• Study Resource
  • Explore
    • Arts & Humanities
    • Business
    • Engineering & Technology
    • Foreign Language
    • History
    • Math
    • Science
    • Social Science

    Top subcategories

    • Advanced Math
    • Algebra
    • Basic Math
    • Calculus
    • Geometry
    • Linear Algebra
    • Pre-Algebra
    • Pre-Calculus
    • Statistics And Probability
    • Trigonometry
    • other →

    Top subcategories

    • Astronomy
    • Astrophysics
    • Biology
    • Chemistry
    • Earth Science
    • Environmental Science
    • Health Science
    • Physics
    • other →

    Top subcategories

    • Anthropology
    • Law
    • Political Science
    • Psychology
    • Sociology
    • other →

    Top subcategories

    • Accounting
    • Economics
    • Finance
    • Management
    • other →

    Top subcategories

    • Aerospace Engineering
    • Bioengineering
    • Chemical Engineering
    • Civil Engineering
    • Computer Science
    • Electrical Engineering
    • Industrial Engineering
    • Mechanical Engineering
    • Web Design
    • other →

    Top subcategories

    • Architecture
    • Communications
    • English
    • Gender Studies
    • Music
    • Performing Arts
    • Philosophy
    • Religious Studies
    • Writing
    • other →

    Top subcategories

    • Ancient History
    • European History
    • US History
    • World History
    • other →

    Top subcategories

    • Croatian
    • Czech
    • Finnish
    • Greek
    • Hindi
    • Japanese
    • Korean
    • Persian
    • Swedish
    • Turkish
    • other →
 
Profile Documents Logout
Upload
Representations of GL_2(A_Q^\infty)
Representations of GL_2(A_Q^\infty)

... tool in the study of smooth representations of TD groups. The idea is simple through the following analogy: Hecke algebras are to TD groups as group algebras are to finite groups. Namely, the Hecke algebra H (G) of a TD group G is made so that, essentially Repsm (G) = Modsm (H (G)) where the supersc ...
Moduli of elliptic curves
Moduli of elliptic curves

... curves. The first method is Igusa’s (cf. [Igu59] and [KM85]): it is based on the fact that the j-invariant construction gives a map Y1 (N ) → A1Z[1/N ] . The target is compactified by A1 ֒→ P1 , and X1 (N ) is defined as the normalization of P1Z[1/N ] in the fraction field of Y1 (N )/Q. One studies ...
The Critical Thread:
The Critical Thread:

... its negation “there exists at least one...”, “agrees with ϕ on ...”, et cetera. Though these concepts are assumed, a few of the above are briefly stated or reviewed for the reader for issues of clarity. Finally, this text in no way is exhaustive or even sufficient in its description in the topics it ...
Algebra I (Math 200)
Algebra I (Math 200)

... with it its inverse. So, H contains a positive integer. Let n be the smallest positive integer contained in H. We will show that H = nZ. First, since n ∈ H also n + n, n + n + n, . . . ∈ H. Also −n, −n + (−n), . . . ∈ H. Thus, nZ 6 H. On the other hand take an arbitrary element h of H and write it a ...
DIALGEBRAS Jean-Louis LODAY There is a notion of
DIALGEBRAS Jean-Louis LODAY There is a notion of

... [L4]. The next step would consist in computing the dialgebra homology of the augmentation ideal of K[GL(A)], for an associative algebra A. Here is the content of this article. In the first section we introduce the notion of associative dimonoid, or dimonoid for short, and develop the calculus in a ...
Contents Lattices and Quasialgebras Helena Albuquerque 5
Contents Lattices and Quasialgebras Helena Albuquerque 5

Algebra II (MA249) Lecture Notes Contents
Algebra II (MA249) Lecture Notes Contents

... Definition Let G and H be two (multiplicative) groups. We define the direct product G × H of G and H to be the set {(g, h) | g ∈ G, h ∈ H} of ordered pairs of elements from G and H, with the obvious component-wise multiplication of elements (g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ) for g1 , g2 ∈ G and ...
MA135 Vectors and Matrices Samir Siksek
MA135 Vectors and Matrices Samir Siksek

... the real number line. Examples or real numbers are 0, −1, 1/2, π, 2 . . .. The set of real numbers is given the symbol R. Below we list some of their properties. There is no doubt that you are thoroughly acquainted with all these properties, and that you use these properties in your manipulations (b ...
The structure of Coh(P1) 1 Coherent sheaves
The structure of Coh(P1) 1 Coherent sheaves

... Definition 1.15. Let O → O(d) be the map given by multiplication by a homogeneous degree d polynomial f . Then the cokernel sheaf is called Of , and the stalk at any point p is a vector space over k whose dimension is the order of f at p. In the case where f has degree 1 (for example, f = x), we cal ...
VISIBLE EVIDENCE FOR THE BIRCH AND SWINNERTON
VISIBLE EVIDENCE FOR THE BIRCH AND SWINNERTON

A Cut-Invariant Law of Large Numbers for Random Heaps
A Cut-Invariant Law of Large Numbers for Random Heaps

... large numbers states the convergence of these ergodic means on the one hand, and that this limit is independent of the AST that has been considered on the other hand. It is proved for additive cost functions and under additional conditions for sub-additive cost functions as well. In both cases, the ...
The Kauffman Bracket Skein Algebra of the Punctured Torus by Jea
The Kauffman Bracket Skein Algebra of the Punctured Torus by Jea

