For a simply laced and hyperbolic Kac–Moody group G = G(R) over a commutative ring R with 1, we consider a map from a finite presentation of G(R) obtained by Allcock and Carbone to a representation–theoretic construction G(R) corresponding to an integrable representation V λ with dominant integral weight λ. When R = Z, we prove that this map extends to a group homomorphism ρλ,Z : G(Z)→ G(Z). We...

1 Algebras In this first section we will consider some common features of familiar algebraic structures such as groups, rings, lattices, and boolean algebras to arrive at a definition of a general algebraic structure. Recall that a group G consists of a nonempty set G, along with a binary operation · : G → G, a unary operation −1 : G → G, and a constant 1 G such that • x · (y · z) = (x · y) · z...

and Applied Analysis 3 Theorem 4. Assume that (H 1 )–(H 6 ) and (7) hold. If there exists a function ξ(t) ∈ C rd(T , (0,∞)) such that for any positive numberM, lim t→∞ ∫ t t0 (ξ (s) p (s) − Q (s)) Δs = ∞, (12) where p (s) = q (s) [1 − p (δ (s))] β , Q (s) = αM(R (σ (s))) α−β r (δ (s)) ((ξ Δ (s)) + ) α+1 (α + 1) α+1 βξ (s) (δ (s)) α , (ξ Δ (s)) + := max {ξΔ (s) , 0} , (13) then (1) is oscillator...

Given a formal power series f(z) ∈ C[[z]] we define, for any positive integer r, its rth Witt transform, W (r) f , by W (r) f (z) = 1 r ∑ d|r μ(d)f(z d)r/d, where μ denotes the Möbius function. The Witt transform generalizes the necklace polynomials, M(α;n), that occur in the cyclotomic identity 1 1− αy = ∞

The global regularity problem for the periodic NavierStokes system ∂tu+ (u · ∇)u = ∆u−∇p ∇ · u = 0 u(0, x) = u0(x) for u : R×(R/Z) → R and p : R×(R/Z) → R asks whether to every smooth divergence-free initial datum u0 : (R/Z) 3 → R there exists a global smooth solution. In this note we observe (using a simple compactness argument) that this qualitative question is equivalent to the more quantita...

Suppose q(z) is a smooth function on [0,∞) whose odd order derivatives are zero at z=0. Take r = (x, y, z) and let U(r, t) be the solution of the the IBVP Utt − Uxx − Uyy − Uzz + q(z)U = 0 for r ∈ R, z ≥ 0, t ∈ R Uz(x, y, z=0, t) = δ(x, y, t), U(r, t) = 0, for t < 0 We show that q(z) may be recovered from a knowledge of U(a, b, 0, t) for t varying over an interval and fixed a, b, by reducing th...

We introduce a method to obtain the envelopes of eccentric orbits in axially symmetric potentials, $\Phi(R,z)$, endowed with $z$-symmetry reflection. By making transformation $z\rightarrow a+\sqrt{a^{2}+ z^{2}}$, $a>0$, we compute resulting mass density, referred here as \emph{effective density} $\rho_{\rm ef}(R,z;a)$, order calculate $Z(R)$ meridional plane $(R,z)$. find that they obey approxi...

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and Applied Analysis 3 Let n r, f or n r, ∞̂ denote the number of poles of f z in |z| ≤ r and let n r, a, f denote the number of a-points of f z in |z| ≤ r, counting with multiplicities. Define the volume function associated with E-valued meromorphic function f z by V ( r, ∞̂, f V r, f 1 2π ∫ Cr log ∣∣ ∣ ∣ r ξ ∣∣ ∣ ∣Δ log ∥ ∥f ξ ∥ ∥dx ∧ dy, ξ x iy, V ( r, a, f ) 1 2π ∫ Cr log ∣ ∣ ∣ ∣ r ξ ∣ ∣ ∣ ∣Δ...