Projective Space Research Articles

Polarity-driven three-dimensional spontaneous rotation of a cell doublet

AbstractMechanical interactions between cells play a fundamental role in the self-organization of organisms. How these interactions drive coordinated cell movement in three dimensions remains unclear. Here we report that cell doublets embedded in a three-dimensional extracellular matrix undergo spontaneous rotations. We investigate the rotation mechanism and find that it is driven by a polarized distribution of myosin within cell cortices. The mismatched orientation of this polarized distribution breaks the doublet mirror symmetry. In addition, cells adhere at their interface through adherens junctions and with the extracellular matrix through focal contacts near myosin clusters. We use a physical theory describing the doublet as two interacting active surfaces to show that rotation is driven by myosin-generated gradients of active tension whose profiles are dictated by interacting cell polarity axes. We also show that three-dimensional shape symmetries are related to broken symmetries of the myosin distribution in cortices. To test for the rotation mechanism, we suppress myosin clusters using laser ablation and generate new myosin clusters by optogenetics. Our work clarifies how polarity-oriented active mechanical forces drive collective cell motion in three dimensions.

Limits of conical Kähler–Einstein metrics on rank one horosymmetric spaces

AbstractWe consider families of conical Kähler–Einstein metrics on rank one horosymmetric Fano manifolds, with decreasing cone angles along a codimension one orbit. At the limit angle, which is positive, we show that the metrics, restricted to the complement of that orbit, converge to (the pull‐back of) the Kähler–Einstein metric on the basis of the horosymmetric homogeneous space, which is a projective homogeneous space. Then we show that, on the symmetric space fibers, the rescaled metrics converge to Stenzel's Ricci flat Kähler metric.

Average Degree of the Essential Variety

AbstractThe essential variety is an algebraic subvariety of dimension 5 in real projective space $$\mathbb R\mathrm P^{8}$$ R P 8 which encodes the relative pose of two calibrated pinhole cameras. The 5-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension 5. The degree of the essential variety is 10, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group $$\textrm{O}(9)$$ O ( 9 ) acting on linear spaces in $$\mathbb R\mathrm P^{8}$$ R P 8 . In this case, the expected number of real intersection points is equal to 4. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes $$\mathbb R\mathrm P^2\times \mathbb R\mathrm P^2$$ R P 2 × R P 2 uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval $$(3.95 - 0.05,\ 3.95 + 0.05)$$ ( 3.95 - 0.05 , 3.95 + 0.05 ) .

Sign involutions on para-abelian varieties

We study the so-called sign involutions on twisted forms of abelian varieties, and show that such a sign involution exists if and only if the class in the Weil–Châtelet group is annihilated by two. If these equivalent conditions hold, we prove that the Picard scheme of the quotient is étale and contains no points of finite order. In dimension one, such quotients are Brauer–Severi curves, and we analyze the ensuing embeddings of the genus-one curve into twisted forms of Hirzebruch surfaces and weighted projective spaces.

Hopf algebroids and twists for quantum projective spaces

We study the relationship between antipodes on a Hopf algebroid ▪ in the sense of Böhm–Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H-Hopf–Galois extensions B⊆A and related Ehresmann–Schauenburg bialgebroid. In particular, we find that the twists are in one-to-one correspondence with H-comodule algebra automorphism of A. We work out in detail the U(1)-extension O(CPqn−1)⊆O(Sq2n−1) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K-theory of the base algebra O(CPqn−1).

Convex co‐compact representations of 3‐manifold groups

AbstractA representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton’s equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton’s equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton’s and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton’s, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton’s equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

Optimizing infrastructure development through BIM: A comprehensive analysis of lifecycle benefits and applications

This comprehensive study delves into the multifaceted impacts of Building Information Modeling (BIM) across different phases of infrastructure projects, encompassing project planning and design, construction, as well as operations and maintenance. By offering a granular analysis of BIM's contributions to design efficiency, enhanced collaboration, sustainability considerations, improved project management, risk mitigation, quality control, asset management, lifecycle analysis, and space management and utilization, the paper presents quantifiable evidence of BIM's transformative potential. The findings underscore the substantial benefits of BIM in reducing project timeframes, minimizing material costs, enhancing operational efficiency, and contributing to sustainability goals. Through detailed quantitative analysis, this article illustrates how BIM technology facilitates a more integrated, efficient, and sustainable approach to infrastructure development, challenging traditional practices and fostering a new paradigm in construction and facility management.

