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alternate case: upper half-plane

Schwarz triangle (10,972 words) [view diff] no match in snippet view article find links to article

angles is given in. In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For

(t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.} When D {\displaystyle D} is the upper half plane the Loewner equation differs from this by changes of variable and is

in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which

described by Émile Picard in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space by the projective special linear group

special open subset of complex affine space generalizing the Siegel upper half plane studied by Siegel (1939). They were introduced by Piatetski-Shapiro (1959

uniformization of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the surface) in a way that makes sense

{\displaystyle y\geq 0} . The set X {\displaystyle X} can be termed the closed upper half plane. To give the set X {\displaystyle X} a topology means to say which

Maass–Selberg relations for the case of real analytic Eisenstein series on the upper half plane. Atle Selberg extended the relations to symmetric spaces of rank 1

a copy H of the upper half plane (of pairs (x, y) with y > 0). Then glue the open upper half of each plane Xa to the upper half plane H by identifying

A level 1 weakly holomorphic modular form is a function f on the upper half plane with the properties: f transforms like a modular form: f ( ( a τ +

is in the Siegel upper half plane of degree g. The function f is originally defined only for elements of the Siegel upper half plane of degree r, but

McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at θ is the Cauchy density on the real

subspace of R 2 {\displaystyle \mathbb {R} ^{2}} consisting of the open upper half plane together with the rationals on the x-axis, namely, X = ( R × ( 0 ,

Vladimirovich Nemytskii. If Γ{\displaystyle \Gamma } is the (closed) upper half-plane Γ={(x,y)∈R2|y≥0}{\displaystyle \Gamma =\{(x,y)\in \mathbb {R} ^{2}|y\geq

in a book titled "Chiral Algebras." Drinfeld reciprocity Drinfeld upper half plane Manin–Drinfeld theorem Quantum group Chiral algebra Quasitriangular

simple expressions for the other Heegner numbers. The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the

Baily and Armand Borel (1964, 1966). If C is the quotient of the upper half plane by a congruence subgroup of SL2(Z), then the Baily–Borel compactification

function F, of exponential type τ/2, having all its zeros in the (closed) upper half plane, such that f ( z ) = F ( z ) F ( z ¯ ) ¯ {\displaystyle f(z)=F(z){\overline

non-tangential maximal function takes a function F defined on the upper-half plane R + n + 1 := { ( x , t )   :   x ∈ R n , t > 0 } {\displaystyle \mathbf

z){\displaystyle \phi (\tau ,z)} of two complex variables (with τ in the upper half plane) such that ϕ(aτ+bcτ+d,zcτ+d)=(cτ+d)ke2πimcz2cτ+dϕ(τ

{\gamma }}''\right)^{2}}}}\,.} For a semi-circle of radius a in the upper half-plane with R = | − a | = a , {\textstyle R=|-a|=a\,,} y = a 2 − x 2 y ′ =

choose as the integration contour for f {\displaystyle f} the union of upper half-plane semicircles of radii ε {\displaystyle \varepsilon } and R {\displaystyle

way are known as Fuchsian groups. The quotient space H2‍/‍Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the

n≥2 there is a polyhedron P with an infinite number of sides. The upper half plane model of n+1 dimensional hyperbolic space in Rn+1 projects to Rn, and

on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane (2nd ed.). New York, NY: Springer. p. 37. ISBN 978-1461479710. Retrieved

mapping is a local isometry, and thus exhibits the portion y ≥ 1 of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping

on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincare Upper Half-Plane (Second ed.). Springer. p. 67. ISBN 978-1-4614-7971-0. "Interview with

the quotient surface M=Γ\H. The following description refers to the upper half-plane model of the hyperbolic plane. This is a hyperbolic surface, in fact

conjugate pairs. For each conjugate pair, pick the zero or pole in the upper half plane and accumulate these to obtain q ( t ) {\displaystyle q(t)} . The uniqueness

