Complete Intersection Research Articles

Improved YOLOv7-based steel surface defect detection algorithm.

In response to the limited detection ability and low model generalization ability of the YOLOv7 algorithm for small targets, this paper proposes a detection algorithm based on the improved YOLOv7 algorithm for steel surface defect detection. First, the Transformer-InceptionDWConvolution (TI) module is designed, which combines the Transformer module and InceptionDWConvolution to increase the network's ability to detect small objects. Second, the spatial pyramid pooling fast cross-stage partial channel (SPPFCSPC) structure is introduced to enhance the network training performance. Third, a global attention mechanism (GAM) attention mechanism is designed to optimize the network structure, weaken the irrelevant information in the defect image, and increase the algorithm's ability to detect small defects. Meanwhile, the Mish function is used as the activation function of the feature extraction network to improve the model's generalization ability and feature extraction ability. Finally, a minimum partial distance intersection over union (MPDIoU) loss function is designed to locate the loss and solve the mismatch problem between the complete intersection over union (CIoU) prediction box and the real box directions. The experimental results show that on the Northeastern University Defect Detection (NEU-DET) dataset, the improved YOLOv7 network model improves the mean Average precision (mAP) performance by 6% when compared to the original algorithm, while on the VOC2012 dataset, the mAP performance improves by 2.6%. These results indicate that the proposed algorithm can effectively improve the small defect detection performance on steel surface defects.

On local Galois deformation rings

Abstract We show that framed deformation rings of mod p representations of the absolute Galois group of a p-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application, we prove density results of loci with prescribed p-adic Hodge theoretic properties.

RISTrack: Robust Infrared Ship Tracking With Modified Appearance Feature Extraction and Matching Strategy

Infrared (IR) ship tracking is becoming increasingly important in various applications. However, it remains a challenging task as the information that can be obtained from infrared images is limited. Aiming at enhancing IR ship tracking accuracy, we propose an innovative approach by presenting feature integration module (FIM) and backup matching module (BMM). FIM takes appearance feature, complete intersection over union (CIoU), and motion direction metrics into account. Regarding appearance feature extraction, an end-to-end characteristic learning strategy with a cross-guided multi-granularity fusion network is proposed to obtain more integral appearance features and enhance re-identification accuracy, which helps to distinguish individual IR ship targets better. Besides, a backup matching strategy is then used to match the unmatched tracks and detections after cascaded matching. Virtual trajectories are generated for the matched tracks to optimize parameters by parameter optimization module (POM). The accumulation of errors caused by the lack of observations in the Kalman filter is reduced. Thus, the position of IR ships can be estimated more accurately, and more robust IR ship tracking can be achieved. In addition, we present a sequential frame IR ship tracking dataset, providing the first public benchmark for testing IR ship tracking performance. Experimental results indicate that the MOTA, MOTP and IDs of the proposed method are 73.441, 80.826, and 32, respectively, outperforming other state-of-the-art methods. This demonstrates the superior robustness of the proposed method, particularly when the IR ships are occluded or the target texture information is lacking. Our dataset is available at https://github.com/echo-sky/SFIST.

Number of triple points on complete intersection Calabi–Yau threefolds

Number of triple points on complete intersection Calabi–Yau threefolds

A new proof of Stanley’s theorem on the strong Lefschetz property

A standard graded artinian monomial complete intersection algebra $A=\Bbbk [x_1,\ldots ,x_n]/(x_1^{a_1},\ldots ,x_n^{a_n})$, with $\Bbbk $ a field of characteristic zero, has the strong Lefschetz property defined by Stanley in 1980. In this paper, we give

Values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs

A weight system is a function on the chord diagrams that satisfies Vassiliev's $4$-term relation. Using the Lie algebra $\mathfrak{sl}_2$ we can construct the simplest nontrivial weight system. The resulting $\mathfrak{sl}_2$ weight system takes values in the space of polynomials of one variable and is completely determined by the Chmutov-Varchenko recurrence relations. Although the definition of the $\mathfrak{sl}_2$ weight system is rather simple, calculations of its values are laborious, and therefore concrete values are only known for a small number of chord diagrams. As concerns the explicit form of values at chord diagrams with complete intersection graphs, Lando stated a conjecture, which initially could only be proved for the coefficients at linear terms of the values of the weight system. We prove this conjecture in full using the Chmutov-Varchenko recurrence relations and the linear operators we introduce for adding a chord to a share, which is the subset of chords of the diagram with endpoints on two selected arcs. Also, relying on the generating function of the values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs, we prove that the quotient space of shares modulo the recurrence relations is isomorphic to the space of polynomials in two variables. Bibliography: 10 titles.

Syzygies in Hilbert schemes of complete intersections

Let e1,…,ec be positive integers and let Y⊆Pn be the monomial complete intersection defined by the vanishing of x1e1,…,xcec. In this paper, we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points Hilbd(Y), and discuss applications to the Hilbert scheme of points Hilbd(X) of arbitrary complete intersections X⊆Pn.

On holomorphic foliations admitting invariant CR manifolds

We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.

Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections

We compute the expectation of the number of linear spaces on a complete intersection in $p$-adic projective space. Here random means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a complete intersection tends to $1$. In the case of the number of lines on a cubic in three-space and on the intersection of two quadrics in four-space, we give an explicit formula for this expectation.

The Kähler different of a 0-dimensional scheme

Given a 0-dimensional scheme [Formula: see text] in the projective [Formula: see text]-space [Formula: see text] over a field [Formula: see text], we are interested in studying the Kähler different of [Formula: see text] and its applications. Using the Kähler different, we characterize the generic position and Cayley–Bacharach properties of [Formula: see text] in several certain cases. When [Formula: see text] is in generic position, we prove a generalized version of the Apéry–Gorenstein–Samuel theorem about arithmetically Gorenstein schemes. We also characterize 0-dimensional complete intersections in terms of the Kähler different and the Cayley–Bacharach property.

