We give an example of an amenable, radical Banach algebra, relying on results from non-abelian harmonic analysis due to H. Leptin, D. Poguntke and J. Boidol. Let A be a Banach algebra, and let E be a Banach A-module. A bounded linear map D : A → E is called a a derivation if D(ab) = a.Db+ (Da).b (a, b ∈ A). A derivation D : A → E is said to be inner if Da = x.a− a.x (a ∈ A) for some x ∈ E. For ...

The Advanced Encryption Standard (AES) is the most widely used block cipher. The high level structure of AES can be viewed as a (10-round) key-alternating cipher, where a t-round key-alternating cipher KAt consists of a small number t of fixed permutations Pi on n bits, separated by key addition: KAt(K,m) = kt ⊕ Pt(. . . k2 ⊕ P2(k1 ⊕ P1(k0 ⊕m)) . . . ), where (k0, . . . , kt) are obtained from ...

Abstract Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell +1}=\bar z_\ell + \textrm {Re}\sum _{k=1}^K\bar b_{\ell k}\,e^{\textrm {i}\omega _{\ell k}\bar }+ c_{\ell ^{\prime}_{\ell k}\cdot x}$. An optimal distribution frequencies $(\omega k},\omega k})$ $e^{\textrm }$ and x}$ is derived. This derivation based on ...

In this paper we study planar polynomial differential systems of this form: dX dt = Ẋ = A(X, Y ), dY dt = Ẏ = B(X, Y ), where A, B ∈ Z[X, Y ] and degA ≤ d, degB ≤ d, ‖A‖∞ ≤ H and ‖B‖∞ ≤ H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D = A(X, Y )∂X + B(X, Y )∂Y . Darboux polynomials are usually comp...

Numerical integration is the study of how the numerical value of an integral can be found. Also called quadrature, which refers to nding a square whose area is the same as the area under a curve, it is one of the classical topics of numerical analysis. Of central interest is the process of approximating a deenite integral from values of the in-tegrand when exact mathematical integration is not ...

Axiomatically, the definition of the joint prob­ ability of several continuous variables is based on a cumulative distribution function, F(x, y, z, K), whose properties are analogous to the case of a single variable, and the probability density function derives from it. However, in practice, the probability P is never defined from a cumulative function, F, but from a density function, f, assum­...

There are several kinds of hyperrings, for example, Krasnerhyperrings, multiplicative hyperring, general hyperrings and$H_v$-rings. In a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf Keywords:} multiplic...

As the interaction between black holes and highly energetic infalling charged matter receives quantum corrections, basic laws of hole mechanics have to be carefully rederived. Using covariant phase space formalism, we generalize first law mechanics, both "equilibrium state" "physical process" versions, in presence nonlinear electrodynamics fields, defined by Lagrangians depending on quadratic e...