Then a Cartan–Eilenberg resolution of A ∗ {\displaystyle A_{*}} is an upper half-plane double complex P ∗ , ∗ {\displaystyle P_{*,*}} (i.e., P p , q = 0 {\displaystyle

analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way. Hardy space John B. Garnett, Bounded

L^{g}}\exp(\pi iTr(\lambda T\lambda ^{t}))} where T is an element of the Siegel upper half plane of degree g. This is a Siegel modular form of degree d and weight dim(L)/2

{H} =\{z\in \mathbb {C} :\operatorname {Im} (z)>0\},} is the complex upper half-plane. Then we have a bijection: H ⟶ T ( T 2 ) {\displaystyle \mathbb {H}

Iwasawa decomposition, both groups are circle bundles over the complex upper half-plane, and their universal cover S L 2 ( ~ R ) {\displaystyle {\mathrm {S}

diagram Schwarz–Christoffel mapping – a conformal transformation of the upper half-plane onto the interior of a simple polygon Special linear group – transformations

constant negative curvature. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane, a geometry of constant curvature

\cdot \tau } for τ ∈ h {\displaystyle \tau \in {\mathfrak {h}}} in the upper-half plane. Then, the lattice generated by 1 / n , τ / n {\displaystyle 1/n,\tau

times. The pole in the energy difference in the denominator is on the upper half-plane in the case of ψ − {\displaystyle \psi ^{-}} , and so does not lie

[ − 3 , 3 ] 2 {\displaystyle (s,t)\in [-3,3]^{2}} . Points in the upper half plane t > 0 correspond to a linear factor with roots s ± i t {\displaystyle

Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincare Upper Half-Plane (Second ed.). Springer. p. 67. ISBN 978-1-4614-7971-0. v t e

certain class VII surfaces whose universal cover is C×H where H is the upper half plane (so they are quotients of this by a group of automorphisms). They have

its Watson-Crick base pairs represented as non-crossing arcs in the upper half plane. Therefore, a secondary structure is a scaffold having many sequences

ISBN 978-3-540-61087-8, MR 1423131 Goss, David (1980). "The algebraist's upper half-plane". Bull. Amer. Math. Soc. (N.S.). 2 (3): 391–415. doi:10.1090/s0273-0979-1980-14751-5

-{\frac {1}{4z}}} , since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that c 0 ( x ) = c 0 ( 0 ) = 1 {\displaystyle

Dirichlet L-functions with automorphic forms (holomorphic functions on the upper half plane of C {\displaystyle \mathbb {C} } (the complex numbers) that satisfy

t > 0, and for complex numbers z along the arc (which lies in the upper half-plane), the argument φ of z lies between 0 and π. So, | e i t z | = | e i

Dirichlet L-functions with automorphic forms (holomorphic functions on the upper half plane of C {\displaystyle \mathbb {C} } (the complex numbers) that satisfy

{\displaystyle f} is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane. See Operator

A level 1 almost holomorphic modular form is a function f on the upper half plane with the properties: f transforms like a modular form: f ( ( a τ +

action of SL(2, Z) is then the usual action of the modular group on the upper half-plane. Consequently, the period domain is the Riemann sphere. This is the

{\displaystyle (0,\pm 1)} . Further, the portion of this image in the upper half-plane depicts the following situation: The double-loops are a reduced set

analytic extension to the upper and lower complex half planes so that the upper half plane is mapped to itself. Algebraic Riccati equation Sylvester's formula

0 ) {\displaystyle (-a,0)} in two ways, counterclockwise, on the upper half-plane arc of this circle, and clockwise, on the lower half-plane arc. The

Euclidean structures on Rn+1. Hermitian if n = 1, when it is the upper half plane or unit complex disc. Bn I (n ≥ 2) split n(2n + 1) n SO(n)SO(n+1) Non-cyclic

the lower half-plane, whilst the right-hand side is analytic in the upper half-plane, analytic continuation guarantees existence of an entire function which

