LetM be the multiplicative group of integral weight meromorphic modular forms for some character of SL2(Z) with integer coefficients, leading coefficient 1, and whose zeros and poles are supported at cusps and imaginary quadratic irrationals. Denote by Mk,h ⊂ M the subset which consists of modular forms of weight k and for which order of the pole at infinity is h. We have M = ∪k∈Z,h∈ 1 12 ZMk,h...

Division algebras D generated by some finitely generated nilpotent subgroup G of the multiplicative group D* of D are studied and the question to what extent G is determined by D is considered. Trivial examples show that D does not determine G up to isomorphism. However, it is proved that if F denotes the center of D, then the F-subalgebra of D generated by G is in fact determined up to isomorp...

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y , respectively. For f ∈ A, the peripheral spectrum of f , denoted by σπ(f) = {λ ∈ σ(f) : |λ| = ‖f‖}, is the set of spectral values of maximum modulus. A map T : A → B is weakly peripherally multiplicative if σπ(T (f)T (g)) ∩ σπ(fg) 6= ∅ for all f, g ∈ A. We show that if T is a surjective, weakly peripherally mu...

In addition, define certain lattices and subsets: L∨ = X∗(T ) = group of algebraic homomorphisms Gm → T L = Hom(X∗(T ),Z), the dual of L ∨ X(T ) = group of algebraic homomorphisms T → Gm, the weights of T . There is a canonical map from L toX(T )—to λ corresponds the multiplicative character e defined by the formula e(μ(x)) = x 〉 , which makes sense because all algebraic homomorphisms Gm → Gm a...

Continuum mechanics for isotropic finite thermoelastic deformations have been reviewed. Thermal effects on mechanical responses of rubbers captured by the isomorphism continuum stored energy (CSE) functional with multiplicative decomposition deformation gradient while preserving structure symmetry structural deformation. The CSE model fits and predicts experimental data SR NR-C60 at different e...

Let $G$ be a compact connected Lie group and $K$ closed subgroup. Assume that the order of any torsion element in integral cohomology is invertible given principal ideal domain $k$. It known this case homogeneous space $G/K$ with coefficients $k$ product $H^{*}(BK)$ over $H^{*}(BG)$ are isomorphic as $k$-modules. We show isomorphism multiplicative natural pair $(G,K)$ provided 2 The proof uses ...

Graphical calculi provide an intuitive, compositional way to express and manipulate quantum states and processes. They also provide a bridge to automated techniques for reasoning and computation via graph rewriting. The power of these calculi stems from the fact that they subsume a wide range of symmetries in the structure of quantum operations such as the Choi-Jamiolkowski isomorphism. One suc...

Denote the free group on 2 letters by F2 and the SL(2,C)-representation variety of F2 by R = Hom(F2, SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multipli...

Let K be a p-adic local field and E an elliptic curve defined over K. The component group of E is the group E(K)/E0(K), where E0(K) denotes the subgroup of points of good reduction; this is known to be finite, cyclic if E has multiplicative reduction, and of order at most 4 if E has additive reduction. We show how to compute explicitly an isomorphism E(K)/E0(K) ∼= Z/NZ or E(K)/E0(K) ∼= Z/2Z× Z/2Z.

A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resembles Girard’s linear logic with the differences attributable to its focus on behavior instead of proof. As wi...