Intersection Problem Research Articles

Approximation by juntas in the symmetric group, and forbidden intersection problems

A family of permutations F⊂Sn is said to be t-intersecting if any two permutations in F agree on at least t points. It is said to be (t−1)-intersection-free if no two permutations in F agree on exactly t−1 points. If S,T⊂{1,2,…,n} with |S|=|T|, and π:S→T is a bijection, then the π-star in Sn is the family of all permutations that agree with π on S. An s-star is a π-star such that π is a bijection between sets of size s. Friedgut and Pilpel, and independently the first author, showed that if F⊂Sn is t-intersecting, and n is sufficiently large depending on t, then |F|≤(n−t)!; this proved a conjecture of Deza and Frankl from 1977. Here, we prove a considerable strengthening of the Deza–Frankl conjecture, namely, that if n is sufficiently large depending on t, and F⊂Sn is (t−1)-intersection-free, then |F|≤(n−t)!, with equality iff F is a t-star. The main ingredient of our proof is a “junta approximation” result, namely, that any (t−1)-intersection-free family of permutations is essentially contained in a t-intersecting junta (a “junta” being a union of boundedly many O(1)-stars). The proof of our junta approximation result relies, in turn, on (i) a weak regularity lemma for families of permutations (which outputs a junta whose stars are intersected by F in a weakly pseudorandom way), (ii) a combinatorial argument that “bootstraps” the weak notion of pseudorandomness into a stronger one, and finally (iii) a spectral argument for highly pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature and one being algebraic. The connection we demonstrate between these notions of pseudorandomness may find further applications.

Precomputed fast rejection ray-triangle intersection

We propose a ray-triangle intersection algorithm with fast-rejection strategies. We intersect the ray with the triangle plane, then transform the intersection problem into 2D by applying a transformation matrix to the ray-plane intersection point. For 2D transformation, we study two different approaches. The first approach uses a transformation matrix which transforms the triangle into a unit triangle. Then, simple 2D tests are performed. The second approach transforms the triangle into a 2D triangle while preserving similarity. This allows us to prune (i.e., to clip away) areas surrounding the triangle, determining whether the transformed intersection point lies within the triangle. We discuss several optimizations for this pruning approach. We implemented both approaches into the CPU-based ray-tracing framework PBRT, version 3, and we performed a time-based comparison against PBRT’s default intersection algorithm and Baldwin and Weber’s algorithm. The results show that our algorithms are faster than the default algorithm. They are comparable to or slightly slower than Baldwin and Weber’s algorithm, however, the pruning approach produces watertight results and may be further optimized. Moreover, additional CPU/GPU experiments outside of PBRT document promising speedup over the standard Möller–Trumbore algorithm in areas like ray-casting or collision detection.

A New Fast Intersection Algorithm for Sorted Lists on GPU

Set intersection algorithms between sorted lists are important in triangles counting, community detection in graph analysis and in search engines where the intersection is computed between queries and inverted indexes. Many researches use GPU techniques for solving this intersection problem. The majority of these techniques focus on improving the level of parallelism by reducing redundant comparisons and distributing the workload among GPU threads. In this paper, we propose the GPU Test with Jumps (GTWJ) algorithm to compute the intersection between sorted lists using a new data structure. The idea of GTWJ is to group the data, of each sorted list, into a set of sequences. A sequence is identified by a key and is handled by a thread. Intersection is computed between sequences with the same key. This key allows skipping data packets in parallel if the keys do not match. A counter is used to avoid useless tests between cells of sequences with different lengths. Experiments on the data used in this filed show that GTWJ is better in terms of execution time and number of tests.

Approximation by lexicographically maximal solutions in matching and matroid intersection problems

We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly 2. Furthermore, we prove that if the ratio is less than 2, say α, then a lexicographically maximal solution achieves (α/2)-approximation, and this bound is tight.

