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Some Basics about First-Order Equations
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
Functions of one variable are typically defined on intervals of the X–axis. For functions of two variables, we must replace the concept of an interval with that of a “region”. For our purposes, a region (in the XY–plane) refers to the collection of all points enclosed by some curve or set of curves on the plane (with the understanding that this curve or set of curves actually does enclose some collection of points in the plane). If we include the curves with the enclosed points, then we say the region is closed; if the curves are all excluded, then we refer to the region as open. This corresponds to the distinction between a closed interval [a, b] (which does contain the endpoints), and an open interval (a, b) (which does not contain the endpoints).
Basic Data Structures
Published in Charles J. Alpert, Dinesh P. Mehta, Sachin S. Sapatnekar, Handbook of Algorithms for Physical Design Automation, 2008
Even though an interval lies on a line that is a one-dimensional space, it is actually a two-dimensional datum because it has two independent parameters. An interval starting at a and ending at b is represented by [a, b]. It is not possible to have a total order over the set of intervals. The idea of the interval tree is to partition the set of intervals into three groups based on a given point x:intervals to the left of the point L(x), intervals to the right of the point R(x), and intervals overlapping with the point C(x). The subsets L(x) and R(x) of intervals can be recursively represented. The subset C(x) also needs to be organized for the queries. Even though C(x) could include all the intervals in the original set, organizing them is much simpler: they can be ordered both on their left points and on their right points. If the query point q < x, only the left points of C(x) need to be checked in increasing order; if q > x, only the right points of C(x) need to be checked in decreasing order. To balance L(x) and R(x), thus to have a short tree, it is desired to use the median of all the endpoints as x. Figure 4.2 shows an interval tree for a set of intervals, where the intervals in C(x) are organized in two lists according to their left and right points.
Mac Protocols in Cognitive Radio Networks
Published in Mohamed Ibnkahla, Cooperative Cognitive Radio Networks, 2018
PCF and DCF can coexist within one cell. It works by carefully defining the interframe time interval. This has been defined in [15] as follows. After a frame has been sent, a certain amount of dead time is required before any station may send a frame. Four different intervals are defined, each for a specific purpose. The shortest interval is short interframe spacing (SIFS) It is used to allow the parties in a single dialog to go first. For example, this includes letting a receiver send a CTS as a response to an RTS, letting a receiver send an ACK for a fragment or full data frame, and letting the sender of a fragment burst transmit the next fragment without sending an RTS again. There is only one station that is entitled to respond after an SIFS interval. If it fails to make use of the channel and a PCF interframe spacing (PIFS) time elapses, the BS may send a beacon frame or poll frame. This mechanism allows a station sending a data frame or fragment sequence to finish its frame without anyone else getting in the way, but gives the BS a chance to grab the channel when the previous sender is done without having to compete with eager users. If the BS has nothing to send, and a DCF interframe spacing (DIFS) time elapses, any station may attempt to acquire the channel. The usual contention rules apply. The last time interval, extended interframe spacing (EIFS), is used only by a station that needs to report a bad or unknown frame. Giving this event the lowest priority comes from the fact that the receiver does not know what to do with the frame; therefore, it should wait a substantial time to avoid interference with an ongoing dialog between two stations. This is illustrated in Figures 7.3 and 7.4.
Optimal experimental design for the assessment of thermophysical properties in existing building walls
Published in Heat Transfer Engineering, 2023
Suelen Gasparin, Julien Berger, Giampaolo D’Alessandro, Filippo de Monte, Dariusz Ucinski
This article proposes the search for the OED in terms of sensor positioning in a wall façade for a thermal conductivity inverse two-dimensional heat transfer problem. The direct problem is solved using a reduced spectral method. It is verified using an analytical solution, highlighting a satisfactory accuracy and computational cost. Then, the OED is investigated by solving an optimization problem of the D-optimum criterion considering two strategies. The first one is based on an exchange algorithm. For the second strategy, a convex relaxation approach is adopted. The decision elements to place the sensors are considered as real number in the unit interval. A (discrete) probability distribution of the sensor position is obtained. The first strategy is the most efficient from a computational point of view. In a few iterations, an local optimal solution is retrieved. The convex relaxation has a higher computational cost but still very low compared to the exhaustive search. Furthermore, it provides complementary probabilistic information for the experimenter. Future works should focus on multi-layer configurations with the thermal conductivity of each layer to retrieve.
Interval uncertain optimization for interior ballistics based on Chebyshev surrogate model and affine arithmetic
Published in Engineering Optimization, 2021
Fengjie Xu, Guolai Yang, Liqun Wang, Quanzhao Sun
In interval mathematics, there are two interval numbers and , and the basic interval arithmetic rules are as follows: Although the interval arithmetic is simple to calculate, it will lead to the phenomenon of interval expansion. That is, when the range of interval functions is estimated by interval arithmetic, the interval of the calculated results will be overestimated owing to the uncertainty in the independent variables of the function. For example, consider ; where , the interval arithmetic results would be instead of the actual result . Interval expansion is caused by variable correlation.
A novel method of combined interval analysis and homotopy continuation in indoor building reconstruction
Published in Engineering Optimization, 2019
Ali Jamali, Francesc Antón Castro, Darka Mioc
In this section, an interval-valued homotopy model of the measurement of horizontal angles by the magnetometer component of the rangefinder is presented. This model blends interval analysis and homotopy continuation. Interval analysis is a well-known method for computing the bounds of a function, having been given bounds on the variables of that function (Ramon Moore and Cloud 2009). The basic mathematical object in interval analysis is the interval instead of the variable. The operators need to be redefined to operate on intervals instead of real variables. This leads to interval arithmetic. In the same way, most usual mathematical functions are redefined by an interval equivalent. Interval analysis allows one to certify computations on intervals by providing bounds on the results. The uncertainty of each measure can be represented using an interval defined either by a lower bound and a higher bound or by a midpoint value and a radius.