Maximum Number Of Intersections Research Articles

Changes to GFAP Immunoreactive Astrocytes in Medial Prefrontal Cortex Following Exposure to Chronic Stress and Antioxidant Supplementation in Rat Model

Introduction: Astrocytes are responsible for many essential functions of neurons in CNS. It has been recognised that chronic stress affects the morphology of astrocyte. Natural antioxidant such as honey has been used as one of the therapeutic strategies to lessen the damaging effect of chronic stress on our body. Therefore, the aim of the study is to explore the effect of natural antioxidant, Tualang honey (TH) on the morphology of astrocytes following chronic stress exposure. Methods: Thirty-two male rats were randomly divided into the 4 groups: (i) control, (ii) stress, (iii) honey, (iv) stress plus honey groups.TH was administered via oral gavage at dose of 1.0 g/kg body weight pre and post experiment. Chronic stress was exposed to animals in group (ii) and (iv) for consecutive 21 days. Anti GFAP immunohistochemistry method was employed to label astrocytes in the medial prefrontal cortex. The number of GFAP+ astrocytes and several parameters related to astrocyte processes were measured. Results: The present study showed that chronic stress reduced the GFAP immunoreactive astrocyte number and percentage of GFAP immunoreactive material. Chronic stress also caused a reduction in astrocyte process ramification as indicated by a reduction in astrocyte total number of processes, average length of processes and maximum number of intersections. However, antioxidant treatment using TH could not reverse these stress-induced changes to the astrocytes. Conclusion: These results demonstrate that chronic stress decreases the number of GFAP immunoreactive astrocyte and cause shrinking of astrocyte processes in stress-sensitive brain region, but these changes cannot be reversed by antioxidant treatment.

Estimate of The Bézout Number For Linear Piecewise Algebraic Curves Over Arbitrary Triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper gives an upper bound of the Bézout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, over arbitrary triangulations.

The Bezout number for linear piecewise algebraic curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper discusses the Bezout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, on regular triangulations. We give an upper bound of the Bezout number for linear piecewise algebraic curves ( B N ( 1 , 0 ; 1 , 0 ; Δ ) ) on the triangulation with an odd interior vertex. For the triangulations which satisfy a vertex coloring condition, we compute the exact value of the Bezout number B N ( 1 , 0 ; 1 , 0 ; Δ ) .

The Complexity of the Union of $(\alpha,\beta)$-Covered Objects

An $(\alpha,\beta)$-covered object is a simply connected planar region $c$ with the property that for each point $p\in\partial c$ there exists a triangle contained in $c$ and having $p$ as a vertex, such that all its angles are at least $\alpha>0$ and all its edges are at least $\beta\cdot{\rm \diam}(c)$-long. This notion extends that of fat convex objects. We show that the combinatorial of the union of $n$ $(\alpha,\beta)$-covered objects of constant description complexity is $O(\lambda_{s+2}(n) \log^2n\log\log n)$, where $s$ is the maximum number of intersections between the boundaries of any pair of given objects, and $\lambda_s(n)$ denotes the maximum length of an $(n,s)$-Davenport--Schinzel sequence. Our result extends and improves previous results concerning convex $\alpha$-fat objects.

Buffon's Needle on Caustics and Torus Quantization

The probability of n + 1 intersections of a long needle in the Buffon problem is the eikonal of a Hankel function which is the principal term in the uniform asymptotic expansion in powers of the small distance between the parallel lines. Evaluating this probability using the torus quantization conditions shows that in the physically meaningful region, where a closed convergence of rays covers the caustic circle, the probability is greater than unity. In addition, the method of steepest descent shows that the caustic and reflection indices are more general than the ones given by torus quantization. The distance from the light source to the center of caustic circle corresponds to the length of the needle, and n times the distance between the parallel lines is the radius of the caustic. Unlike diffraction problems, the solution cannot be extended to the shadow zone since the angles become imaginary. In the continuum limit where the distance between the parallels tends to zero, the number of intersections is governed by an arc sine law in which maximum number of intersections, or the maximum chord length in a circle of a given radius, is most probable.

THE BEZOUT NUMBER FOR PIECEWISE ALGEBRAIC CURVES

The computation of the Bezout number, the maximum number of intersection points between two piecewise algebraic curves whose common points are finite, is considered. A piecewise algebraic curve is a curve determined by a bivariate spline function. It is found that the maximum number of intersections depends not only on the degrees and the differentiability of the spline functions, but also on the structure of the partition on which the spline functions are defined.

3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting—preprocess a set ϰ of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure—preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching— preprocess a set C of n convex fat polygons, so that the k objects intersecting a “not-too-large” query polygon can be reported efficiently, and (iv) bounded-size segment shooting—preprocess a set C as in (iii), so that the first object (if exists) hit by a “not-too-long” oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(λs(n)log3n), where s is the maximum number of intersections between the boundaries of the (xy-projections) of any pair of objects, and λs(n) is the maximum length of (n, s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(λs(n)log2n). The query time in the first problem is O(log4n), the query time in the second and third problems is O(log3n + klog2n), and the query time in the fourth problem is O(log3n).We also present a simple algorithm for computing a depth order for a set ϰ as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for ϰ, if exists, is a linear order of ϰ, such that, if K1, K2 ∈ ϰ and K1 lies vertically above K2, then K1 precedes K2.) Unlike the algorithm of Agarwal et al. (1995) that might output a false order when a depth order does not exist, the new algorithm is able to determine whether such an order exists, and it is often more efficient in practical situations than the former algorithm.

Computing depth orders for fat objects and related problems

Let K be a set of n non-intersecting objects in 3-space. A depth order of K , if it exists, is a linear order < of the objects in K such that if K, L ϵ K and K lies vertically below L then K < L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If K is a set of n triangles whose xy-projections are all ‘fat’, then a depth order for K can be computed in time O( n log 5 n). (ii) If K is a set of n convex and simply-shaped objects whose xy-projections are all ‘fat’ and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(nλ s 1 2 (n) log 4n) , where s is the maximum number of intersections between the boundaries of the xy-projections of any pair of objects in K , and λ s ( n) is the maximum length of ( n, s) Davenport-Schinzel sequences.

The maximum number of intersections between two plane rectangular paths

We show that the maximum number of intersections between two plane rectangular paths with lengthsm andn, 2≤m≤n, is 4n+2, ifm=4 andn≡1(mod 3); and it ismn+1 otherwise.

Almost tight upper bounds for lower envelopes in higher dimensions

We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d?3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(nd?1+?), for any ?>0; the constant of proportionality depends on ?, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2?q(n)) for some constantq depending on the shape and degree of the surfaces (where ?q(n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n2+?), and give several applications of the new bounds.