| To each number > 1 there corresponds a discrete countable set of numbers denoted by Z and named set of beta-integers. The set Z is precisely the set of real numbers which are polynomial in when they are written in \basis ", and Z = Zwhen 2 N. We prove here a list of arithmetic properties of Z : addition, multiplication, relation with integers, when is a quadratic Pisot-Vijayaraghavan unit (qu...

Abstract We generalize the greedy and lazy $\beta $ -transformations for a real base to setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , which were recently introduced by first second authors as particular case Cantor bases. As in case, these new transformations, denoted $T_{{\boldsymbol }}}$ $L_{{\boldsymbol respectively, can be iterated order generate digi...

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...

Abstract For a quasi-smooth hypersurface X in projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb })\rightarrow H^p(X)$$ i ∗ : H</...

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 ...

Write $$\mathrm {ord}_p(\cdot )$$ for the multiplicative order in $${\mathbb {F}}_p^{\times }$$ . Recently, Matthew Just and second author investigated problem of classifying pairs $$\alpha , \beta \in {\mathbb {Q}}^{\times }\setminus \{\pm 1\}$$ which {ord}_p(\alpha ) > \mathrm {ord}_p(\beta holds infinitely many primes p. They called such order-dominant. We describe an easily-checkable suffic...

Let p $p$ be a fixed prime. We show that the number of isomorphism classes finite rings order n $p^n$ is α $p^\alpha$ , where = 4 27 3 + O ( 5 / 2 ) $\alpha =\frac{4}{27}n^3+\mathnormal {O}(n^{5/2})$ . This result was stated (with weaker error term) by Kruse and Price in 1969; problem with their proof pointed out Knopfmacher 1973. also commutative β $p^\beta$ $\beta =\frac{2}{27}n^3+\mathnormal...

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...