1 Similarity Metrics Many recommendation algorithms employ some form of similarity metric in the generation of ratings predictions. Similarity metrics are often associated with some form of distance measure. Definition 1.0.1. Let δ be a function δ : R × R → R. Let x,y, z ∈ R. Then δ is a distance measure if it satisfies the following four properties. (d1) δ(x,y) ≥ 0 (no negative distances). (d2...

We introduce another powerful method of solving PDEs. First, we need to consider some preliminary definitions and ideas. Definition [Delta Function] The δ-function is defined by the following three properties, δ (x) = 0, ∞, x � = 0, x = 0, ∞ δ (x) dx = 1 −∞ ∞ f (x) δ (x − a) dx = f (a) −∞ where f is continuous at x = a. The last is called the sifting property of the δ-function. To make proofs w...

A natural question to ask is whether this result has an analogue in the context of large cardinals. The purpose of this article is to provide the strongest such analogue not ruled out by limitations imposed by the existence of Woodin cardinals. To describe the latter limitations we consider the forcing P , described as follows. Let δ be inaccessible and consider the language L(δ): (a) n ∈ R bel...

We construct explicit δ-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these δ-bracketing covers are bounded from above by δ−2+o(δ−2). A lower bound for the cardinality of arbitrary δ-bracketing covers for d-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-paralle...

In this paper, we introduce a new class of sets, namely semi*δ-open sets, using δ-open sets and the generalized closure operator. We find characterizations of semi*δ-open sets. We also define the semi*δ-interior of a subset. Further, we study some fundamental properties of semi*δ-open sets and semi*δ-interior.

A connected graph G is optimal-κ if the connectivity κ(G) = δ(G), where δ(G) is the minimum degree of G. It is super-κ if every minimum vertex cut isolates a vertex. An optimal-κ graph G is moptimal-κ if for any vertex set S ⊆ V (G) with |S| ≤ m, G−S is still optimal-κ. The maximum integer of such m, denoted by Oκ(G), is the vertex fault tolerance of G with respect to the property of optimal-κ....

k x[k]δ[n − k] : convolution of x[n] and δ[n] = · · · + x[−2]δ[n + 2] + x[−1]δ[n + 1] + x[0]δ[n] + x[1]δ[n − 1] + · · · : any sequence can be decomposed as a sum of scaled and shifted impulses. (a) Scale (by x[k]) and shift (from δ[n] to δ[n− k]) imply linearity and time-invariance, respectively. (b) Envision x[n] : {..., x[−1], x[0], x[1], ...} as a signal vector, and δ[n − k] : {..., 0, δ[n −...