By investigating the extent to which variation in the coefficients of a convex combination of unitaries in a unital $J{B}^{*}$-algebra permits that combination to be expressed as convex combination of fewer unitaries of the same algebra, we generalise various results of R. V. Kadison and G. K. Pedersen. In the sequel, we shall give a couple of characterisations of $J{B}^{*}$-algebras of $tsr1$.

J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $\Omega \subset {ℝ}^d$ is a smooth bounded domain, 1 ≤ p < ∞ and $f\in W^{1,p}\left(\Omega \right)$, then $li{m}_{s\nearrow 1}(1-s){\int }_{\Omega }{\int }_{\Omega }{\left(\right|f\left(x\right)-f\left(y\right)|}^p{)/(\left|\right|x-y\left|\right|}^{d+sp})dxdy=K{\int }_{\Omega }{\left|\nabla f\left(x\right)\right|}^{p}dx$, where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_p^{s,p}\left(\Omega \right)$. The purpose of this paper is to obtain analogous asymptotic formulae for some...

The theory of Mellin analytic functionals with unbounded carrier is developed. The generalized Mellin transform for such functionals is defined and applied to solve the Laplace-Beltrami type singular equations on a hyperbolic space. Then the asymptotic expansion of solutions is found.

We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.

A new criterion of asymptotic periodicity of Markov operators on $L^1$, established in [3], is extended to the class of Markov operators on signed measures.

Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T(X₁,...,Xₙ)=n^{-1}{\sum }_{1\le iLet{q}_i\left(i\ge 1\right)beeigenvaluesoftheHilbert-Schmidtoperatorassociatedwiththekernelh(x,y),andq₁bethelargestinabsolutevalueone.Weprovethat}$Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12)$,$where $G_i$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and ${\beta }^{\text{'}}:=E\left|h(X,X₁)\right|³+{E\left|h(X,X₁)\right|}^{18/5}+E\left|g\left(X\right)\right|³+{E\left|g\left(X\right)\right|}^{18/5}+1<\infty$.

Motivated by the Watts-Strogatz model for a complex network, we introduce a generalization of the Erdős-Rényi random graph. We derive a combinatorial formula for the moment sequence of its spectral distribution in the sparse limit.

Two new examples are given for illustrating the method of quantum decomposition in the asymptotic spectral analysis for a growing family of graphs. The odd graphs form a growing family of distance-regular graphs and the two-sided Rayleigh distribution appears in the limit of vacuum spectral distribution of the adjacency matrix. For a spidernet as well as for a growing family of spidernets the vacuum distribution of the adjacency matrix is the free Meixner law. These distributions are calculated...

We derive the asymptotic spectral distribution of the distance k-graph of N-dimensional hypercube as N → ∞.