# Rudenskaya O.G.'s 4-Quasiperiodic Functions on Graphs and Hypergraphs PDF

By Rudenskaya O.G.

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**Extra resources for 4-Quasiperiodic Functions on Graphs and Hypergraphs**

**Sample text**

11 (Uniform binomial point process) Let Φ = {x1, …, xn} ⊂ W be a point process with a fixed and finite number n of points on the compact set W. Φ is a uniform BPP if and only if Φ, viewed as a random vector, is uniformly distributed in Wn. DEFINITION From the independence assumption it follows that for Borel subsets A1, …, Anof W, The name of this process comes from the fact that the number of points in A ⊂ W is binomially distributed with parameters n = Φ(W) and p = | A |⁄| W |. Hence the intensity is λ(x)=n1W(x) ⁄ | W |, and the intensity measure is 35 Since the total number of points is fixed, the numbers of points in Φ(A) and Φ(W\A) are not independent, in contrast to the PPP.

This is a straightforward generalization to the inhomogeneous case. The intensity Φ(x) needs to be normalized by Λ(B) in order for it to be a proper probability density. In general, the intensity (function) λ(x) need not exist. 7) is the volume of the unit ball. 6 Properties of point processes In this section, we introduce and define several important properties of point processes. First, we introduce notation for a translated point process. } is the point process translated by x ∈ d. Similarly, for an event E, the translated event Ex is defined as E ≜ {φ ∈ : φ−x ∈ E} E ∈ So, for example, if E is the event that Φ ⊂ 2 does not have any points in the ball b(o, r), then E(1, 0) is the event that Φ does not have any points in the ball b((1, 0), r).

The interarrival intervals Si are iid exponential with parameter λ. 2. The generalized interarrival intervals xk+i – xi for k > 0 are Erlang (gamma) distributed with parameters k and λ for all i. 3. The probability of two or more arrivals in a given interval is asymptotically of smaller order than the length of the interval: 30 (N([a, a + h)) ≥ 2) = o(h), h → 0. 4. 2) where +n = {(t1, …, tn) ∈ n : 0 < t1 < t2 < … < tn} is the order cone (a hyperoctant or orthant) in n dimensions. 2). We may also define the inhomogeneous or non-uniform PPP, where the intensity is a function of the location, denoted as λ(t) ≥ 0, in a similar way.

### 4-Quasiperiodic Functions on Graphs and Hypergraphs by Rudenskaya O.G.

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