Pairwise Intersections Research Articles

Intersection-union dual-stream cross-attention Lova-SwinUnet for skin cancer hair segmentation and image repair

Skin cancer images have hair occlusion problems, which greatly affects the accuracy of diagnosis and classification. Current dermoscopic hair removal methods use segmentation networks to locate hairs, and then uses repair networks to perform image repair. However, it is difficult to segment hair and capture the overall structure between hairs because of the hair being thin, unclear, and similar in color to the entire image. When conducting image restoration tasks, the only available images are those obstructed by hair, and there is no corresponding ground truth (supervised data) of the same scene without hair obstruction. In addition, the texture information and structural information used in existing repair methods are often insufficient, which leads to poor results in skin cancer image repair. To address these challenges, we propose the intersection-union dual-stream cross-attention Lova-SwinUnet (IUDC-LS). Firstly, we propose the Lova-SwinUnet module, which embeds Lovasz loss function into Swin-Unet, enabling the network to better capture features of various scales, thus obtaining better hair mask segmentation results. Secondly, we design the intersection-union (IU) module, which takes the mask results obtained in the previous step for pairwise intersection or union, and then overlays the results on the skin cancer image without hair to generate the labeled training data. This turns the unsupervised image repair task into the supervised one. Finally, we propose the dual-stream cross-attention (DC) module, which makes texture information and structure information interact with each other, and then uses cross-attention to make the network pay attention to the more important texture information and structure information in the fusion process of texture information and structure information, so as to improve the effect of image repair. The experimental results show that the PSNR index and SSIM index of the proposed method are increased by 5.4875 and 0.0401 compared with the other common methods. Experimental results unequivocally demonstrate the effectiveness of our approach, which serves as a potent tool for skin cancer detection, significantly surpassing the performance of comparable methods.

Positive links with arrangements of pseudocircles as shadows

An arrangement of pseudocircles A is a collection of Jordan curves in the plane that pairwise intersect (transversally) at exactly two points. How many non-equivalent links have A as their shadow? Motivated by this question, we study the number of non-equivalent positive oriented links that have an arrangement of pseudocircles as their shadow. We give sharp estimates on this number when A is one of the three unavoidable arrangements of pseudocircles.

Enhanced Tomato Pest Detection via Leaf Imagery with a New Loss Function

Pests have caused significant losses to agriculture, greatly increasing the detection of pests in the planting process and the cost of pest management in the early stages. At this time, advances in computer vision and deep learning for the detection of pests appearing in the crop open the door to the application of target detection algorithms that can greatly improve the efficiency of tomato pest detection and play an important technical role in the realization of the intelligent planting of tomatoes. However, in the natural environment, tomato leaf pests are small in size, large in similarity, and large in environmental variability, and this type of situation can lead to greater detection difficulty. Aiming at the above problems, a network target detection model based on deep learning, YOLONDD, is proposed in this paper. Designing a new loss function, NMIoU (Normalized Wasserstein Distance with Mean Pairwise Distance Intersection over Union), which improves the ability of anomaly processing, improves the model’s ability to detect and identify objects of different scales, and improves the robustness to scale changes; Adding a Dynamic head (DyHead) with an attention mechanism will improve the detection ability of targets at different scales, reduce the number of computations and parameters, improve the accuracy of target detection, enhance the overall performance of the model, and accelerate the training process. Adding decoupled head to Head can effectively reduce the number of parameters and computational complexity and enhance the model’s generalization ability and robustness. The experimental results show that the average accuracy of YOLONDD can reach 90.1%, which is 3.33% higher than the original YOLOv5 algorithm and is better than SSD, Faster R-CNN, YOLOv7, YOLOv8, RetinaNet, and other target detection networks, and it can be more efficiently and accurately utilized in tomato leaf pest detection.

Development and Validation of a Deep-Learning-Based Algorithm for Detecting and Classifying Metallic Implants in Abdominal and Spinal CT Topograms.

To develop and validate a deep-learning-based algorithm (DLA) that is designed to segment and classify metallic objects in topograms of abdominal and spinal CT. DLA training for implant segmentation and classification was based on a U-net-like architecture with 263 annotated hip implant topograms and 2127 annotated spine implant topograms. The trained DLA was validated with internal and external datasets. Two radiologists independently reviewed the external dataset consisting of 2178 abdomen anteroposterior (AP) topograms and 515 spine AP and lateral topograms, all collected in a consecutive manner. Sensitivity and specificity were calculated per pixel row and per patient. Pairwise intersection over union (IoU) was also calculated between the DLA and the two radiologists. The performance parameters of the DLA were consistently >95% in internal validation per pixel row and per patient. DLA can save 27.4% of reconstruction time on average in patients with metallic implants compared to the existing iMAR. The sensitivity and specificity of the DLA during external validation were greater than 90% for the detection of spine implants on three different topograms and for the detection of hip implants on abdominal AP and spinal AP topograms. The IoU was greater than 0.9 between the DLA and the radiologists. However, the DLA training could not be performed for hip implants on spine lateral topograms. A prototype DLA to detect metallic implants of the spine and hip on abdominal and spinal CT topograms improves the scan workflow with good performance for both spine and hip implants.

