We define Sobolev norms of arbitrary real order for a Banach representation $(\pi, E)$ Lie group, with regard to single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on growth rate representation. In particular, we obtain spectral gap $D$ space smooth vectors $E$. The main tool novel factorization ...

1 Basics 2 Linear Equations 3 Linear Maps 4 Rank One Matrices 5 Algebra of Matrices 6 Eigenvalues and Eigenvectors 7 Inner Products and Quadratic Forms 8 Norms and Metrics 9 Projections and Reflections 10 Similar Matrices 11 Symmetric and Self-adjoint Maps 12 Orthogonal and Unitary Maps 13 Normal Matrices 14 Symplectic Maps 15 Differential Equations 16 Least Squares 17 Markov Chains 18 The Expo...

Let dν be the metric associated with a strictly positive submeasure ν on some boolean algebra P. If dν is bounded from above by 1, Eν = 1 − dν is a (fuzzy) similarity relation on P at least w.r.t. the Lukasiewicz t-norm, but possibly also w.r.t. numerous further t-norms. In this paper, we show that under certain assumptions on P and ν, we may associate with ν in a natural way a continuous t-nor...

Let Φ be a trace-preserving, positivity-preserving linear map on the algebra of complex 2 × 2 matrices, and let Ω be any finite-dimensional completely positive map. For p = 2 and p ≥ 4, we prove that the maximal p-norm of the product map Φ⊗Ω is the product of the maximal p-norms of Φ and Ω. Restricting Φ to the class of completely positive maps, this settles the multiplicativity question for al...

In this note first we define a BCK‐algebra on the states of a deterministic finite automaton. Then we show that it is a BCK‐algebra with condition (S) and also it is a positive implicative BCK‐algebra. Then we find some quotient BCK‐algebras of it. After that we introduce a hyper BCK‐algebra on the set of all equivalence classes of an equivalence relation on the states of a deterministic finite...

The purpose of this article is to initiate a systematic study dimension-free relations between basic communication and query complexity measures various matrix norms. In other words, our goal obtain inequalities that bound parameter solely as function another parameter. This in contrast perhaps the more common framework where poly-logarithmic dependencies on number input bits are tolerated. Dim...

This paper investigates the use of general bases with fixed poles for the purposes of robust estimation. These bases, which generalise the common FIR, Laguerre and two–parameter Kautz ones, are shown to be fundamental in the disc algebra provided a very mild condition on the choice of poles is satisfied. It is also shown, that by using a min–max criterion, these bases lead to robust estimators ...

Let M be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace τ . Let d be an injective positive measurable operator with respect to (M, τ ) such that d is also measurable. Define Lp(d) = {x ∈ L0(M) : dx+ xd ∈ Lp(M)} and ‖x‖Lp(d) = ‖dx+ xd‖p . We show that for 1 6 p0 < p1 6 ∞, 0 < θ < 1 and α0 > 0, α1 > 0 the interpolation equality (Lp0(d 0), Lp1(d α))θ = Lp(d ) hol...

This paper gives double angle theorems that bound the change in an invariant subspace of an inde nite Hermitian matrix in the graded form H = D AD subject to a perturbation H ! e H = D (A + A)D. These theorems extend recent results on a de nite Hermitian matrix in the graded form (Linear Algebra Appl., 311 (2000), 45{60) but the bounds here are more complicated in that they depend on not only r...

As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They gave a complete characterization of T-measures for a Frank triangular norm T. Here we characterize T-measures with respect to non-Frank strict triangular norms. We show the speciic roles of Frank triangular norms and a newly introduced family, nearly Frank triangular norm...