Download Spectral Theory for SL2(Z)\SL2(R)/SO2(R) by Garrett P. PDF

By Garrett P.

Similar algebra books

Groebner bases algorithm: an introduction

Groebner Bases is a method that gives algorithmic suggestions to quite a few difficulties in Commutative Algebra and Algebraic Geometry. during this introductory instructional the fundamental algorithms in addition to their generalization for computing Groebner foundation of a collection of multivariate polynomials are provided.

The Racah-Wigner algebra in quantum theory

The advance of the algebraic facets of angular momentum thought and the connection among angular momentum idea and exact issues in physics and arithmetic are coated during this quantity.

Wirtschaftsmathematik für Studium und Praxis 1: Lineare Algebra

Die "Wirtschaftsmathematik" ist eine Zusammenfassung der in den Wirtschaftswissenschaften gemeinhin benötigten mathematischen Kenntnisse. Lineare Algebra führt in die Vektor- und Matrizenrechnung ein, stellt Lineare Gleichungssysteme vor, berichtet über Determinanten und liefert Grundlagen der Eigenwerttheorie und Aussagen zur Definitheit von Matrizen.

Extra resources for Spectral Theory for SL2(Z)\SL2(R)/SO2(R)

Example text

13This (reasonable) statement is the axiom of choice, cf. 3. 18 I. 11. 10, it is not unreasonable to use 2A to denote the set of functions from an arbitrary set A to a set with 2 elements (say {0, 1}). Prove that there is a bijection between 2A and the power set of A (cf. 2). 3] 3. Categories The language of categories is aﬀectionately known as abstract nonsense, so named by Norman Steenrod. This term is essentially accurate, and not necessarily derogatory: categories refer to nonsense in the sense that they are all about the ‘structure’, and not about the ‘meaning’, of what they represent.

Now assume that f is a monomorphism. This says something about arbitrary sets Z and arbitrary functions Z → A; we are going to use a microscopic portion of this information, choosing Z to be any singleton {p}. Then assigning functions α , α : Z → A amounts to choosing to which elements a = α (p), a = α (p) we should send the single element p of Z. For this particular choice of Z, the property deﬁning monomorphisms, f ◦α =f ◦α =⇒ α = α becomes f ◦ α (p) = f ◦ α (p) =⇒ α = α , that is . f (a ) = f (a ) =⇒ α = α Now two functions from Z = {p} to A are equal if and only if they send p to the same element, so this says: f (a ) = f (a ) =⇒ a = a .

These categories are called discrete. As another example, consider the category corresponding to endowing Z with the relation ≤: for example, G3 G5 UG 7 2b bb ppÐpÐd p p bb 13 pp Ð bb ppp ÐÐÐ 0  pppp Ð G4 3 22 I. Preliminaries: Set theory and categories is a (randomly chosen) commutative diagram in this category. It would still be a (commutative) diagram in this category if we reversed the vertical arrow 3 → 3, or if we added an arrow from 3 to 4; while we are not allowed to draw an arrow from 4 to 3, since 4 ≤ 3.