STRUCTURES OF THE SETS OF BOREL AND PICARD EXCEPTIONAL VECTORS OF ANALYTIC CURVES

Previously, we have described the structure of the sets of Picard and Borel exceptional vectors for a transcendental p-dimensional integer curve with linearly independent components without common zeros. It was proved that the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces of dimension not higher than p-1 of a p-dimensional complex Euclidean space. In addition, the sum of the dimensions of all these subspaces does not exceed p and any pairwise intersection of these subspaces contains only the zero vector. The set of Picard exceptional vectors has the same structure. For an integer curve of non-integer or zero order, the set of Borel exceptional vectors together with the zero vector is a single subspace of dimension no higher than p-1, which is not the case for the set of Picard exceptional vectors of an integer curve of non-integer order. The sufficiency of these results is confirmed to some extent by the corresponding examples. In this paper, we show that the above results on the structure of the set of Borel and Picard exceptional vectors for integer curves can be transferred to the case of analytic curves in the circle. We have also constructed an analytic curve in the circle, which confirms, to some extent, the sufficiency of the results. However, it was not possible to find out whether the corresponding result is transferred to the case of analytic curves of non-integer or zero order.

Bounds on piercing and line-piercing numbers in families of convex sets in the plane

A family of sets has the (p,q) property if among any p members of it some q intersect. It is shown that if a finite family of compact convex sets in R2 has the (p+1,2) property then it is pierced by ⌊p2⌋+1 lines. A colorful version of this result is proved as well. As a corollary, the following is proved: Let F be a finite family of compact convex sets in the plane with no isolated sets, and let F′ be the family of its pairwise intersections. If F has the (p+1,2) property and F′ has the (r+1,2) property, then F is pierced by (⌊r2⌋2+⌊r2⌋)p points when r⩾2, and by p points otherwise. The proofs use the topological KKM theorem.

Coloring unions of nearly disjoint hypergraph cliques

AbstractWe consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are generally known to exist only when the number of cliques is exponential in the clique size (Glock, Kühn, Lo, and Osthus, Mem. Amer. Math. Soc. 284 (2023) v+131 pp; Keevash, Preprint; Rödl, Eur. J. Combin. 6 (1985) 69–78). We construct near designs where the number of cliques is polynomial in the clique size, and show that they have large chromatic number. The case when the cliques have pairwise intersections of size at most one seems particularly challenging. Here, we give lower bounds by analyzing a random greedy hypergraph process. We also consider the related question of determining the maximum number of caps in a finite projective/affine plane and obtain nontrivial upper and lower bounds.

Minimum spanning tree cycle intersection problem on outerplanar graphs

We study the minimum spanning tree cycle intersection (MSTCI) problem on outerplanar graphs in this paper. Consider a connected simple graph G=(V,E) and any spanning tree T=(V,ET) of G, it is well-known that each non-tree edge e∈E∖ET induces a fundamental cycle in T∪{e}. We say that two distinct fundamental cycles intersect if they share at least one edge. The MSTCI problem seeks a spanning tree such that the number of pairwise cycle intersections is minimized. We provide a linear-time algorithm to solve the problem on outerplanar graphs by exploiting the cycle-cut duality on plane graphs. Besides, we show that the minimum cycle intersection number is bounded by O(n) for outerplanar graphs with n vertices.

Cooperative concurrent targeting for planar arrays of point sources

Motivated by practical applications in satellite formations and directional antenna arrays, the problem of targeting a planar array of point sources at one common object of interest is proposed and then solved using novel distributed coordination strategies. These point sources are arbitrarily located on the x–y plane and the boundary point sources, whose either x or y coordinate is extremal, already orient to the non-coplanar target point. The only global information shared among the remaining point sources is the positive directions of the global coordinate axes x and y, and hence, they have to rely on sensing the changes of the orientation angles of their nearest neighbors to adjust their own orientation. We prove that under our control law, the orientation lines will asymptotically intersect at the same point of concurrency as the boundary point sources. The main idea behind the designed control law is the intuitive argument from Euclidean geometry that reducing the differences between the distances to the x–y plane of the pairwise intersection points of the orientation lines leads to realizing concurrent targeting. We further show the boundedness and exponential convergence speed of the orientation angles.