Intermediate dimensions under self-affine codings

Intermediate dimensions were recently introduced by Falconer et al. (Math Z 296:813–830, 2020) to interpolate between the Hausdorff and box-counting dimensions. In this paper, we show that for every subset E of the symbolic space, the intermediate dimensions of the projections of E under typical self-affine coding maps are constant and given by formulas in terms of capacities. Moreover, we extend the results to the generalized intermediate dimensions introduced by Banaji (Monatsh Math 202: 465–506, 2023) in several settings, including the orthogonal projections in Euclidean spaces and the images of fractional Brownian motions.

Algebraic dependences of meromorphic mappings sharing few moving hyperplanes with truncated multiplicity

In this article, we will prove an algebraic dependence theorem for meromorphic mappings into a complex projective space sharing few moving hyperplanes with different truncated multiplicity. Moreover, we also consider the weaker condition: \[\nu _{(f,{a_i}), \le k_i}\le \nu _{(g,{a_i}), \le k_i} \text{ instead of } \nu _{(f,{a_i}), \le k_i}= \nu_{(g,{a_i}), \le k_i}\] for some moving hyperplanes among the given moving hyperplanes. In order to implement this, besides using the technique reported by S. D. Quang in (Two meromorphic mappings having the same inverse images of some moving hyperplanes with truncated multiplicity, Rocky Mountain J. Math., vol. 52, no. 1, pp. 263–273, 2022) we have to separate the 2n + 2 moving hyperplanes from the given p+1 moving hyperplanes. After that, we count multiples of the intersection of the inverse images of the mappings f and g sharing these moving hyperplanes. Our result is an improvement of many previous results in this topic.

Complex submanifolds of indefinite complex space forms

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.

Slower but more accurate mental rotation performance in aphantasia linked to differences in cognitive strategies

Mental rotation tasks are frequently used as standard measures of mental imagery. However, aphantasia research has brought such use into question. Here, we assessed a large group of individuals who lack visual imagery (aphantasia) on two mental rotation tasks: a three-dimensional block-shape, and a human manikin rotation task. In both tasks, those with aphantasia had slower, but more accurate responses than controls. Both groups demonstrated classic linear increases in response time and error-rate as functions of angular disparity. In the three-dimensional block-shape rotation task, a within-group speed-accuracy trade-off was found in controls, whereas faster individuals in the aphantasia group were also more accurate. Control participants generally favoured using object-based mental rotation strategies, whereas those with aphantasia favoured analytic strategies. These results suggest that visual imagery is not crucial for successful performance in classical mental rotation tasks, as alternative strategies can be effectively utilised in the absence of holistic mental representations.

DESIGN & FABRICATION OF RETRACTABLE SEATING AND DRIVING MECHANISM ARRANGEMENT FOR TELESCOPING 3 WHEEL GO-KART

Most of the recreation parks are using 4-wheel petrol (or) diesel powered go-carts which are expensive, heavy in weight and pollutes the environment. As the fossil fuels are depreciating and cost of the fuels is also increasing rapidly in global markets, it is very much essential to develop new go-kart designs with less weight, runs on electric energy (battery) so that it is not dependent on costly imported fuels like petrol / diesel and also consumes less energy. 3 Wheel electric (battery powered) Go Kart is one of the less weight designs that consumes less energy and adds no pollution to the environment. As a part of our project dissertation, we are planning to fabricate chassis for telescopic 3-wheel go-kart. Telescopic 3-wheel go-kart has fallowing advantages when compared to 4-wheel 2-seater go- karts; Less in mass hence gives good energy economy No pollution Easy to transport as it occupies less space Project Involves; Study of various types of chassis, materials used for chassis, structural elements like pipes, plates and flats, fabrication operations like welding, cutting, drilling, etc.

DESIGN & FABRICATION OF CHASSIS FOR TELESCOPING 3-WHEEL GO-KART

Most of the recreation parks are using 4-wheel petrol (or) diesel powered go-carts which are expensive, heavy in weight and pollutes the environment. As the fossil fuels are depreciating and cost of the fuels is also increasing rapidly in global markets, it is very much essential to develop new go-kart designs with less weight, runs on electric energy (battery) so that it is not dependent on costly imported fuels like petrol / diesel and also consumes less energy. 3 Wheel electric (battery powered) Go Kart is one of the less weight designs that consumes less energy and adds no pollution to the environment. As a part of our project dissertation, we are planning to fabricate chassis for telescopic 3-wheel go-kart. Telescopic 3-wheel go-kart has fallowing advantages when compared to 4-wheel 2-seater go- karts; Less in mass hence gives good energy economy No pollution Easy to transport as it occupies less space Project Involves; Study of various types of chassis, materials used for chassis, structural elements like pipes, plates and flats, fabrication operations like welding, cutting, drilling, etc.