Analytic Solution to the Piecewise Linear Interface Construction Problem and Its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes

The plane–cube intersection problem has been discussed in the literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding computing time, so a faster analytic solution to the plane–cube intersection would greatly reduce the computing time for such simulations. We derive an analytic solution for all intersection cases and compare it to the previous solution from Scardovelli and Zaleski (Scardovelli, R.; Zaleski, S. Analytical relations connecting linear interfaces and volume fractions in rectangular grids. J. Comput. Phys.2000, 164, 228–237), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding computing time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton–Raphson is fastest with compiler optimization enabled, while analytic solutions perform better than iterative solutions without. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC—curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.

A Control Method with Reinforcement Learning for Urban Un-Signalized Intersection in Hybrid Traffic Environment.

To control autonomous vehicles (AVs) in urban unsignalized intersections is a challenging problem, especially in a hybrid traffic environment where self-driving vehicles coexist with human driving vehicles. In this study, a coordinated control method with proximal policy optimization (PPO) in Vehicle-Road-Cloud Integration System (VRCIS) is proposed, where this control problem is formulated as a reinforcement learning (RL) problem. In this system, vehicles and everything (V2X) was used to keep communication between vehicles, and vehicle wireless technology can detect vehicles that use vehicles and infrastructure (V2I) wireless communication, thereby achieving a cost-efficient method. Then, the connected and autonomous vehicle (CAV) defined in the VRCIS learned a policy to adapt to human driving vehicles (HDVs) across the intersection safely by reinforcement learning (RL). We have developed a valid, scalable RL framework, which can communicate topologies that may be dynamic traffic. Then, state, action and reward of RL are designed according to urban unsignalized intersection problem. Finally, how to deploy within the RL framework was described, and several experiments with this framework were undertaken to verify the effectiveness of the proposed method.

ХУДОЖНИК-НАИВИСТ ИВАН СЕЛИВАНОВ В ПОЭЗИИ И ПРОЗЕ ЛЮБОВИ НИКОНОВОЙ

The article examines the dialogue of words and painting in the works of Kuzbass authors – writer L.A. Nikonova and naive artist I. Selivanov. Theoretical issues of perception of painting, problems of intersection of pictorial and verbal creativity, etc. are revealed in connection with the analysis of metatext, with an emphasis on L.A. Nikonova’s understanding of the image of I.E. Selivanov as an artist close to her in spirit.

The orbit intersection problem in positive characteristic

We study the orbit intersection problem for a linear space and an algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi _1, \Phi _{2}: K^d \rightarrow K^{d}$ be affine maps, $\Phi _i(

A Safety Evaluation Model of Intersections under Mixed Traffic Conditions Using Traffic Conflicts and Cloud Model

Safety evaluation of traffic conflict is a very important and challenging issue in evaluating intersection safety under incomplete traffic accident data conditions and is also one of the main safety surrogate measures of analyzing accident data recently. It helps to analyze and solve intersection problems comprehensively and deeply. From there, it helps to improve traffic safety as well as reduce the risk of traffic accidents at intersections. Various evaluation methods based on traffic conflict have been proposed to make conflict safety levels at intersections more consistent and objective. However, a major concern is that many existing measurements are still subjective and are not easy to obtain uniformly. This study aimed to develop a model for safety evaluation at intersections in a comprehensive way that may be expected to directly link to the severity of the accident from different evaluation indicators. First, the three factors, including time to collision (TTC), conflicting speed (CS), and deceleration rate (DR) to avoid a crash, are introduced into safety evaluation of conflicts as the indicators. And then, as regards the fuzziness and randomness of the evaluation indicators, the qualitative concept has to be converted into a quantitative one utilizing cloud model, which implements the natural transformation between the qualitative concept of the safety level of traffic conflict and the membership degree of the evaluation indicators corresponding to the different safety levels. Finally, an indicator weight model is built based on the information entropy and the AHP method to determine the safety level. We illustrate the practical implementation of the proposed method using actual data of a typical signalized intersection from Hanoi City of Vietnam. The results indicate that traffic conflict analyzed by the proposed method was appropriate with actual state of the intersection, and the proposed method is simple, effective, and feasible, so it has a certain application value.