Impact of antibiotic-coated sutures on surgical site infections: a second-order meta-analysis.

Surgical site infections (SSIs) pose a global challenge, impacting patients and healthcare expenditures. This second-order meta-analysis endeavors to assess the efficacy of antibiotic sutures in averting SSIs by amalgamating data from various meta-studies. This research adhered to the PRISMA 2020 guidelines. The quality and comprehensiveness of the encompassed meta-analyses were assessed through the QUOROM checklist and AMSTAR techniques. The primary study overlap was evaluated via measures such as pairwise intersection heat maps, corrected covered area, and the citation matrix of evidence. The statistical power at the study-level was determined utilizing the meta-meta package. Data synthesis employed random and fixed effects models at a 95% CI. A meta-regression analysis was conducted to explore potential correlations between the CDC classification of SSIs, trial types, and the observed effect sizes in the studies. This investigation revealed a significant reduction in SSI rates due to antimicrobial-coated sutures, evidenced by a relative risk (RR) of 0.68 (95% CI: 0.59-0.76), with a prediction interval of 0.38-1.19. The analysis encompassed 18 studies with 22 meta-analyses, demonstrating a median QUOROM score of 13.6 out of 18 and an AMSTAR score of 9.1 out of 11. The presence of moderate heterogeneity was noted ( Q =106.611, I2 =54.038%), with nonrandomized controlled trials exhibiting an RR of 0.56 (95% CI: 0.39-0.80), and RCTs displaying an RR of 0.71 (95% CI: 0.63-0.81). Subgroup analysis unveiled variable RR reductions for specific surgical procedures. Antimicrobial-coated sutures offer a promising approach to mitigating SSIs risk. However, their efficacy is optimally realized when employed in conjunction with other robust practices.

Almost affinely disjoint subspaces and covering Grassmannian codes

A family of k-dimensional subspaces of Fqn with pairwise trivial intersection is called L-almost affinely disjoint (AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. Liu et al. introduced the notion of AAD family, and investigated the lower and upper bounds of maximal such sets and conjectured that for any k, n and a large enough L=L(n,k), there exists an [n,k,L]q-AAD family with size qn−2k in [Finite Fields Appl. 75 (2021), 101879]. Etzion and Zhang introduced covering Grassmannian codes (CGCs) for generalized combination networks [IEEE Trans. Inf. Theory, 65 (2019), 4131–4142.]. An α-(n,k,δ)qc covering Grassmannian code C is a set of k-dimensional subspaces of Fqn, such that every set of α codewords of C spans a subspace of dimension at least k+δ. In this paper, we give a construction of AAD families and CGCs based on maximum rank metric codes and caps in projective geometries, and a recursive construction based on maximum rank metric codes. As a consequence, we prove that Liu et al.'s conjecture is still true for k≥3 and n=4k, and improve lower bounds on maximum sizes of AAD families and lower bounds on maximum sizes of 3-(n,k,2k)qc covering Grassmannian codes.

The crossing number of the generalized Petersen graph P(3k,k) in the projective plane

ABSTRACT The crossing number of a graph G in a surface Σ, denoted by c r Σ ( G ) , is the minimum number of pairwise intersections of edges in a drawing of G in Σ. Let k be an integer satisfying k ≥ 3 , the generalized Petersen graph P ( 3 k , k ) is the graph with vertex set V ( P ( 3 k , k ) ) = { u i , v i ∣ i = 1 , 2 , … , 3 k } and edge set E ( P ( 3 k , k ) ) = { u i u i + 1 , u i v i , v i v k + i ∣ i = 1 , 2 , … , 3 k } , the subscripts are read modulo 3 k . In this paper we investigate the crossing number of P ( 3 k , k ) in the projective plane. We determine the exact value of c r N 1 ( P ( 3 k , k ) ) is k–2 when 3 ≤ k ≤ 7 , moreover, for k ≥ 8 , we get that k − 2 ≤ c r N 1 ( P ( 3 k , k ) ) ≤ k − 1.

Parameter Identification of Bivariate Fractal Interpolation Surfaces by Using Convex Hulls

The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented.

