Titlebook: Combinatorial Number Theory and Additive Group Theory; Alfred Geroldinger,Imre Z. Ruzsa Textbook 2009 Birkh�user Basel 2009 Graph.Graph th

cardiac-arrest

https://doi.org/10.1007/1-4020-4709-6Let . and . be sets in a (mostly commutative) group. We will call the group operation addition and use additive notation. The . of these sets is .∈ .∈

Melodrama

Marc Hein,Anna B. Roehl,René H. TolbaLet . be sets in a group, |.|=., |.|=.. The cardinality of A+B can be anywhere between max(.) and .. Our aim is to understand the connection between this size and the structure of these sets.

留恋

Designing a Small Animal Imaging CenterThis chapter is about questions of the following kind. Assume we have finite sets . in a group .. What can we say about . if we know the structure of ., or we have some information about how these sets are situated within .? The “what” will be in most cases a lower estimate for the cardinality.

强行引入

Nicolau Beckmann,Birgit LedermannA finite set is naturally measured by its cardinality. A set of reals is naturally measured by its Lebesgue measure (non-measurable sets do exist, just we never meet them). There is no similarly universal way to measure and compare infinite sets of integers. The most naturally defined one is the asymptotic density.

amenity

替代品

Stuart Foster,Catherine TheodoropoulosThe next collection of exercises is deliberately vague. The number of points is my estimate for the difficulty. This naturally depends on expertise; some of them were told in the course; then naturally the difficulty turns to 0.

Exposure

How to Choose the Right Imaging ModalityLet . and . denote sets of integers and let . + . = { . + . : . ∈ ., . ∈ . } be the sumset and . = | . : . ∈ ., . ∈ . the product set.

星球的光亮度

NotationOur notation and terminology is consistent with [71]. We briefly gather some key notions. We denote by ℕ the set of positive integres, and we put ℕ. = ℕ ∪ { 0 }. For real numbers, . ∈ ℝ we set [.] = { . ∈ ℤ | . , and we define sup Ø = max Ø = min Ø = 0.

贫困

放纵

The structure of sets of lengthsSets of lengths in Krull monoids and in noetherian domains are finite and nonempty. Furthermore, either all sets of lengths are singletons or sets of lengths may become arbitrarily large (see Lemma 1.0.1).