Remarkable Characteristics of a Novel Path Interval Determination in Filleted End Milling

This study describes remarkable characteristics of a novel path interval determination in filleted end milling with a tool inclination. CNC milling machine is one of the core technologies in practical manufacturing. Computer-aided technologies have contributed to the technological advancement. Tool path generation in computer-aided manufacturing is really important for CNC milling process. Although there are a lot of parameters treated in tool path generation, path interval is one of the influential factors in considering the balance between manufacturing efficiency and machined surface feature. A path interval determination in filleted end milling commonly entails the intersection problems with mathematical complexities in essence. The related studies have been reported so far, while there has scarcely been a procedure to cope effectively with the complexities. Hence, this study focused on a novel path interval determination proposed in our previous study. After the analytical discussions were made with the computational and experimental results, it was acquired from the explicit evidences that the novel procedure possessed remarkable characteristics contributable for a path interval determination in multi-axis filleted end milling.

Optimum Hydraulic Design of Inverted Syphon Structure in Sewer System with a Case Study of Al-Hashmea Sewer Project

An inverted syphon is a hydraulic structure utilized to convey open channel flow under obstructions. This study focuses on identifying the safest and optimum design of a sewage inverted syphon to avoid intersection problems with immovable infrastructures. Hydraulic design requirements, inlet and outlet distribution rooms features and protection of expected loads on syphons’ pipes were taken under consideration in formulation the optimization model. Further, different pipes’ material, many levels of the design variables and parameters were incorporated in the model to highlight the effect of each variable on the optimum design. The objective function was the minimum construction cost, and the design requirements were presented as constraints in the model. Results demonstrated that the optimum design for discharges ranging from 0.38 m3/sec to 0.95 m3/sec was Poly Vinyl Chloride (PVC) syphon, but for other discharges the reinforced concrete type was the optimum. Compared to traditional design, the optimum solutions provide a significant cost reduction. Concrete inverted syphon was the optimum possible solution to solve intersection problem of Al-Hashmea sewer project to pass sewage flow under the railway.

An improvement on Furstenberg’s intersection problem

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

Layout analysis and optimization design for Zhengding Bus Terminal

As an important component of road transportation, the bus terminal has always been the focus of research. Therefore, this article uses a combination of quantitative and qualitative analysis to study Zhengding third-level bus terminal from the perspective of layout analysis. On the basis of actual investigation, the relevant indicators of the bus terminal scale are calculated. At the same time, the analysis of the three functional areas and process flow lines inside the passenger station revealed that the passenger station has the problems of intersection of pedestrians and vehicles and the unreasonable and imperfect layout of the ticket waiting hall. Finally, a specific optimization plan is proposed and optimized for the problems existing in Zhengding bus terminal.

A quantum protocol for private substitution problem

Private substitution (PS) is a problem for replacing one party’s binary string with the other party’s binary string in a privacy preserving manner. It is a new application of secure multi-party computation. In this paper, a protocol for PS using Pauli gates X, Z and SWAP gate is put forward. A quantum secure direct communication protocol is also used to transmit some classic information. Correctness analysis shows that parties can get the correct substitution results. The security of the proposed protocol is also analyzed, which shows that the protocol can resist attacks from outside and participants. The quantum private substitution protocol can be used to solve quantum private set intersection problem and quantum private set union problem.

Geometric Analysis of Ground-Target Coverage from a Satellite by Field-Mapping Method

This paper proposes an analytical calculation method of the ground point visibility from a satellite in a circular orbit for an arbitrary sensor field considering the perturbation based on the geometric analysis of the sensor, satellite, and ground-point target. First, to describe an arbitrary field of view accurately, the concept of the field-element vector is introduced to parameterize the field discretely. Also, based on the geometric projection, the field-element equations related to the satellite orbit, field parameters, and ground point, which can be solved analytically, are derived. Furthermore, using a two-dimensional map composed of the argument of latitude and ascending node, the problem of calculating visible intervals is transformed into the intersection problem of a group of parallel lines and single or double regions generated by the field of view and the target ground point, which can be solved rapidly. The proposed method is verified by simulations considering different types of sensor fields. The simulation results show that the proposed method is rapid, efficient, intuitive, and accurate, which makes it be suitable for different types of fields and have wide application prospects.