Octopuses in the Boolean cube: Families with pairwise small intersections, part I

Let F1,…,Fℓ be families of subsets of {1,…,n}. Suppose that for distinct k,k′ and arbitrary F1∈Fk,F2∈Fk′ we have |F1∩F2|⩽m. What is the maximal value of |F1|…|Fℓ|? In this work we find the asymptotic of this product as n tends to infinity for constant ℓ and m.This question is related to a conjecture of Bohn et al. that arose in the 2-level polytope theory and asked for the largest product of the number of facets and vertices in a two-level polytope. This conjecture was recently resolved by Weltge and the first author.The main result can be rephrased in terms of colorings. We give an asymptotic answer to the following question. Given an edge coloring of a complete m-uniform hypergraph into ℓ colors, what is the maximum of ∏Mi, where Mi is the number of monochromatic cliques in i-th color?

Coloring directed hypergraphs

Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertex set of each hyperedge is partitioned into two parts, a tail and a head. We present a conjecture of D. Pálvölgyi and the author, which states that directed hypergraphs with a certain restriction on their pairwise intersections can be properly colored with two colors. Besides other contributions, our main result is a proof of this conjecture for 3-uniform directed hypergraphs. This result can be phrased equivalently such that if a 3-uniform directed hypergraph avoids a certain directed hypergraph with two hyperedges, then it admits a proper 2-coloring. Previously, only extremal problems regarding the maximum number of edges of directed hypergraphs that avoid a certain hyperedge were studied.

Large 1-systems of curves in nonorientable surfaces

A longstanding avenue of research in orientable surface topology is to create and enumerate collections of curves in surfaces with certain intersection properties. We look for similar collections of curves in non-orientable surfaces. A surface is non-orientable if and only if it contains a Mobius band. We generalize a construction of Malestein-Rivin-Theran to non-orientable surfaces to exhibit a lower bound for the maximum number of curves that pairwise intersect 0 or 1 times in a generic non-orientable surface.

Plücker-Type Inequalities for Mixed Areas and Intersection Numbers of Curve Arrangements

Abstract Any collection of $n$ compact convex planar sets $K_1,\dots , K_n$ defines a vector of ${n\choose 2}$ mixed areas ${\operatorname {V}}(K_i,K_j)$ for $1\leq i<j\leq n$. We show that for $n\geq 4$ these numbers satisfy certain Plücker-type inequalities. Moreover, we prove that for $n=4$, these inequalities completely describe the space of all mixed area vectors $({\operatorname {V}}(K_i,K_j)\,:\,1\leq i<j\leq 4)$. For arbitrary $n\geq 4$, we show that this space has a semialgebraic closure of full dimension. As an application, we show that the pairwise intersection numbers of any collection of $n$ tropical curves satisfy the Plücker-type inequalities. Moreover, in the case of four tropical curves, any homogeneous polynomial relation between their six intersection numbers follows from the corresponding Plücker-type inequalities.

Joints tightened

Given a set of lines in $\Bbb{R}^3$, a {\it joint} is a point passed through by three lines not all lying in some plane. The joints problem asks for the maximum number of joints formed by $L$ lines. Using the polynomial method, Guth and Katz proved that the answer is $O(L^{3/2})$, which is best possible up to a constant factor. We prove a new upper bound of $(\sqrt{2}/3)L^{3/2}$, which matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic planes, and use their pairwise intersections to form $\smash{\big(\matrix{k\cr 2}\big)}$ lines and their triple-wise intersections to form $\smash{\big(\matrix{k\cr 3}\big)}$ joints. Guth conjectured that this construction is optimal. Our result generalizes to arbitrary dimensions and fields.

Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.

Four Major Service Industries and Economic Growth in Nigeria: An Empirical Analysis

This study has investigated the influence of four major service industries in Nigeria on the economic growth. Data for the study composed of value gross domestic product (GDP), value added of trade, value added of information and communication, value added of financial and insurance, value added of real estate, and indicators of financial deepening. Data were sourced from the Central Bank of Nigeria Statistical Bulletin. All the series were subjected to Unit root test and cointegration test. The results of the Augmented Dickey Fuller showed that all the series were stationary at first difference and there was existence of long run relationship among the series as confirmed by the results of the Johansen’ cointegration test. The estimations were carried out using Ordinary Least Squares (OLS) and Dynamic Ordinary Least Squares (Dynamic OLS). The findings of the study showed that trade and real estates were the movers of economic growth in the period under study. Also, the findings showed that pairwise intersections of trade and financial and insurance; financial and insurance, and real estate; and trade and real estate contributed significantly to economic growth during the period under study. The authors recommended that there should be concrete policies on diversification as to ensure that the impact of trade and real estate is spilled- over to other sectors.