We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R and δ a σderivation of R. We recall that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-invariant and δ-invariant ideal I (i.e. σ(I) ⊆ I and δ(I) ⊆ I) of R. A ring R is called a δ-divided ring...

Proof. Clearly (A ∩ B)[G] ⊆ A[G] ∩ B[G]. Let us show the opposite inclusion. It is enough to deal with ordinals. Note that B[G]∩ V = B, since Q ⊆ B. Let δ ∈ A[G]∩B[G]∩On = A[G]∩B ∩On. Pick a canonical name δ ∼ ∈ A such that δ ∼ = δ. It is of the form ⟨⟨qi, τ̌i⟩ | i < ρ ≤ |Q|⟩, where τi’s are in B and ⟨qi | i < ρ⟩ is a maximal antichain. Now, δ ∼ ∈ B, since |Q|B ⊆ B. So, δ ∼ ∈ A∩B. Hence, δ = δ ∼...

We define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I (n)denote the set of all involutions on [n](= {1, 2, . . . , n}) and let F(2n)denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let [n]q = 1+q+· · ·+qn−1 and let (k)q denote the q-binomial coefficient....

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.