Flexibility of Analysis Serving Computational Polynomial Algebra or Arithmetics

Fundamental concepts in algebraic geometry such as algebraic cycles (with respect to intersection problems) or residues (with respect to division or interpolation ones) gain in flexibility once they are interpreted within the frame of currents in complex geometry ; such interpretation may even be transposed to the frame of arithmetics thanks to recent developments in analytic geometry over algebraically closed fields equipped with an ultrametric (instead of archimedean) absolute value. This survey intends to revisit from such currential interpretation (transposed if possible to the arithmetic setting, for example on Berkovich analytic spaces in place of complex spaces) improper intersection theory, together with the notions of local or global multiplicity, as well as multivariate residue theory with its articulations with Cauchy integral representation theory and Lagrange interpolation formula.

On the intersection problem for linear sets in the projective line

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.

Grouting Mechanism in Water-Bearing Fractured Rock Based on Two-Phase Flow

Grouting is always used in mine water plugging, reinforcement, and other disaster prevention projects. The diffusion mechanism of slurry in fractured rock is affected by geological environment and slurry performance, which should be revealed and characterized better. Based on the two-phase flow diffusion theory, a slurry diffusion model considering flowing water condition was established for a blocking area of a fracture zone in one case from China. The feasibility of two-phase flow model in grouting diffusion calculation was analyzed. The diffusion model in dynamic water environment was studied, and the diffusion range varying with time in the grouting area of Zhangji Coal Mine was explored. The optimization method of multi grouting holes was put forward, and the influence of water flowing was discussed. The results show that the slurry diffusion calculated by the two-phase flow model was feasible and consistent with the experimental study. The dynamic water can change the conventional circular diffusion state of slurry, but its pattern was oval and leaf type. There were different penetration distances in directions, and typical grouting voids were made on the side and upstream. When the single-hole grouting was carried out, the predetermined value can be achieved in the height range, but it was only about 15 m on the side because of the water flowing, which cannot meet the requirements. The optimization scheme of grouting was put forward, which adopted multiple grouting holes in the long side, and grouting in different directions and periods to avoid the possible problems of multihole intersection. The rationality and effectiveness of the proposed optimization method were verified through the calculation of water yield and analysis of cement composition from the drilling core in the grouted zone. In the grouting process, the water flowing has double effects, which has a significant role in promoting and scouring along the flow direction, but there is a significant weakness in the side diffusion. It is very important to realize the rational use of the dynamic water through the optimization scheme. This study is an important basic work of grouting mechanism, and it is expected to promote the development of grouting technology and application of two-phase fluid-solid coupling theory.

Characterization and computation of feasible trajectories for an articulated probe with a variable-length end segment

An articulated probe is modeled in the plane as two line segments, ab and bc, joined at b, with ab being very long, and bc of some small length r. We investigate a trajectory planning problem involving the articulated two-segment probe where the length r of bc can be customized. Consider a set P of simple polygonal obstacles with a total of n vertices, a target point t located in the free space such that t cannot see to infinity, and a circle S centered at t enclosing P. The probe initially resides outside S, with ab and bc being collinear, and is restricted to the following sequence of moves: a straight line insertion of abc into S followed by a rotation of bc around b. The goal is to compute a feasible obstacle-avoiding trajectory for the probe so that, after the sequence of moves, c coincides with t.We prove that, for n line segment obstacles, the smallest length r for which there exists a feasible probe trajectory can be found in O(n2+ϵ) time using O(n2+ϵ) space, for any constant ϵ>0. Furthermore, we prove that all values r for which a feasible probe trajectory exists form O(n2) intervals, and can be computed in O(n5/2) time using O(n2+ϵ) space. We also show that, for a given r, the feasible trajectory space of the articulated probe can be characterized by a simple arrangement of complexity O(n2), which can be constructed in O(n2) time. To obtain our solutions, we design efficient data structures for a number of interesting variants of geometric intersection and emptiness query problems.

Minimum Spanning Tree Cycle Intersection problem

Consider a connected graph G and let T be a spanning tree of G. Every edge e∈G−T induces a cycle in T∪{e}. The intersection of two distinct such cycles is the set of edges of T that belong to both cycles. We consider the problem of finding a spanning tree that has the least number of such non-empty intersections. In this article we analyze the particular case of complete graphs, and formulate a conjecture for graphs that have a universal vertex.