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Algebraic Aspects
Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019
Marlos A. G. Viana, Vasudevan Lakshminarayanan
and clearly φ(cz)=cφ(z) for all c∈ℝ. That is, φ is a linear mapping of real linear spaces and φ(cz)=cφ(z). We then say that φ is an algebra homomorphism.
Summation Kernels for Orthogonal Polynomial Systems
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
Frank Filbir, Rupert Lasser, Josef Obermaier
If we define f* = (f̃)–, where f̃(x) = f(x̃), then (l1(h), *, +) is a Banach *-algebra with unit ϵe. The map f → fh is an isometric isomorphism from the Banach space l1(h) onto M(K) and also a *- algebra homomorphism. Of course, if (K, *, ~) is commutative, then the Banach algebras l1(h) and M(K) are commutative, too.
Electromagnetism and Quantum Field Theory
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
and therefore, on using quantum Ito’s formula and the fact that jt is an algebra homomorphism, we get Q[1]R[n]=P[n+1],Q[1]S[n]=Q[n+1],S[1]R[n]=R[n+1],S[1]S[n]=S[n+1],n≥1
A variety of algebras closely related to subordination algebras
Published in Journal of Applied Non-Classical Logics, 2022
Let and be subordination algebras and consider their corresponding pseudo-subordination algebras and . A Boolean algebra homomorphism h from to is a subordination algebra morphism from to if and only if for all ,a strong subordination algebra morphism from to if and only if for all .
On the symmetry of induced norm cones
Published in Optimization, 2022
Historically, formally-real Jordan algebras were the objects of interest. In finite dimensions, they're equivalent to Euclidean Jordan algebras, but the latter is an easier definition to start with. However, one consequence of that retroactive definition is that ‘isomorphism’ refers only to an invertible linear Jordan-algebra homomorphism, and not necessarily to an isometry. It could mean nothing else, because in a formally-real Jordan algebra, there may not be an inner product to preserve. In what follows, we investigate what this means for the family of Jordan spin algebras. To avoid perpetuating the confusion, we use the term Jordan isomorphism to indicate an invertible linear Jordan-algebra homomorphism.
The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits
Published in Dynamical Systems, 2018
Proposition 3.1 is related to a reformulation in our setting of a familiar algebraic result. Namely, is isomorphic, both as -algebras and as -modules, to . One defines the algebra homomorphism θ from to by sending x to β: this is clearly surjective, since is spanned as a -vector space by the powers of β. The kernel of θ is precisely <p(x) >, given that p(x) is the minimal polynomial of β. So, by the First Isomorphism Theorem for algebra homomorphisms, . This also shows that multiplication of an element of by x corresponds to multiplication of the corresponding element of by β. Finally, admits a -basis {1, x, …, xd − 1}, which defines an isomorphism between and . The pair becomes a -module if we define for , hence multiplication by β in corresponds to the linear map in (see [1, p. 478]).