... This dissertation studies the Kauffman bracket skein algebra of the punctured torus. The first chapter contains the historical background on the Kauffman bracket skein algebra and its applications. The second chapter contains the multiplication rule for the Kauffman bracket skein algebra of the cylinder ...
Dyadic Tensor Notation
Dyadic Tensor Notation

... VIII). Tensors have many applications in theoretical physics, not only in polarization problems like the ones described below, but in dynamics, elasticity, uid mechanics, quantum theory, etc.. The mathematical apparatus for dealing with tensor problems may not yet be familiar, so this sheet contain ...
Sylow`s Subgroup Theorem
Sylow`s Subgroup Theorem

... Tharatorn Supasiti February 2, 2010 Note: This is not my work. Most comes from Suzuki [2] and Wilkins [3]. ...
Notes on Measure Theory Definitions and Facts from Topic 1500
Notes on Measure Theory Definitions and Facts from Topic 1500

... < {(a, b)|a, b ∈ Q, a < b} >σ • A subset of R is measurable if it’s an element of the completion of {Borel sets in R} w.r.t. Lebesgue measure. It’s hard to make non-measurable sets, or even non-Borel sets. This follows from the earlier definition of a Borel set, which was essentially defined as the ...
noncommutative polynomials nonnegative on a variety intersect a
noncommutative polynomials nonnegative on a variety intersect a

... 1.4. Behavior of Polynomials on Real Zero Sets. One of our main themes is taking into account behavior of zero sets. For the free algebra Rhx, x∗ i, there is a “Real Nullstellensatz”. S g Let p1 , . . . , pk , q ∈ Rhx, x∗ i. If q(X)v = 0 for every (X, v) ∈ n∈N (Rn×n ) × Rn such that p1 (X)v = · · · ...
UNIFORMLY CONTINUOUS FUNCTIONS ON NON
UNIFORMLY CONTINUOUS FUNCTIONS ON NON

... Hausdorff second countable groupoids one can construct systems of measures that satisfy ”left invariance” condition ([1]). But the continuity assumption has topological consequences for the groupoid. It entails that the range map (and hence the domain map) is open ([9, Proposition I. 4]). On the oth ...
Hankel Matrices: From Words to Graphs
Hankel Matrices: From Words to Graphs

... Definability of numeric graph parameters, IV Usually, summation is allowed over second order variables, whereas products are over first order variables. L is typically Second Order Logic or a suitable fragment thereof. We are especially interested in MSOL and CMSOL, Monadic Second Order Logic, possi ...
2.1. Functions on affine varieties. After having defined affine
2.1. Functions on affine varieties. After having defined affine

... In fact, the statement is false if the ground field is not algebraically closed, as you can see from the example of the function x21+1 that is regular on all of A1 (R), but not polynomial. Example 2.1.12. Probably the easiest case of an open subset of an affine variety X that is not of the form X f ...
Mathematical structures
Mathematical structures

... Apart from the purely aesthetical appeal of a transparent setting where a given relation is clearly understood, the reward when we trivialize a relation is that the structures we uncover in this process often are of much more general applicability than the settings we originally were given. A trivia ...
Subject: Mathematics Lesson: Isomorphism and Theorems on
Subject: Mathematics Lesson: Isomorphism and Theorems on

... An isomorphism from a group G to a group G is one to one mapping from G to G that preserved the group operation. Value Addition: Note Two groups G and G are called isomorphic, written as G1 ~ G2. If there is an isomorphism from Gonto G' i.e. two groups g and g are isomorphic if there exist a ma ...
AN INTRODUCTION TO (∞,n)-CATEGORIES, FULLY EXTENDED
AN INTRODUCTION TO (∞,n)-CATEGORIES, FULLY EXTENDED

... (1) for each (n − 1)-manifold M , a vector space F (M ), such that F (∅) = k and disjoint unions of manifolds are mapped to the tensor products of the corresponding vector spaces; (2) for each bordism B : M → N , a k-linear map F (M ) → F (N ) satisfying the usual coherence axioms for categories. Re ...
ASSOCIATIVE GEOMETRIES. I: GROUDS, LINEAR RELATIONS
ASSOCIATIVE GEOMETRIES. I: GROUDS, LINEAR RELATIONS

... term). Just as groups are generalized by semigroups, grouds are generalized by semigrouds which are simply sets with a ternary map satisfying (G3). By work dating back at least to that of V.V. Vagner, e.g. [Va66], it is already known that this concept has important applications in geometry and algeb ...
S11MTH 3175 Group Theory (Prof.Todorov) Quiz 5 (Practice) Name
S11MTH 3175 Group Theory (Prof.Todorov) Quiz 5 (Practice) Name

... S11MTH 3175 Group Theory (Prof.Todorov) ...
Algebra I: Section 6. The structure of groups. 6.1 Direct products of
Algebra I: Section 6. The structure of groups. 6.1 Direct products of

... is the cyclic group a direct product of smaller subgroups? Answer: No. In G′ every element g 6= e has order order o(g) = 2; in fact, writing (+) for the operation in Z2 × Z2 we have (a, b) + (a, b) = (2 · a, 2 · b) = (0, 0). In contrast, G has a cyclic generator a such that o(a) = 4. That’s impossib ...
< 1 2 3 4 5 6 7 8 9 ... 39 >

Oscillator representation

In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself. The contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended.
  • studyres.com © 2025
  • DMCA
  • Privacy
  • Terms
  • Report