholds but with increased noise and reduced uniformity. In image space, charge summing correction improved iodine quantification and CNR at high energy thresholds.

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.

Symmetric Point Sets with Few Intersection Numbers in PG(r,q)

In 1965, H. Retkin and E. Stein defined a symmetric point set as a set of points with the same intersection numbers. In this paper, we perform a detailed analysis of symmetric point sets of the finite projective space which shows that the class of symmetric sets is very broad including caps, two-character sets and transitive sets. We derive necessary conditions for the existence of such sets. Since the most studied sets are caps and two-character sets and not much seems to be known in the general case of sets, which are different from caps, with more than two intersection numbers, by using incidence-preserving group actions, symmetric point sets with few intersection numbers are provided. The results indicate that any finite projective space contains symmetric sets with few intersection numbers.

Elliptic log symplectic brackets on projective bundles

Let X be the product of a complex projective space and a polydisc. We study Poisson brackets on X that are log symplectic, that is, generically symplectic and such that the inverse 2-form has only first order poles. We propose a method of constructing such Poisson brackets that additionally are elliptic, in a precise sense. Our method relies on the local Torelli theorem for log symplectic manifolds of Pym, Schedler and the author, and uses combinatorics of smoothing diagrams. We demonstrate effectiveness of the method on a series of examples, recovering, in particular, all log symplectic cases of elliptic Feigin–Odesskii Poisson brackets qn,k on Pn-1.

Optimal Transport Between Algebraic Hypersurfaces

Abstract What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces. First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. This distance is finer than the Fubini–Study one. The innner Wasserstein distance is complete and geodesic: geodesics corresponds to optimal deformations of one algebraic hypersurface into another one. Outside the discriminant this distance is induced by a smooth Riemannian metric, which is the real part of an explicit Hermitian structure. Moreover, this Hermitian structure is Kähler and the corresponding metric is of Weil–Petersson type. To prove these results we develop new techniques, which combine complex and symplectic geometry with optimal transport, and which we expect to be relevant on their own. We discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.

Hypersurfaces with large automorphism groups

We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to isomorphism and provide explicit generators for the automorphism group.

Information Geometry for Decoding

We revisit the idea of using elements of information geometry for decoding low-density parity-check (LDPC) codes, as introduced by Ikeda et al. In this work, we explicitly compute the m-projection to an e-flat submanifold, in the case of a binary symmetric channel and the Gaussian channel. We exemplify the algorithm by testing moderate size Gallager codes. To approach decoding problems, we show general theorems based on alternating projections in the framework of information geometry, inspired by von Neumann’s theorem for the convergence of alternating projections in Hilbert spaces. More precisely, consider the manifold S of the probability distributions on the n-dimensional hypercube (i.e., the set of binary sequences of length n). Let p be in S. In the case of two intersecting m-flat or e-flat submanifolds, the method of alternating projections on the two submanifolds converges to the projection of p on their intersection. This result is also generalized to a finite family of submanifolds of S.

Structuring data analysis projects in the Open Science era with Kerblam!

Background Structuring data analysis projects, that is, defining the layout of files and folders needed to analyze data using existing tools and novel code, largely follows personal preferences. Open Science calls for more accessible, transparent and understandable research. We believe that Open Science principles can be applied to the way data analysis projects are structured. Methods We examine the structure of several data analysis project templates by analyzing project template repositories present in GitHub. Through visualization of the resulting consensus structure, we draw observations regarding how the ecosystem of project structures is shaped, and what salient characteristics it has. Results Project templates show little overlap, but many distinct practices can be highlighted. We take them into account with the wider Open Science philosophy to draw a few fundamental Design Principles to guide researchers when designing a project space. We present Kerblam!, a project management tool that can work with such a project structure to expedite data handling, execute workflow managers, and share the resulting workflow and analysis outputs with others. Conclusions We hope that, by following these principles and using Kerblam!, the landscape of data analysis projects can become more transparent, understandable, and ultimately useful to the wider community.

Riemannian Shape Optimization of Thin Shells Using Isogeometric Analysis

ABSTRACTIn this contribution, we consider the optimal shape design of thin elastic shell structures based on a linearized shell model of Koiter's type, whose shape can be described by a surface immersed in three‐dimensional Euclidean space. We regard the set of unparametrized immersions of the surface as an infinite‐dimensional Riemannian shape space and perform optimization in this setting using the Riemannian shape gradient. Nonuniform rational basis splines (NURBS) are employed to discretize the shell and numerically solve the underlying equations that govern its mechanical behavior via isogeometric analysis. By representing NURBS patches as B‐spline patches in real projective space, NURBS weights can also be incorporated into the optimization routine. We discuss the practical implementation of the method and demonstrate our approach on the compliance minimization of a half‐cylindrical shell under static load and fixed area constraint.

Six-dimensional Supermultiplets from Bundles on Projective Spaces

The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to P1×P3. We use this fact, together with the pure spinor superfield formalism, to study supermultiplets in six dimensions, starting from vector bundles on projective spaces. We classify all multiplets whose derived invariants for the supertranslation algebra form a line bundle over the nilpotence variety; one can think of such multiplets as being those whose holomorphic twists have rank one over Dolbeault forms on spacetime. In addition, we explicitly construct multiplets associated to natural higher-rank equivariant vector bundles, including the tangent and normal bundles as well as their duals. Among the multiplets constructed are the vector multiplet and hypermultiplet, the family of O(n)-multiplets, and the supergravity and gravitino multiplets. Along the way, we tackle various theoretical problems within the pure spinor superfield formalism. In particular, we give some general discussion about the relation of the projective nilpotence variety to multiplets and prove general results on short exact sequences and dualities of sheaves in the context of the pure spinor superfield formalism.