On stability behaviors of 5D M-theory black objects

Using N = 2 supergravity formalism, we investigate certain behaviors of five-dimensional black objects from the compactification of M-theory on a Calabi–Yau three-fold. The manifold has been constructed as the intersection of two homogeneous polynomials of degrees (ω + 2, 1) and (2, 1) in a product of two weighted projective spaces given by WP4(ω,1,1,1,1)×P1 . First, we determine the allowed electric charge regions of the BPS and non BPS black holes obtained by wrapping M2-branes on appropriate two cycles in such a proposed Calabi–Yau three-fold. After that, we calculate the entropy of these solutions which takes a maximal value corresponding to ω = 1 defining the ordinary projective space P4 . For generic values of ω, we show that the non BPS states are unstable. Then, we conduct a similar study of five-dimensional black strings. Concerning the allowed magnetic charge regions of the BPS and non BPS black stringy solutions derived from M5-branes on dual divisors, we calculate the tension taking a minimal value for P4 . By determining the recombination factor, we show that the non-BPS black string states are stable in the allowed regions in the magnetic charge space.

Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Let P:=U(p+q)/U(p)×U(q) be the complex Grassmann manifold and F:T1,0P→[0,+∞) be an arbitrary U(p+q)-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U(p+q)-invariant Kähler metric on P. In particular on the complex projective space CPn=U(n+1)/U(n)×U(1), there exists no U(n+1)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U(p+q)-invariant Kähler metrics on P, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.

Terracini Loci, Linear Projections, and the Maximal Defect

We continue the study of Terracini loci formed by x points of a variety embedded in a projective space. Our main results are a refined study of Terracini loci arising from linear projections, the description of the maximal x with a non-empty Terracini locus for Hirzebruch surfaces, and the maximal “weight”, “corank”, or “defect” in several cases. For low x, we even show which defects can occur.

Query reformulation system based on WordNet and word vectors clusters

With tremendous evolution in the internet world, the internet has become a household thing. Internet users use search engines or personal assistants to request information from the internet. Search results are greatly dependent on the entered keywords. Casual users may enter a vague query due to lack of knowledge of the domain-specific words. We propose a query reformulation system that determines the context of the query, decides on keywords to be replaced and outputs a better-modified query. We propose strategies for keyword replacements and metrics for query betterment checks. We have found that if we project keywords into the vector space of word projection using word embedding techniques and if the keyword replacement is correct, clusters of a new set of keywords become more cohesive. This assumption forms the basis of our proposed work. To prove the effectiveness of the proposed system, we applied it to the ad-hoc retrieval tasks over two benchmark corpora viz TREC-CDS 2014 and OHSUMED corpus. We indexed Whoosh search engine on these corpora and evaluated based on the given queries provided along with the corpus. Experimental results show that the proposed techniques achieved 9 to 11% improvement in precision and recall scores. Using Google’s popularity index, we also prove that the reformulated queries are not only more accurate but also more popular. The proposed system also applies to Conversational AI chatbots like ChatGPT, where users must rephrase their queries to obtain better results.

EEG-eye movement based subject dependence, cross-subject, and cross-session emotion recognition with multidimensional homogeneous encoding space alignment

The joint learning of multimodal is helpful to extract the general information cross-modality in improving the performance of multimodal emotion recognition. However, focusing on a single common pattern can cause multimodal data to deviate from its original distribution and fail to fully capture the potential representation of the data. Therefore, we propose a multi-dimensional homogenous encoding spatial alignment (MHESA) method, which consists of two parts: multi-modal joint learning and modal knowledge transfer. To obtain a common projection space of EEG-EM features, we use a multimodal joint space encoder to learn the homogeneous joint space of EEG-Eye Movement (EM). To obtain a homogeneous encoding space based on modal knowledge, the knowledge transfer module learns the spatial distribution of EM features while retaining the original EEG features. The output of each module is used to construct a multidimensional homogeneous encoding space. The weight and multi-task loss function of the multi-dimensional homogeneous encoding space are dynamically adjusted by the Multi-task Joint Optimization Strategy (MJOS). By analyzing the effect of multi-task optimization, we found that compared with the subject dependence scene, the cross-subject scene has an advantage in the construction of joint encoding space, and the modal knowledge transfer feature has a higher contribution degree in cross-session. The experimental results show the MHESA method can make the model achieve more stable performance in three emotion recognition scenes.

Spectral metrics on quantum projective spaces

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dąbrowski. In particular, we establish that the Connes metric metrizes the weak-⁎ topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podleś spheres.