An Improved YOLOX Model and Domain Transfer Strategy for Nighttime Pedestrian and Vehicle Detection

Aimed at the vehicle/pedestrian visual sensing task under low-light conditions and the problems of small, dense objects and line-of-sight occlusion, a nighttime vehicle/pedestrian detection method was proposed. First, a vehicle/pedestrian detection algorithm was designed based on You Only Look Once X (YOLOX). The model structure was re-parameterized and lightened, and a coordinate-based attention mechanism was introduced into the backbone network to enhance the feature extraction efficiency of vehicle/pedestrian targets. A feature-scale fusion detection branch was added to the feature pyramid, while a loss function was designed, which combines Complete Intersection Over Union (CIoU) for target localization and Varifocal Loss for confidence prediction to improve the feature extraction ability for small, dense, and low-illumination targets. In addition, in order to further improve the detection accuracy of the algorithm under low-light conditions, a training strategy based on data domain transfer was proposed, which fuses the larger-scale daylight dataset with the smaller-scale nighttime dataset after low-illumination degrading. After low-light enhancement, training and testing were performed accordingly. The experimental results show that, compared with the original YOLOX model, the improved algorithm trained by the proposed data domain transfer strategy achieved better performance, and the mean Average Precision (mAP) increased by 5.9% to 82.4%. This research provided effective technical support for autonomous driving safety at night.

A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves

We show that a bound of Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety X ⊆ P r is sharp exactly for complete intersections, provided the variety X is cut out scheme-theoretically by several hypersurfaces in P r . This generalizes a result of Bertram-Ein-Lazarsfeld.

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

Foliations by curves on threefolds

AbstractWe study the conormal sheaves and singular schemes of one‐dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ‐stable whenever the tangent bundle is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on and on a smooth quadric hypersurface . Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.

Configurations of eigenpoints

This note is motivated by Question 16 of http://cubics.wikidot.com (see also [16]): Which configurations of 15 points in P3 arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in Pn is the complete intersection of two suitable smooth determinantal curves on a smooth determinantal surface. Moreover, we prove that the converse result holds if n=3, providing an answer in any degree to the cited question. Finally, we show that any general set of points in P3 can be enlarged to an eigenscheme of a partially symmetric tensor.

Detection of River Floating Garbage Based on Improved YOLOv5

The random dumping of garbage in rivers has led to the continuous deterioration of water quality and affected people’s living environment. The accuracy of detection of garbage floating in rivers is greatly affected by factors such as floating speed, night/daytime natural light, viewing angle and position, etc. This paper proposes a novel detection model, called YOLOv5_CBS, for the detection of garbage objects floating in rivers, based on improvements of the YOLOv5 model. Firstly, a coordinate attention (CA) mechanism is added to the original C3 module (without compressing the number of channels in the bottleneck), forming a new C3-CA-Uncompress Bottleneck (CCUB) module for improving the size of the receptive field and allowing the model to pay more attention to important parts of the processed images. Then, the Path Aggregation Network (PAN) in YOLOv5 is replaced with a Bidirectional Feature Pyramid Network (BiFPN), as proposed by other researchers, to enhance the depth of information mining and improve the feature extraction capability and detection performance of the model. In addition, the Complete Intersection over Union (CIoU) loss function, which was originally used in YOLOv5 for the calculation of location score of the compound loss, is replaced with the SCYLLA-IoU (SIoU) loss function, so as to speed up the model convergence and improve its regression precision. The results, obtained through experiments conducted on two datasets, demonstrate that the proposed YOLOv5_CBS model outperforms the original YOLOv5 model, along with three other state-of-the-art models (Faster R-CNN, YOLOv3, and YOLOv4), when used for river floating garbage objects detection, in terms of the recall, average precision, and F1 score achieved by reaching respective values of 0.885, 90.85%, and 0.8669 on the private dataset, and 0.865, 92.18%, and 0.9006 on the Flow-Img public dataset.

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients.

Hybrid models for homological projective duals and noncommutative resolutions

We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding $f$. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space $B$ or, equivalently, as the derived category of the sheaf of $\mathcal{A}$-modules on $B$, where $\mathcal{A}$ is an $A_{\infty}$ algebra. This latter interpretation corresponds to a noncommutative resolution of $B$. We compute explicitly the algebra $\mathcal{A}$ by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an $A_{\infty}$ algebra with a cyclic group. Then we apply our results to the HPD of $f$ corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in $\mathbb{P}^{n}$.

On the Topology of Random Real Complete Intersections

Given a smooth real projective variety X and m ample line bundles \(L_1,\dots L_m\) on X also defined over \(\mathbf {R}\), we study the topology of the real locus of the complete intersections defined by global sections of \(L_1^{\otimes d}\oplus \cdots \oplus L^{\otimes d}_m\). We prove that the Gaussian measure of the space of sections defining real complete intersections with high total Betti number (for example, maximal complete intersections) is exponentially small, as d grows to infinity. This is deduced by proving that, with very high probability, the real locus of a complete intersection defined by a section of \(L_1^{\otimes d}\oplus \dots \oplus L^{\otimes d}_m\) is isotopic to the real locus of a complete intersection of smaller degree.

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let Z be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let X be a smooth ample divisor in Z, then X is a Mori dream space as well. The second result is: Let Z be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let X be a smooth anti-canonical hypersurface in Z, which is a smooth Calabi–Yau variety, then the unique minimal model of X up to isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X. The third result is: Let P:=Pn×⋯×Pn be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of n+1 hypersurfaces of multidegree (1,…,1) in P with dim⁡X≥3. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.