Stable sheaves on bielliptic surfaces: from the classical to the modern

Bielliptic surfaces are the last family of Kodaira dimension zero algebraic surfaces without a classification result for the Chern characters of stable sheaves. We rectify this and prove such a classification using a combination of classical techniques, on the one hand, and derived category and Bridgeland stability techniques, on the other. Along the way, we prove the existence of projective coarse moduli spaces of objects in the derived category of a bielliptic surface that Bridgeland semistable with respect to a generic stability condition. By systematically studying the connection between Bridgeland wall-crossing and birational geometry, we show that for any two generic stability conditions τ,σ, the two moduli spaces Mτ(v) and Mσ(v) of objects of Chern character v that are semistable with respect to τ (resp. σ) are birational. As a consequence, we show that for primitive v, the moduli space of stable sheaves of class v is birational to a moduli space of stable sheaves whose Chern character has one of finitely many easily understood “shapes".

Line Spreads That Produce Projective Planes

We explicitly classify those line spreads of projective 5-space over a field that have the property that the given spread induces a spread in the 3-space generated by any pair of spread lines. We determine their fix groups and conclude that there exist such spreads with a trivial fix group. Also, we characterise regular line spreads among all line spreads of projective 3-space by their projectivity group and also by a weakening of the regularity condition.

Fine‐grained multi‐path channel estimation and matched reception by single metasurface antenna

AbstractThis paper investigates methods to enhance antenna's channel information extraction capabilities in wireless communications, particularly in challenging multi‐path environments. Focusing on the issues of channel information loss due to multi‐path superposition at the receiver and multi‐path fading during wireless channel propagation, this paper utilizes metasurface antennas to rapidly reconfigure antenna patterns and construct high‐dimensional projection spaces, thus improving information sensing resolution. Based on this, a channel‐matched receiving pattern design method is proposed to mitigate the communication performance loss caused by multi‐path fading. Furthermore, this paper analyses the comprehensive impact of various metasurface parameters on the performance of the proposed scheme, providing theoretical guidance for engineering implementation. Simulation results demonstrate that the proposed scheme can achieve the performance of multi‐path channel estimation and reception close to those of multi‐antenna arrays using a single radio frequency link.

Mechanisms of alignment of lamellar-forming block copolymer under shear flow.

The potential applications of block copolymer thin films, utilising their self-assembly capabilities, are enhanced when achieving long-range ordering. In this study we explain the experimental alignment of lamellae under shear flow findings [S. Pujari et al. Soft Matter, 2012, 8, 5258] and classify the alignment mechanisms based on shear rate and segregation, uncovering similarities to the systems subjected to electric fields, suggesting a common pathway of lamellae orientations. However, the presence of thin films surfaces introduces distinct features in the lamellae orientation under shear compared to electric fields. Notably, we observe the emergence of a three-dimensional rotation alongside the conventional two-dimensional rotation. Furthermore, a transient regime has been identified within the melting mechanism, which confirms the existence of the checkboard pattern proposed by Schneider et al. [Macromolecules, 2018, 51, 4642]. These findings significantly enhance our understanding of block copolymer alignments and shed light on the intricate interplay between external fields and the lamellar structure.

A quantitative version of Northcott's theorem on points of bounded height: The function field case

AbstractLet be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let denote an algebraic closure of . For given integers and we count points in projective space with absolute logarithmic height and generating an extension of degree over . Specifically, we derive an asymptotic estimate for the number of such points as when and orders of growth for the number of such points when .

КОМФОРТНАЯ ГОРОДСКАЯ СРЕДА В КОНТЕКСТЕ РАЗВИТИЯ РЕКРЕАЦИОННОЙ СИСТЕМЫ РЕГИОНОВ

The article selects and analyzes scientific literary sources on the improvement of public spaces. The analysis of the world experience of landscaping and modern specifics of urban development of public spaces from the point of view of the relationship of green areas with urban development is carried out. The main planning problems of public spaces and the problems of their reconstruction are investigated. The best techniques and features of public space projects have been identified.

Movable cones of complete intersections of multidegree one on products of projective spaces

We study Calabi–Yau manifolds which are complete intersections of hypersurfaces of multidegree 1 in an m-fold product of n-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi–Yau manifold X is infinite and a free product of copies of Z. Moreover, we give an explicit description of the boundary of the movable cone Mov¯(X). In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of Bir(X).

On (i)-Curves in Blowups of Pr

In this paper, we study (i)-curves with i∈{−1,0,1} in the blown-up projective space Pr in general points. The notion of (−1)-curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar (−1)-curves in order to construct a counterexample to Hilbert’s 14th problem. We introduce the notion of classes of (0)- and (1)-curves in Pr with s points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, F (which we will refer to as the anticanonical curve class). For Mori Dream Spaces, we prove that (−1)-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, (0)- and (1)-Weyl lines give the extremal rays for the cone of movable curves in Pr with r+3 points blown up. As an application, we use the technique of movable curves to reprove that if F2≤0 then Y is not a Mori Dream Space, and we propose to apply this technique to other spaces.