} The branch of the logarithm is chosen to be single-valued in the upper half-plane and to agree with the natural log along the positive real axis. As

be realized as follows. Considering the action of SL(2, R) on the upper half-plane model H2 of the hyperbolic plane by Möbius transformations, the affine

picture: In this case, the Tits cone is not the whole plane, but only the upper half plane. Taking the quotient by the positive reals then yields another copy

}{2\pi \mathbb {Z} }}} . The corresponding one-point functions on the upper half-plane { ℑ z > 0 } {\displaystyle \{\Im z>0\}} are ⟨ V ( n , w ) ( z ) ⟩ Dirichlet

sphere with [−1,1] removed onto the upper half-plane. The Möbius transformation (t − 1)/(t + 1} carries the upper half-plane onto the unit disk. Composition

plane with its Poincaré metric. It acts by Möbius transformations. The upper half-plane can be identified with the unit disc by the Cayley transform. Under

q=e^{i\pi \tau }} , for τ {\displaystyle \tau } a complex number in the upper half plane, and define the theta function: θ ( τ ) = ∑ n q n 2 . {\displaystyle

isomorphisms SO+(1, 2) ≅ PSL(2, R) ≅ PSU(1, 1) allow one to interpret the upper half plane model as the quotient SL(2, R)/SO(2) and the Poincaré disc model as

transform ci which as a Möbius transformation maps the unit disk onto the upper half plane. Given a subset I of indices of the strongly orthogonal family ψ1,

d\xi } is the fundamental solution of the Laplace equation in the upper half-plane. It represents the electrostatic potential in a semi-infinite plate

example, on the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it. A mapping of the identity component

upper half space chart. This is closely analogous to the well known upper half plane chart for the hyperbolic plane H2, which is familiar to generations

y {\displaystyle u=\ln {\sqrt {x/y}}} produce a point (u,v) in the upper half-plane. The hyperbolic metric in the quadrant depends on the Poincaré half-plane

)\leq {\ell (\gamma ) \over \cosh s}.} Applying an isometry in the upper half plane, it may be assumed that the geodesic line is the positive imaginary

former (latter) has all its poles and discontinuities in the lower (upper) half-plane. The thermal propagator G ( ω n ) {\displaystyle {\mathcal {G}}(\omega

Zagier (1981) constructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to

-\infty <z\leq 0\,} , while the fluid with the label 'G' lives in the upper half-plane 0 ≤ z < ∞ {\displaystyle 0\leq z<\infty \,} . To specify the solution

circles that are tangent to the x-axis at the origin. The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise

BRST quantized string takes the form (in radial quantization in the upper half plane), | Ψ ⟩ = ∫ d 26 p ( T ( p ) c 1 e i p ⋅ X | 0 ⟩ + A μ ( p ) ∂ X μ

point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane,

be associated with Q ⊂ R ⊂ C, the rationals in the extended complex upper-half plane. The group of homographies on P1(Z / nZ) is called a principal congruence

theorem Schwarz–Christoffel mapping – a conformal transformation of the upper half-plane onto the interior of a simple polygon. Conformal radius The existence

0 ) ] {\displaystyle \left[Re(-1/2),Im(0)\right]} which is in the upper half-plane ( I m ( Γ ( ω ) ) > 0 ) {\displaystyle \left(Im(\Gamma (\omega ))>0\right)}

to the lower half plane of t. The integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles

closed subgroup of finite covolume, since this subgroup acts on the upper half plane with a fundamental domain of finite hyperbolic area. The group SL(2

domain corresponds to the open unit disk and the tube domain to the upper half plane. Both these spaces have transitive groups of biholomorphisms given

principal series. The group G = SL(2,R) acts transitively on the Poincaré upper half plane H 2 = { x + r i ∣ r > 0 } {\displaystyle {\mathfrak {H}}^{2}=\{x+ri\mid