Problem: For horizontal propagation between transmitting and receiving antennas situated near the ground reasonably far from each other, the distances traveled by tne direct and reflected rays are nearly the same. There is almost no relative phase shift. If the emission is polarized vertically, the two fields (direct and reflected) add and there is maximum of received signal. If the signal is polarized horizontally, the two signals subtract and the received signal is largely cancelled. The vertical plane radiation patterns are shown in the image at right. With vertical polarization there is always a maximum for =0, horizontal propagation (left pattern). For horizontal polarization, there is cancellation at that angle. Note that the above formulae and these plots assume the ground as a perfect conductor. These plots of the radiation pattern correspond to a distance between the antenna and its image of 2.5. As the antenna height is increased, the number of lobes increases as well.
What is also increased along with the antenna's height?
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Answer: lobes

Problem: In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.
What is the name used to describe the four conditions of closure, associativity, identity and invertibility?
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Answer: group axioms

Problem: From the middle of the 3rd century Germanic tribes regularly invaded the Roman territories. Around 275 the Romans could no longer maintain the northern border and Utrecht was abandoned. Little is known about the next period 270650. Utrecht is first spoken of again several centuries after the Romans left. Under the influence of the growing realms of the Franks, during Dagobert I's reign in the 7th century, a church was built within the walls of the Roman fortress. In ongoing border conflicts with the Frisians this first church was destroyed.
How was the the church destroyed
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Answer:
In ongoing border conflicts with the Frisians this first church was destroyed.